4.13. Using a theorem of Howe and Moore, we also show that interpretable
Hilbert spaces generated by asymptotically free complete types capture all
representations of algebraic groups over local fields of characteristic $0$. _

_In Section 4.3 we review the special case of $\omega$-categorical structures
and we recover the theorem of [Tsa12] which classifies all unitary
representations of oligomorphic groups._

###  4.1. Unitary representations of automorphism groups

_In this section, we show how interpretable Hilbert spaces give rise to
unitary representations of automorphism groups. We show that irreducibility is
a property of the interpretable Hilbert space $\mathcal{H}$. Using a general
continuous logic construction and a theorem of Howe and Moore, we also show
that interpretable Hilbert spaces provide an interesting framework for
studying unitary representations of arbitrary groups._

We start by recalling some basic definitions. See the appendix of [BdlHV08]
for more background on unitary representations.

######  Definition 4.1.

Let $G$ be a group and $H$ a Hilbert space, real or complex. A unitary
representation $\sigma$ of $G$ on $H$ is a group action $G\times H\to H$ such
that for every $g\in G$ $\sigma(g)$ is a unitary map if $H$ is a complex
Hilbert space and $\sigma(g)$ is an orthogonal map if $H$ is a real Hilbert
space.

When $G$ is a topological group, we say that $\sigma$ is continuous if
$\sigma:G\times H\to H$ is continuous. This is equivalent to $\sigma(\cdot,v)$
being continuous for every $v\in V$.

Convention: In this paper, we only consider continuous unitary representations
of topological groups, so we just say ‘representation’ instead of ‘continuous
unitary representation’.

######  Definition 4.2.

Let $G$ be a group and let $\sigma,\sigma^{\prime}$ be two representations of
$G$ on the Hilbert spaces $H$ and $H^{\prime}$ respectively. $\sigma$ and
$\sigma^{\prime}$ are equivalent if there is a surjective isometry $U:H\to
H^{\prime}$ such that for all $g\in G$ and $v\in H$,
$U(\sigma(g)v)=\sigma^{\prime}(g)U(v)$. We say that $U$ intertwines $\sigma$
and $\sigma^{\prime}$.

Let $T$ be an arbitrary continuous logic theory and let $M\models T$. Then
$\mathrm{Aut}(M)$ is a topological group with the topology of pointwise
convergence. $\mathrm{Aut}(M)$ has a basis at the identity consisting of
subsets of the form $\\{g\in\mathrm{Aut}(M)\mid d(gA,A)<\epsilon,A\subseteq
M\text{ finite},\epsilon>0\\}$. When $T$ is a classical logic theory, this is
a basis of subgroups. Note that we can add imaginary sorts to $M$ without
changing the topology on $\mathrm{Aut}(M)$.

Suppose $\mathcal{H}$ is an interpretable Hilbert space in $T$. For any
$M\models T$, $\mathrm{Aut}(M)$ has a _canonical unitary representation_ $\pi$
on $H(M)$ given by $\pi(g)hx=h(gx)$ where $h$ is the direct limit map on any
piece of $\mathcal{H}$. Since we can add the pieces of $\mathcal{H}$ to $M$
without affecting the topology on $\mathrm{Aut}(M)$, $\pi$ is a continuous
representation.

The following lemma is an easy definition chase:

######  Lemma 4.3.

If $\mathcal{H}$, $\mathcal{H^{\prime}}$ are isomorphic interpretable Hilbert
spaces in $T$, then for any $M\models T$, the representations of
$\mathrm{Aut}(M)$ on $H(M)$ and $H^{\prime}(M)$ are equivalent.

We turn to a discussion of irreducibility for interpretable Hilbert spaces.

######  Definition 4.4.

Let $\mathcal{H}_{0}$ be an $\bigwedge$-interpretable subspace of
$\mathcal{H}$. We say that $\mathcal{H}_{0}$ is irreducible if there do not
exist complete types $q,q^{\prime}$ in $\mathcal{H}_{0}$ such that $q(x)\cup
q^{\prime}(y)$ implies $\langle x,y\rangle=0$.

We will study irreducibility for $\bigwedge$-interpretable Hilbert spaces in
relation with the notion of $\omega$-near-homogeneity See Definition 2.25. As
in the discussion following Definition 2.25, when discussing a $\omega$-near-
homogeneous structure with an interpretable Hilbert spaces $\mathcal{H}$, we
will always assume that the $\omega$-near-homogeneity property applies to the
pieces of $\mathcal{H}$.

######  Lemma 4.5.

Let $\mathcal{H}_{0}$ be a $\bigwedge$-interpretable subspace of
$\mathcal{H}$.

  1. (1)

if $\mathcal{H}_{0}$ is irreducible then for any $\omega$-near-homogeneous
$M\models T$ the canonical representation $\pi$ of $\mathrm{Aut}(M)$ on
$H_{0}(N)$ is irreducible.

  2. (2)

if there is some $M\models T$ realising all types of $T$ such that the
canonical representation $\pi$ of $\mathrm{Aut}(M)$ on $H_{0}(N)$ is
irreducible, then $\mathcal{H}_{0}$ is irreducible

###### Proof.

(1) Let $M\models T$ be $\omega$-near-homogeneous. Let $v,w\in H(M)$ be two
nonzero vectors. Write $q_{1},q_{2}$ for their respective types. By
irreducibility of $\mathcal{H}_{0}$ and the assumption following Definition
2.25, $d(q_{1},q_{2})^{2}<\|v\|^{2}+\|w\|^{2}$. By $\omega$-near-homogeneity,
we can find $g\in\mathrm{Aut}(M)$ such that $\|gv-w\|$ is arbitrarily close to
$d(q_{1},q_{2})$. Then $gv$ and $w$ are not orthogonal and the representation
of $\mathrm{Aut}(M)$ on $H_{0}(M)$ is irreducible.

(2) Take $M\models T$ as in the statement. Let $p,q$ be types of
$\mathcal{H}_{0}$ and let $v,w$ be realisations in $H_{0}(M)$. There is
$g\in\mathrm{Aut}(M)$ such that $\langle gv,w\rangle\neq 0$ so
$\mathcal{H}_{0}$ is irreducible. ∎

The following proposition shows that irreducibility is a local notion, when
the interpretable Hilbert space is generated by a complete type. See also
Proposition 4.13 for a qualitatively similar statement.

######  Proposition 4.6.

Let $p$ be a complete type in $\mathcal{H}$. Take $M\models T$ $\omega$-near-
homogeneous and realising all types.

Let $a\models p$ in $H(M)$, let $K$ be the subgroup of $\mathrm{Aut}(M)$
fixing $\mathrm{bdd}(a)$ setwise and let $A=H_{p}(M)\cap\mathrm{bdd}(a)$. If
the canonical representation of $K$ on $A$ is irreducible, then
$\mathcal{H}_{p}$ is irreducible.

###### Proof.

Let $q,q^{\prime}$ be two types in $\mathcal{H}_{p}$. Let $v,w$ be
realisations of $q,q^{\prime}$ in $H(M)$. Choosing
$\mathrm{Aut}(M)$-conjugates of $v$ and $w$ if necessary, we can assume that
$P_{A}v$ and $P_{A}w$ are nonzero. Conjugating $v$ by an element of $K$, we
can assume that $\langle P_{A}v,P_{A}w\rangle\neq 0$.

Consider the nonforking extension $r$ of $\mathrm{tp}(v/A)$ to
$\mathrm{bdd}(Aw)$ with respect to the inner product maps of $\mathcal{H}$. If
$z$ realises $r$ in an $\omega_{1}$-saturated elementary extension $N$ of $M$,
we have $\langle z,w\rangle=\langle P_{A}v,P_{A}w\rangle\neq 0$. Let
$r^{\prime}=\mathrm{tp}(z,w)$.

By our assumption on $M$, $r^{\prime}$ is realised in $M$ by some pair
$(z^{\prime},w^{\prime})$. By $\omega$-near-homogeneity, we can assume that
$w^{\prime}$ is arbitrarily close to $w$. Now
$\mathrm{tp}(z^{\prime})=\mathrm{tp}(v)$ so we can find $g\in\mathrm{Aut}(M)$
taking $v$ arbitrarily close to $z^{\prime}$. Now we have $\langle
gv,w\rangle\approx\langle z^{\prime},w^{\prime}\rangle=\langle z,w\rangle\neq
0$. ∎

We now introduce a construction which allows us to connect the study of
interpretable Hilbert spaces with the study of unitary representations of
arbitrary groups. Given a faithful representation of a group $G$ on a Hilbert
space $H$, we will construct a continuous logic structure with universe
generated by the orbit of an arbitrary vector in $H$ under the action of $G$.
We use the following general continuous logic construction:

######  Definition 4.7.

Let $M$ be a continuous logic structure in a language $\mathcal{L}$ and let
$G$ be a subgroup of $\mathrm{Aut}(M)$. We define an expansion $M_{G}$ of $M$
in a language $\mathcal{L}_{G}$ as follows.

For every $n\geq 1$, for every finite Cartesian product $P$ of sorts of $M$
and for every orbit $O$ of $G$ on $P$, we add a function symbol
$r_{O}:P\to[0,\infty)$. $M_{G}$ is the structure obtained from $M$ by
interpreting each function $r_{O}(x)$ as $d(x,\overline{O})$, the distance
between $x$ and the metric closure of $O$.

We could equivalently define $M_{G}$ as the maximal $G$-invariant expansion of
$M$. The following lemma is straightforward:

######  Lemma 4.8.

Let $M$ be a continuous logic structure, $G$ a subgroup of $\mathrm{Aut}(M)$.
Then $\mathrm{Aut}(M_{G})$ is the closure of $G$ in $\mathrm{Aut}(M)$ with the
topology of pointwise convergence.

We apply the above construction to unitary representations:

######  Lemma 4.9.

Let $G$ be an arbitrary group and let $H$ be a Hilbert space with a cyclic
faithful representation $\sigma$ of $G$. Let $v$ be a cyclic vector and let
$X$ be the closure of the orbit of $v$ in $H$. Let $M$ be the continuous logic
structure with universe $X$ and with the inner product map on $X$ induced from
$H$ as only function symbol. The following hold:

  1. (1)

$M_{G}$ is an atomic $\omega$-near-homogeneous structure (see Definition 2.25)
and $X$ is a complete type in $M_{G}$

  2. (2)

there is an interpretable Hilbert space $\mathcal{H}$ in $Th(M)$ generated by
the sort $X$ such that, writing $\pi\upharpoonright G$ for the restriction of
the canonical representation of $\mathrm{Aut}(M_{G})$ to $G$, we have
$\pi\upharpoonright G\simeq\sigma$.

  3. (3)

$\mathrm{Aut}(M_{G})$ is isomorphic to the closure of $G$ in $U(H)$, the
unitary group of $H$, in the weak operator topology.

###### Proof.

(1) and (2) are trivial. For (3), it is enough to note that the weak operator
topology on $U(H)$ coincides with the pointwise convergence topology on $U(H)$
with respect to the set $X$. ∎

It is also clear that the structure $M_{G}$ above remains atomic and
$\omega$-near-homogeneous under expansion by the pieces of $\mathcal{H}$ so
that the assumption following Definition 2.25 holds.

A classical theorem of [HM79] shows that when $G$ is a connected Lie group,
$\mathrm{Aut}(M_{G})$ is not significantly larger than $G$. We recall this
theorem here:

######  Theorem 4.10 ([HM79], Theorem 2.1).

Let $G$ be a connected Lie group with a faithful continuous unitary
representation on $H$ and write $\overline{G}$ for the closure of $G$ in
$U(H)$, the unitary group of $H$, with the weak operator topology. Then

  1. (1)

The normaliser of $G$ in $U(H)$ is closed and hence $G$ is normal in
$\overline{G}$

  2. (2)

$\overline{G}/G$ is at most 2-step nilpotent

  3. (3)

$\overline{G}/Z(\overline{G})$ is a Lie group and $\overline{G}/G\cdot
Z(\overline{G})$ is Abelian

  4. (4)

If the adjoint group of $G$ is closed, then $\overline{G}=G\cdot
Z(\overline{G})$.

See also Theorem 3.1 in [HM79] for an extension of the above result to
connected groups $G$.

In the next section, we will show that asymptotic freedom is a natural notion
from the point of view of representation theory. We remark here that it is
more difficult to interpret directly our notion of scatteredness in
representation theoretic terms. The main obstacle is that we require
$\omega$-saturation to verify scatteredness. While $\omega$-saturation holds
in ultraproducts of structures $M_{G}$, it does not usually hold for a single
structure.

Nevertheless, since every structure $M_{G}$ is atomic and $\omega$-near-
homogeneous, Proposition 2.27 applies when the weak closure of the orbit $X$
is locally compact. Proposition 2.27 can be easily reformulated in purely
representation theoretic terms:

######  Proposition 4.11.

Let $G$ be a group and let $\sigma$ be a faithful cyclic unitary
representation of $G$ on $H$. Let $v$ be a cyclic vector and let $X$ be the
metric closure of the orbit of $v$ in $H$.

Suppose that the weak closure of $X$ in $H$ is locally compact in the metric
topology. Then $\sigma$ is equivalent to an orthogonal sum of representations
$(\tau_{i})$ such that each $\tau_{i}$ has a cyclic vector $v_{i}$ satisfying
the following: if $w$ is a conjugate of $v_{i}$ such that $\langle
w,v_{i}\rangle\neq 0$, then the orbit of $w$ under the stabilizer of $v_{i}$
is precompact.

###  4.2. Unitary representations with asymptotically free orbits

_In this section we study interpretable Hilbert spaces generated by
asymptotically free complete types from the point of view of representation
theory. We show that asymptotically free complete types give rise to induced
representations, and we prove a Mackey-style irreducibility criterion for
these induced representations._

_We also relate asymptotic freedom to the notion of disintegrated strongly
minimal sets in continuous logic._

_Finally, using a theorem of Howe and Moore, we show that interpretable
Hilbert spaces generated by an asymptotically free complete type capture all
irreducible unitary representations of algebraic groups over a local field of
characteristic $0$._

We begin by recalling the notion of induced representation in the special case
where we induce from an open subgroup. See [BdlHV08] for more details. Let $G$
be a topological group and take $K$ an open subgroup of $G$. Let $\sigma$ be a
representation of $K$ on a Hilbert space $V$. We suppose here that $V$ is a
real Hilbert space (the case of complex Hilbert spaces is similar). Write
$\mathbb{R}G$ for the free vector space on $G$. We define
$G\otimes_{\sigma}V$, the $\sigma$-tensor of $G$ and $V$, to be the vector
space $\mathbb{R}G\otimes V$ quotiented by a suitable subspace so that
$gk\otimes_{\sigma}v=g\otimes\sigma(k)v$ for all $g\in G$ and $k\in K$.

We define an inner product on $G\otimes_{\sigma}V$ as follows. For any
$g,g^{\prime}\in G$ and $v,v^{\prime}\in V$, if $gK\neq g^{\prime}K$, then
$\langle g\otimes_{\sigma}v,g^{\prime}\otimes_{\sigma}v^{\prime}\rangle=0$. If
$gK=g^{\prime}K$, then find $k\in K$ such that $g^{\prime}=gk$ and define
$\langle
g\otimes_{\sigma}v,g^{\prime}\otimes_{\sigma}v^{\prime}\rangle=\langle
v,\sigma(k)v^{\prime}\rangle$. Observe that if we choose a set of coset
representatives for $G/K$, we can identify $G\otimes_{\sigma}V$ with the
orthogonal sum of copies of $V$ indexed by $G/K$.

We will always work with the Hilbert space completion of $G\otimes_{\sigma}V$.
We also write $G\otimes_{\sigma}V$ for this completion. We define the induced
representation of $G$ from $\sigma$, denoted $\mathrm{Ind}_{K}^{G}(\sigma)$,
as the unitary representation of $G$ on $G\otimes_{\sigma}V$ given by
$g\cdot(g^{\prime}\otimes_{\sigma}v)=gg^{\prime}\otimes_{\sigma}v$. Since $K$
is open in $G$, the induced representation is continuous.

######  Proposition 4.12.

Let $T$ be a continuous logic theory with an interpretable Hilbert space
$\mathcal{H}$. Suppose $p$ is an asymptotically free complete type in
$\mathcal{H}$. For $x,y\models p$, write $x\sim y$ if
$\mathrm{bdd}(x)=\mathrm{bdd}(y)$ and write $[x]$ for the equivalence class of
$x$ under $\sim$.

Let $M\models T$, write $G=\mathrm{Aut}(M)$ and suppose that for some (any)
$a\models p$, the orbit of $a$ under $G$ is metrically dense in $p$. Fix
$a\models p$ in $H(M)$ and write $K$ for the open subgroup of elements of $G$
which fix $[a]$ setwise. Then the canonical representation $\pi$ of $G$ on
$H_{p}(M)$ is equivalent to $\mathrm{Ind}_{K}^{G}(\sigma)$ where $\sigma$ is
the restriction of $\pi$ to $K$ on the Hilbert space $V$ spanned by $[a]$.

###### Proof.

Since $p$ is asymptotically free, $p$ is metrically locally compact and hence
there is $\epsilon>0$ such that for any $x,y\models p$, if $d(x,y)<\epsilon$
then $[x]=[y]$. Therefore $K$ is open in $G$. Let $A=[a]$ and write
$\\{A_{i}\mid i\in I\\}$ for the orbit of $A$ under $G$ setwise (we ignore
permutations of $A$). For every $i\in I$ pick $g_{i}\in\mathrm{Aut}(M)$ which
maps $A$ to $A_{i}$. Then $\\{g_{i}\mid i\in I\\}$ is a list of
representatives for the left cosets of $K$ in $\mathrm{Aut}(M)$. Since $p$ is
asymptotically free, the sets $A_{i}$ are pairwise orthogonal in $H_{p}(M)$.

Write $\pi$ for the canonical representation of $\mathrm{Aut}(M)$ on
$H_{p}(M)$ and let $\sigma$ be the restriction of $\pi$ to $K$ on $V$, the
vector space spanned by $A$. Let $\iota=\mathrm{Ind}_{K}^{G}(\sigma)$ and
write $W=G\otimes_{\sigma}V$. We show that $\iota$ and $\pi$ are equivalent.
Write also $g_{i}\otimes_{K}V$ for the subspace of $W$ given by
$\\{g_{i}\otimes_{\sigma}v\mid v\in V\\}$. Take $w\in W$ and write $w=\sum
w_{i}$ where $w_{i}\in g_{i}\otimes_{\sigma}V$. Let
$P_{i}:g_{i}\otimes_{\sigma}V\to V$ be the Hilbert space isomorphism taking
$g_{i}\otimes_{\sigma}v$ to $v$. We define

$U(w)=\sum\pi(g_{i})P_{i}(w_{i}).$

Since $A_{i}\perp A_{j}$ for $i\neq j$, we have
$\pi(g_{i})P_{i}(w_{i})\perp\pi(g_{j})P_{j}(w_{j})$, so $U$ is well-defined,
and it is easy to check that $U$ is in fact a surjective isometry. $U$
intertwines $\iota$ and $\pi$: take $g\in G,i\in I,v\in V$. Then, writing
$gg_{i}=g_{j}k$, we have

$U(\iota(g)(g_{i}\otimes_{K}v))=U(g_{j}\otimes_{K}\sigma(k)v)=\pi(g_{j})(\sigma(k)v)=\pi(g_{j}k)v=\pi(g)U(g_{i}\otimes_{K}v).$

∎

Remark: Taking $a$ and $K$ as in Proposition 4.12, we note that
$\mathrm{Aut}(M/[a])$ is a normal subgroup of $K$ contained in the kernel of
$\sigma$. Write $G_{1}$ for the group of automorphisms of the set $[a]$. Then
$\sigma$ factors through $Aut(M/[a])$ to a faithful representation of a
subgroup of $G_{1}$. Since $[a]$ is a separable locally compact metric space,
the closure of $K/\mathrm{Aut}(M/[a])$ in $G_{1}$ is locally compact with
respect to the topology of pointwise convergence. We say that the canonical
representation of $G$ is obtained from the representation of
$K/\mathrm{Aut}(M/[a])$ by inflation.

We prove a version of Mackey’s irreducibility criterion for the induced
representations arising in Proposition 4.12. General irreducibility criteria
for induced representations have been studied in [Mac51], [God48], [Cor75].
See also Proposition 4.1 in [Tsa12] for an irreducibility criterion in the
case of oligomorphic groups. In all cases, irreducibility of the induced
representation is verified at the level of the commensurator (also called the
quasi-normalizer) of the subgroup $K$. See [Cor75] or [Tsa12] for a
discussion. Our criterion also takes place at the level of the commensurator
of the subgroup we are inducing from.

In all cases cited above, the irreducibility criterion for induced
representations holds only when we induce from _a finite dimensional
representation_. It is shown in [BC03] that the naive generalisation of the
criterion of [Cor75] does not hold for infinite dimensional representations.
Our Proposition 4.13 has the interesting feature that it does not require
finite-dimensionality of the representations we are inducing from.

######  Proposition 4.13.

Suppose $p$ is an asymptotically free complete type in $\mathcal{H}$. For
$x,y\models p$, write $x\sim y$ if $\mathrm{bdd}(x)=\mathrm{bdd}(y)$ and write
$[x]$ for the equivalence class of $x$ under $\sim$.

Let $M\models T$ and fix $a\models p$ in $M$. Write $K$ for the subgroup of
elements of $G$ which fix $[a]$ setwise. Let $\sigma$ be the restriction of
the canonical representation $\pi$ of $\mathrm{Aut}(M)$ to $K$ on the Hilbert
space $V$ spanned by $[a]$. Then

  1. (1)

If $\sigma$ is irreducible, then $\mathcal{H}_{p}$ is irreducible

  2. (2)

If $M$ is $\omega$-near-homogeneous and $\sigma$ is reducible, then
$\mathcal{H}_{p}$ is reducible.

###### Proof.

To see (2), note that by Proposition 4.12 and easy facts about induced
representations, if $\sigma=\sigma_{1}\oplus\sigma_{2}$, then
$\pi=\mathrm{Ind}_{K}^{G}(\sigma_{1})\oplus\mathrm{Ind}_{K}^{G}(\sigma_{2})$,
so $\pi$ is reducible, and this is witnessed by two complete types realised in
$M$. Hence $\mathcal{H}_{p}$ is reducible.

We prove (1). Suppose that $\sigma$ is irreducible. If we move to an
$|M|^{+}$-strongly homogeneous and $|M|^{+}$-saturated elementary extension
$M^{\prime}$ of $M$, then the representation of the subgroup of
$\mathrm{Aut}(M^{\prime})$ which fixes $[a]$ setwise is also irreducible on
$V$. Hence by Lemma 4.5(2), we can assume that $M$ is $\omega_{1}$-strongly
homogeneous.

Recall that in Proposition 4.12 we expressed $H_{p}(M)$ as the orthogonal sum
of subspaces $g_{i}\otimes_{\sigma}V$ where $(g_{i})_{i\in I}$ is a set of
coset representatives of $G/K$. Suppose $0$ is an indexing element in $I$ with
$g_{0}=e$ so that $V=g_{0}\otimes_{\sigma}V$.

Suppose that we have a $G$-invariant subspace $Z$ of $H_{p}(M)$. Then the
orthogonal projection $P_{Z}$ commutes with $G$. Fix a nonzero $v\in V$. We
can write $P_{Z}v=\sum_{j\in J}u_{j}$ where $J\subseteq I$ is countable and
$u_{j}\in g_{j}\otimes_{\sigma}V$ is nonzero. Write $u_{0}$ for the element of
$\\{u_{j}\mid j\in J\\}$ which lies in $V$. Since the $u_{j}$ are pairwise
orthogonal, $u_{0}=0$ would imply that $P_{Z}v=0$. Switching if necessary to
$Z^{\perp}$, we can assume that $P_{Z}v\neq 0$.

Write $J_{0}=\\{j\in J\mid g_{j}a\notin\mathrm{bdd}(a)\\}$ and $J_{1}=\\{j\in
J\mid a\notin\mathrm{bdd}(g_{j}a)\\}$. Then $J=\\{0\\}\cup J_{0}\cup J_{1}$.
We show that $J_{0}=J_{1}=\emptyset$. By $\omega_{1}$-strong homogeneity and
saturation of $M$, we can find a sequence $(\alpha_{n})$ in
$\mathrm{Aut}(M/\mathrm{bdd}(a))$ such for any $n\neq m$

$\alpha_{n}\big{\\{}[g_{j}a]\mid j\in
J_{0}\big{\\}}\cap\alpha_{m}\big{\\{}[g_{j}a]\mid j\in
J_{0}\big{\\}}=\emptyset.$

Hence for all $n\neq m$, the sets $\alpha_{n}\\{u_{j}\mid j\in J_{0}\\}$ and
$\alpha_{m}\\{u_{j}\mid j\in J_{0}\\}$ are orthogonal. For all $n$ and $j\in
J\setminus J_{0}$, we also have $\alpha_{n}u_{j}=u_{j}$. Therefore
$(\alpha_{n}P_{Z}(v))$ converges weakly to $\sum_{j\in J\setminus
J_{0}}u_{j}$. Since $Z$ is $G$-invariant, we have $\sum_{j\in J\setminus
J_{0}}u_{j}\in Z$. Since $\sum_{j\in J\setminus J_{0}}u_{j}$ is at least as
close to $v$ as $\sum_{j\in J}u_{j}$, we conclude that $J_{0}=\emptyset$.

Suppose for a contradiction that we have $j_{1}\in J_{1}$. Let $J_{2}=\\{j\in
J\mid g_{j}a\notin\mathrm{bdd}(g_{j_{1}}a)\\}$. Note that $0\in J_{2}$. By the
same argument as above, we have $\sum_{j\in J\setminus J_{2}}u_{j}\in Z$.
Therefore $\sum_{j\in J_{2}}u_{j}=\sum_{j\in J}u_{j}-\sum_{j\in J\setminus
J_{2}}u_{j}\in Z$. Since $j_{1}\notin J_{2}$, $\sum_{j\in J_{2}}u_{j}$ is an
element of $Z$ closer to $v$ than $P_{Z}v$ and this is a contradiction. Hence
$J_{1}=\emptyset$ and $P_{Z}v\in V$.

Therefore, $Z\cap V$ is a nonempty $K$-invariant subspace of $V$. Since
$\sigma$ is irreducible, we have $Z\cap V=V$. By $G$-invariance of $Z$, we
have $g_{i}\otimes_{\sigma}V\subseteq Z$ for all $g_{i}$ and hence
$Z=H_{p}(M)$. This proves that $\pi$ is irreducible. ∎

Remark: We leave it to the reader to compare Proposition 4.13 with the failure
of the naive criterion considered in [BC03]. In particular, we note that our
criterion in Proposition 4.13 is very much tied to a specific equivalence
relation on the type $p$.

We now relate asymptotic freedom and strongly minimality in continuous logic.
Strong minimality has been defined in [Han20] and we rephrase the definition
here:

######  Definition 4.14.

We say that a continuous logic theory $T$ with a sort $X$ is strongly minimal
if for every $M\models T$, every $M$-definable function $f(x)$ in one variable
into $\mathbb{R}$ has a unique generic value $\alpha$ on $X$, meaning that for
any $\beta\neq\alpha$, the set $\\{f(x)=\beta\\}$ is compact.

######  Lemma 4.15.

Suppose $T$ is a strongly minimal continuous logic theory. Then for any
$M\models T$, $\mathrm{bdd}$ is a pregeometry.

###### Proof.

We check the exchange property, i.e. for $A\subseteq M$ and $a,b\in M$, if
$a\in\mathrm{bdd}(Ab)$ and $a\notin\mathrm{bdd}(A)$, then
$b\in\mathrm{bdd}(Aa)$.

Take $a,b$ in $M$ such that $a\in\mathrm{bdd}(Ab)$ and
$a\notin\mathrm{bdd}(A)$. Recall that a complete type $q$ over $Ab$ is
uniquely determined by the values $f(q)$ where $f$ ranges over $Ab$-definable
functions into $\mathbb{R}$. By strong minimality, there is an $Ab$-definable
function $f$ with generic value $\alpha$ such that the value $f(a)$ is not
generic. By a standard approximation argument, we can find an $A$-definable
function $g(x,y)$ such that $g(a,b)$ is not the generic value of $g(x,b)$.

Observe that there is some $\alpha_{y}$ such that for any
$c\notin\mathrm{bdd}(A)$, $\alpha_{y}$ is the generic value of $g(c,y)$. There
is a corresponding value $\alpha_{x}$ for $g(x,c)$, and it follows that
$\alpha_{x}=\alpha_{y}$. Therefore $g(a,b)$ is not the generic value of
$g(a,y)$ and $b\in\mathrm{bdd}(Aa)$. ∎

######  Definition 4.16.

Let $T$ be strongly minimal. We say that $T$ is disintegrated if the
pregeometry is trivial on $T$, in the sense that for any $M\models T$ and
$A,B\subseteq M$, $\mathrm{bdd}(A\cup B)=\mathrm{bdd}(A)\cup\mathrm{bdd}(B)$

######  Proposition 4.17.

Let $T$ be a continuous logic theory with a single sort $X$, and a single
$\mathbb{C}$-valued binary function $b$ on $X$. Assume $b$ is positive
definite, and that $X$ is an asymptotically free complete type. Then $T$ is
strongly minimal and disintegrated.

###### Proof.

Let $M\models T$. We consider the equivalence relation $\sim$ on $X$ defined
as the transitive closure of the relation $\langle x,y\rangle\neq 0$. For
$x\in X$, write $[x]$ for the $\sim$-class of $x$. Since $X$ is a complete
asymptotically free type, if $a\in X(M)$ then $[a]\subset bdd(a)$; within
$bdd(a)$, $[a]$ is the complement of a $\bigwedge$-definable set over $a$. If
$b\in X(N)$ for some $N\models T$, there exists an elementary isomorphism
$bdd(a)\to bdd(b)$; clearly it carries $[a]$ to $[b]$; so any two
$\sim$-classes are isomorphic and the isomorphism type of these classes does
not depend on the choice of model $M$.

Let $N$ be a model of $T$, and let $\\{x_{i}\mid i\in I\\}$ be a set of
representatives of $\sim$-classes in $X(N)$. Then any bijection $I\to I$ can
be extended to an automorphism of $M$. Hence for $A\subseteq X$,
$\mathrm{bdd}(A)=\bigcup_{a\in A}[a]$. It follows that $T^{-}$ is strongly
minimal and disintegrated. ∎

Finally, we show how asymptotic freedom is a natural representation theoretic
notion. Let $\sigma$ be a representation of a group $G$ on $H$ and take $v\in
H$. We say that the orbit of $v$ is _asymptotically free_ if for every
$\alpha>0$, the set of conjugates $w$ of $v$ such that $|\langle
w,v\rangle|\geq\alpha$ is precompact.

We have already seen that representations with asymptotically free orbits
capture representations of automorphism groups of measurable structures on the
associated $L^{2}$-spaces, and representations of absolute Galois groups on
various associated $L^{2}$-spaces. In Section 4.3, we show that
representatiions with asymptotically free orbits also capture all
representations of automorphism groups. Finally, thanks to Theorem 6.1 in
[HM79], we show that irreducible unitary representations of certain algebraic
groups also have asymptotically free orbits:

######  Proposition 4.18 ([HM79], Theorem 6.1).

Let $\sigma$ be an irreducible representations of an algebraic group over a
local field of characteristic $0$. Then every orbit of $\sigma$ is
asymptotically free.

###### Proof.

Let $G$ be a connected algebraic group over a local field of characteristic
$0$ with a representation $\sigma$ on $H$. Let $P\leq G$ be the preimage under
$\sigma$ of the circle group in $U(H)$. Then Theorem 6.1 in [HM79] shows that
for any $v\in H$, the map $g\mapsto|\langle\sigma(g)v,v\rangle|$ tends to $0$
on $G/P$. Since the action by $P$ does not affect compactness, we deduce that
the orbit of $v$ is asymptotically free.

If $G$ is not connected, we find a connected normal algebraic subgroup $G_{0}$
such that $G/G_{0}$ is finite. Then $\sigma$ splits as a finite orthogonal sum
of irreducible representations of $G_{0}$. For any $v\in H$, the $G$-orbit of
$v$ is a finite union of $G_{0}$ orbits in the irreducible subrepresentations
and hence the $G$-orbit of $v$ is asymptotically free. ∎

See [BM00] for an overview of the Howe-Moore result and its extension to
various additional cases.

Combining Proposition 4.18 and Proposition 4.17, we find:

######  Corollary 4.19.

Let $G$ be an algebraic group over a local field of characteristic $0$. Then
any irreducible representation of $G$ can be understood as a continuous
homomorphism $\rho:G\to\mathrm{Aut}(X)$ for some strongly minimal
disintegrated $X$ possessing a definable positive definite function.

$D$ and $\rho$ depend only on the choice of $G$ orbit in the representation of
$G$.

###  4.3. Unitary representations of automorphism groups of
$\omega$-categorical structures

_In this section, we recall some results of[Tsa12] and [Iba21] about unitary
representations of automorphisms groups of $\omega$-categorical structures and
we show that Corollary 2.24 combined with Proposition 4.12 recovers the
classification theorem in [Tsa12]._

We refer the reader to [BYBHU08] for standard facts concerning
$\omega$-categorical structures in continuous logic. Recall the Ryll-
Nardzewski theorem in continuous logic which says that $T$ is
$\omega$-categorical if and only if every complete type is principal (see
[BYBHU08] 12.10). See also [BYBHU08] 12.11 whcih shows that if $M$ is the
separable model of an $\omega$-categorical theory, then $M$ is $\omega$-near-
homogeneous.

For an $\omega$-categorical continuous logic theory $T$, we define the
expansion $T^{princ}$ as in Definition 1.7 by adding $p$ as a new sort to $T$,
where $p$ is any principal type of $T$. The following result is Lemma 1.1 in
[Iba21] rephrased in the language of this paper.

######  Lemma 4.20.

Let $T$ be an $\omega$-categorical continuous logic theory and let $M$ be an
$\omega$-near-homogeneous model of $T$. Let $\sigma$ be a representation of
$\mathrm{Aut}(M)$ on a Hilbert space $H$. Then there is an interpretable
Hilbert space $\mathcal{H}$ in $T^{princ}$ such that $\sigma$ is equivalent to
the canonical representation of $\mathrm{Aut}(M)$ on $H(M)$.

If $T$ is a classical logic theory, then the $T^{princ}$ construction is not
needed, since a principal type is a definable set $D$ and we can define the
interpretation map outside of $D$ to be the trivial $0$ map. Now the following
lemma is a combination of Lemma 3.1 in [Tsa12] and Lemma 1.1 in [Iba21]:

######  Lemma 4.21.

Let $T$ be a classical logic $\omega$-categorical theory and let $M$ be an
$\omega$-homogeneous model of $T$. Let $\sigma$ be a unitary representation of
$\mathrm{Aut}(M)$ on a Hilbert space $H$. Then there is a strictly
interpretable Hilbert space $\mathcal{H}$ in $T$ such that $\sigma$ is
equivalent to the canonical representation of $\mathrm{Aut}(M)$ on $H(M)$.

Let $T$ be a classical logic $\omega$-categorical theory and let $M$ be an
$\omega$-homogeneous model of $T$. Note that if $p$ is a type in a classical
imaginary sort of $M$, then the relation $x\in\mathrm{acl}(y)$ is symmetric
and transitive on $p$. This is because $\mathrm{acl}(x)\cap p$ is a finite set
with fixed cardinality. Applying Corollary 4.21, Corollary 2.24 and
Proposition 4.12, we deduce directly that every unitary representation of
$\mathrm{Aut}(M)$ is an orthogonal sum of representations obtained by
inflation from representations of groups of partial automorphisms of finite
sets of the form $\mathrm{acl}(a)\cap\mathrm{tp}(a)$, where $a$ is a classical
imaginary element of $M$. This is precisely the classification theorem 5.2 in
[Tsa12].

It remains to be seen if it is possible to build on the techniques developed
in this paper in order to find a classification of the unitary representations
of continuous logic $\omega$-categorical structures. This is an open question
for future research.

##  Appendix A Continuous logic

In _any_ version of continuous logic, one has the notion of a type-definable
set. The collection of type-definable sets is closed under positive Boolean
combinations and under projections. As in the discrete logic case, the
projection $(\exists x)p(x,y)$ of a partial type $p(x,y)$ is the partial type
$q(y)$ such that, in a sufficiently saturated model, $q(b)$ holds iff there
exists $a$ with $p(a,b)$. We can thus freely write formulas or sentences
involving positive first order operations. These describe partial types in any
formulation of continuous logic, regardless of a specific calculus.

With this in mind, we give a presentation of continuous logic based on the
approach of [HI02]. A similar approach was recently used in [GP21]. This
approach uses the syntax of classical logic and uses classical results of
model theory to deduce corresponding results in continuous logic. This
formalism is completely equivalent to the approach of [BYBHU08] and this paper
is written so that it is always easy to translate any argument into the
formalism of [BYBHU08] if one wishes to do so.

In A.1, we define syntax and type-spaces and we discuss briefly the relation
to [HI02]. In A.2, we recall some model theoretic notions which are used in
this paper. This section is mainly based on [BYBHU08]. In A.3, we recall some
classical facts about the model theory of Hilbert spaces.

###  A.1. Continuous logic and type-spaces

_In this section we introduce the basic concepts of continuous logic. Since we
take classical discrete logic as our starting point, we will use the
qualifiers ‘classical logic…’ and ‘continuous logic…’ to highlight the
differences between the two logics. We do not use this terminology elsewhere
in this paper._

In continuous logic, we will work with positive formulas (and their
negations):

######  Definition A.1.

Let $\mathcal{L}$ be an arbitrary language for classical first-order logic. We
say that an $\mathcal{L}$-formula $\phi(x)$ is positive if $\phi(x)$ is
logically equivalent to a formula which uses only the logical connectives
$\wedge$ and $\vee$ and the usual quantifiers $\forall$ and $\exists$.

Note that if $\phi$ and $\psi$ are positive $\mathcal{L}$-formulas, then
$(\neg\phi\rightarrow\psi)$ is positive. In general, positive formulas can
have very weak expressive power but we will always work in certain languages
and theories where they have strong expressive power. The restrictions we
impose on $\mathcal{L}$ are the following.

We always fix a multi-sorted language $\mathcal{L}$ with sorts $(S_{i})_{i\in
I}$ and $(I_{n})_{n\geq 1}$. We refer to the sorts $(S_{i})$ as the _metric
sorts_ and to the sorts $I_{n}$ as the _value sorts_. $\mathcal{L}$ has an
equality relation on every value sort but not on any metric sort. Each sort
$I_{n}$ will be identified with the interval $[-n,n]$, so we add functions
$i_{nm}:I_{n}\to I_{m}$ for $n\leq m$ which will play the role of inclusion
functions. The value sorts are also equipped with the following structure of
the real numbers: functions $+,-,\times,\max$ between the appropriate value
sorts and predicates $=$ and $\leq$ (note that we choose $\leq$ and not $<$,
for reasons which become clear below). In each value sort $I_{n}$ we also add
a constant symbol for each rational number in $[-n,n]$. Each metric sort is
equipped with a function $d_{i}:S_{i}\times S_{i}\to I_{n}$ for some $n$.
$\mathcal{L}$ may contain more function symbols but no other relation symbols.

Whenever we fix $\mathcal{L}$, we also fix a _minimal $\mathcal{L}$-theory_
$T_{\mathcal{L}}$. This is a collection of $\mathcal{L}$-sentences
$T_{\mathcal{L}}=T_{0}\cup T_{\mathbb{R}}$ satisfying the following
conditions:

  1. (1)

$T_{\mathbb{R}}$ says that the value sorts satisfy the full first-order theory
of the real numbers (where $I_{n}$ is identified with $[-n,n]$)

  2. (2)

$T_{0}$ says that every $d_{i}$ is a pseudometric on $S_{i}$ and that $d_{i}$
is bounded by some rational $c_{i}$.

  3. (3)

$T_{0}$ says that every function symbol $f$ in $\mathcal{L}$ is a uniformly
continuous function in the following way: for every rational $\epsilon>0$
there is a rational $\delta>0$ such that $T_{\mathcal{L}}$ contains the
positive sentence

$\forall x,y(d(x,y)<\delta\rightarrow d_{i}(f(x),f(y))\leq\epsilon)$

where $x,y$ are finite tuples of variables appropriate for $f$ and $d$ is the
max-metric on the sorts corresponding to the tuple $x$ and $d_{i}$ is the
metric on the sort of $f(x)$.

When working with $\mathcal{L}$, we only ever consider models of
$T_{\mathcal{L}}$.

######  Definition A.2.

A continuous logic $\mathcal{L}$-structure $M$ is an $\mathcal{L}$-structure
in the usual sense of classical first order logic such that $M\models
T_{\mathcal{L}}$, the value sorts of $M$ are the standard real numbers, and
each metric sort of $M$ is a complete metric space.

A useful way of constructing continuous logic $\mathcal{L}$-structures is to
quotient sufficiently saturated classical $\mathcal{L}$-structures: given any
model $M$ of $T_{\mathcal{L}}$, we define $\tilde{M}$ to be the structure
obtained by quotienting every sort of $M$ by the $\bigwedge$-definable
equivalence relation $E(x,y)$ which says that $x$ and $y$ are within distance
$\leq 1/n$ for every $n\geq 0$. It is easy to check that for every function
symbol $f\in\mathcal{L}$, the interpretation $f^{M}$ of $f$ in $M$ induces a
uniformly continuous function $f^{\tilde{M}}$ on $\tilde{M}$. Therefore
$\tilde{M}$ is a classical $\mathcal{L}$-structure and $\tilde{M}\models
T_{\mathcal{L}}$ which respects the uniform continuity conditions of
$T_{\mathcal{L}}$.

Therefore, if $M\models T_{\mathcal{L}}$ then $\tilde{M}\models
T_{\mathcal{L}}$. If $M$ is $\omega_{1}$-saturated, then the value sorts of
$\tilde{M}$ are the standard real numbers and each sort of $\tilde{M}$ is a
complete metric space. Therefore $\tilde{M}$ is a continuous logic
$\mathcal{L}$-structure. We say that $\tilde{M}$ is the _standardisation_ of
$M$.

The key property of positive formulas is the following: if $\phi(x)$ is any
positive formula and $M\models\phi(a)$ where $a\in M$ and $M$ is a classical
$\mathcal{L}$-structure, then $\tilde{M}\models\phi(\tilde{a})$ where
$\tilde{a}$ is the equivalence class of $a$ in $\tilde{M}$.

For a continuous logic language $\mathcal{L}$, a positive formula $\phi(x)$ in
$\mathcal{L}$, and a rational $\epsilon>0$, we define the _$\epsilon$
-approximation of $\phi(x)$_, written $\phi^{\epsilon}(x)$, as the formula
obtained from $\phi$ by weakening all the bounds mentioned in $\phi$ by
$\epsilon$. This construction is adapted from Section 5 in [HI02]. Explicitly,
$\phi^{\epsilon}$ is defined up to logical equivalence inductively as follows:

  1. (1)

if $\phi(x)$ is of the form $f(x)=r$ where $r$ is a rational and $f(x)$ is an
$\mathcal{L}$-term, then $\phi^{\epsilon}(x)$ is $|f(x)-r|\leq\epsilon$

  2. (2)

if $\phi(x)$ is of the form $f(x)\leq r$, then $\phi^{\epsilon}(x)$ is
$f(x)\leq r+\epsilon$

  3. (3)

if $\phi(x)=\psi(x)\wedge\chi(x)$, then
$\phi^{\epsilon}(x)=\psi^{\epsilon}(x)\wedge\chi^{\epsilon}(x)$, and similarly
if $\phi(x)=\psi(x)\vee\chi(x)$

  4. (4)

if $\phi(x)=(\exists y)\psi(x,y)$, then $\phi^{\epsilon}(x)=(\exists
y)\psi^{\epsilon}(x,y)$, and similarly if $\phi(x)=(\forall y)\psi(x,y)$.

Note that in any $M\models T_{\mathcal{L}}$, we always have
$\phi(M)\subseteq\phi^{\epsilon}(M)$ for any $\epsilon>0$.

In continuous logic, we are usually interested in knowing if a formula
$\phi(x)$ is _approximately satisfied_ by a point $a$ in $M$. This means that
for all $\epsilon>0$, $M\models\phi^{\epsilon}(a)$, where $\models$ is meant
in the usual sense of classical logic. The difference between satisfaction and
approximate satisfaction is the same as the difference between saying
$(\exists y)f(a,y)=0$ and $\inf_{y}|f(a,y)|=0$. [HI02] take the route of
defining a new relation of approximate satisfaction inside a continuous logic
structure (see section 5 in [HI02]). In this paper, we prefer to keep the
usual notion of satisfaction but we work with approximations of formulas.

In Section 5 of [HI02], the authors define the _weak negation_ $neg(\phi)(x)$
of a positive $\mathcal{L}$-formula $\phi(x)$ as follows:

  1. (1)

if $\phi(x)$ is of the form $f(x)=r$, then $neg(\phi)(x)$ is $|f(x)-r|\geq 0$

  2. (2)

if $\phi(x)$ is of the form $f(x)\leq r$, then $neg(\phi)$ is $f(x)\geq r$

  3. (3)

if $\phi(x)=\psi(x)\wedge\chi(x)$ then $neg(\phi)(x)=neg(\psi)(x)\vee
neg(\chi)(x)$, and similarly if $\phi(x)=\psi(x)\vee\chi(x)$.

  4. (4)

if $\phi(x)=(\exists y)\psi(x,y)$, then $neg(\phi)=(\forall y)neg(\psi)(x,y)$,
and similarly if $\phi(x)=(\forall y)\psi(x,y)$.

Note that $neg(\phi)$ is a positive formula and in any model of
$T_{\mathcal{L}}$, $\neg\phi\subseteq neg(\phi)$. It is easy to check that for
any continuous logic structure $M$ and $a\in M$, for any positive formula, for
all $\epsilon>0$, either $M\models\phi^{\epsilon}(a)$ or $M\models
neg(\phi^{\epsilon}(a))$ and that for $\epsilon<\delta$, $\phi^{\epsilon}$ and
$neg(\phi^{\delta})$ are inconsistent. This shows that continuous logic
languages have strong expressive power.

######  Definition A.3.

Let $T$ be a consistent collection of $\mathcal{L}$-sentences. We say that $T$
is a continuous logic $\mathcal{L}$-theory if $T\supseteq T_{\mathcal{L}}$ and
for every $\phi\in T$, either $\phi\in T_{\mathcal{L}}$ or there is some
positive sentence $\psi$ and $\epsilon>0$ such that $\phi=\psi^{\epsilon}$

We say that $T$ is a complete continuous logic $\mathcal{L}$-theory if for
every positive $\mathcal{L}$-sentence $\phi$, for every $\epsilon>0$, $T$
contains either $\phi^{\epsilon}$ or $neg(\phi^{\epsilon})$.

When $M$ is a continuous logic $\mathcal{L}$-structure, we write
$Th(M)=\\{\phi^{\epsilon}\mid M\models\phi,\epsilon>0\\}$

Therefore, a complete continuous logic theory $T$ is not maximal consistent
with respect to positive sentences. Nevertheless, if $M\models T$ is a
continuous logic structure and $N$ is a nonprincipal ultrapower of $M$, then
the set of positive sentences true in $\tilde{N}$ is maximal consistent and
does not depend on $M$.

Let $M,N$ be continuous logic structures. We say that $N$ is an _elementary
extension_ of $M$ (and we write $M\prec N$) if $M$ is a substructure of $N$ in
the usual sense and for any tuple $a$ in $M$ and any positive formula
$\phi(x)$, if $M\models\phi(a)$ then $N\models\phi(a)$. Note that
$Th(M)=Th(N)$ although more positive formulas and sentences may be true in $N$
than in $M$.

######  Definition A.4.

Let $T$ be a continuous logic theory. A continuous logic type-definable set
$p(x)$ is a collection of positive formulas consistent with $T$ such that for
every $\phi(x)\in p(x)$ there is some positive $\psi(x)$ and $\epsilon>0$ and
$\phi=\psi^{\epsilon}$.

A continuous logic complete type $p(x)$ is a continuous logic type-definable
set such that for any positive formula $\phi(x)$ and $\epsilon>0$, $p$
contains either $\phi^{\epsilon}$ or $neg(\phi^{\epsilon})$.

When $M\models T$ is a continuous logic structure, $a\in M$ and $A\subseteq
M$, we write $\mathrm{tp}(a/A)$ for the continuous logic type of $a$ over $A$.
This is the set of formulas $\phi^{\epsilon}(x)$ such that $\phi(x)$ is a
positive formula over $A$ satisfied by $a$.

We write $S_{x}(T)$ for the space of continuous logic complete types in $x$.

Equivalently, we could define a continuous logic type $p(x)$ as a continuous
logic type-definable set such that there is a maximal consistent set $q(x)$ of
positive $\mathcal{L}$-formulas with $p=\\{\phi^{\epsilon}\mid\phi\in q\\}$.

Saturation is defined as expected for continuous logic structures, with
respect to continuous logic types. Note that the existence of saturated
continuous logic structures follows from the existence of saturated models in
classical first order logic: if $M\models T$ is $\kappa$-saturated in the
usual sense of classical logic, it is easy to show that the standardisation
$\tilde{M}$ is $\kappa$-saturated as a continuous logic structure. Homogeneous
continuous logic structures are constructed in the same way.

We know from classical logic that $S_{x}(T)$ is a compact topological space. A
basis of closed sets of $S_{x}(T)$ is given by the formulas contained in the
continuous logic types of $T$. Moreover, our discussion of the weak negation
$neg(\phi)$ shows that $S_{x}(T)$ is Hausdorff.

For the purpose of this appendix, write $S_{pos}(T)$ for the set of maximal
consistent types of positive formulas. For every $p(x)\in S_{pos}(T)$, define
$\tilde{p}=\\{\phi^{\epsilon}\mid\phi\in p\\}\in S_{x}(T)$. When $M\models T$
is an $\omega_{1}$-saturated continuous logic structure, every tuple $a\in M$
realises some $p\in S_{pos}(T)$. Therefore, $S_{pos}(T)$ and $S_{x}(T)$ are
homeomorphic topological spaces via the map $p\mapsto\tilde{p}$.

In this paper we often work in $\omega_{1}$-saturated continuous logic
structures. In this context, continuous logic is equivalent to working with
the fragment of positive formulas in classical logic, as is clear from the
definitions we have laid down. Arguments about the type space $S_{x}(T)$ can
be streamlined by discussing the space $S_{pos}(T)$ and working with arbitrary
positive formulas. Whenever deducing results about non-saturated continuous
logic structures, we are careful to introduce approximations of formulas to
transfer the results appropriately.

  * _In the remainder of this appendix, we work in continuous logic, so we say ‘structure’ instead of ‘continuous logic structure’, ‘theory’ instead of ‘continuous logic theory’, ‘ type’ instead of ‘ continuous logic type’, etc._

###  A.2. Standard facts and definitions in continuous logic

Many basic results in classical logic go through to continuous logic
unchanged. We record some definitions and results which are used in this
paper. In this section, $T$ always denotes a complete continuous logic theory.

####  A.2.1. Bounded and definable closure

######  Definition A.5.

Let $M\models T$ and take $A\subseteq M$. We say that a tuple $b\in M$ is in
the bounded closure of $A$ if for every elementary extension $N$ of $M$, there
is no infinite indiscernible sequence realising $\mathrm{tp}(b/A)$ in $N$. We
write $\mathrm{bdd}(A)$ for the bounded closure of $A$ in $M$.

We say that a tuple $c\in M$ is in the definable closure of $A$ if for every
elementary extension $N$ of $M$, $c$ is the only realisation of
$\mathrm{tp}(c/A)$ in $N$. We write $\mathrm{dcl}(A)$ for the definable
closure of $A$ in $M$.

See 10.7 and 10.8 in [BYBHU08] for standard results about definable and
bounded closure. Note in particular that if $M\prec N$ and $A\subseteq M$ then
$\mathrm{bdd}(A)$ is the same set in $M$ and in $N$, and similarly for
$\mathrm{dcl}(A)$.

####  A.2.2. Definable functions

######  Definition A.6.

Let $M$ be a $\kappa$-saturated model of $T$ and take $A\subseteq M$ of size
$<\kappa$. Let $X$ and $Y$ be finite Cartesian products of sorts of $T$ and
let $p$ be a type-definable subset of $X$ over $A$. We say that a function
$f:p\to Y$ is definable over $A$ if the set $\\{(x,f(x))\mid x\in p\\}$ is
type-definable over $A$.

When $M$ contains $A$ but isn’t sufficiently saturated, we say that $f$ is
definable over $A$ if there is a sufficiently saturated elementary extension
$N$ of $M$ and a definable function on $N$ which restricts to $f$.

When $f:p\to Y$ is definable over $A$, we often identify $f$ with its graph in
the type space $S_{xy}(A)$. $f$ is definable on $p$ over $A$ if and only if
the function $p\times Y\to\mathbb{R},(x,y)\mapsto d(f(x),y)$ is definable on
$p$ over $A$. See 9.24 in [BYBHU08].

In the special case where $Y$ is the value sort of $T$ corresponding to the
interval $[-n,n]$, the definable functions $p\to Y$ over $A$ are exactly the
continuous functions $S_{x}(A)\cap p\to[-n,n]$. This is because the type-space
of a $Y$ can always be identified with the interval of real numbers $[-n,n]$.

Since type-spaces are compact Hausdorff topological spaces, any complete type
$q$ in $S_{x}(A)$ is uniquely determined by the values $f(q)$ where $f$ ranges
over the $A$-definable functions $X\to\mathbb{R}$. Urysohn’s lemma also
entails that any definable $f:p\to\mathbb{R}$ extends to a continuous function
$S_{x}(A)\to\mathbb{R}$ so the local definition of $f$ on $p$ is not usually
relevant. This is a significant difference with general definable functions
$p\to Y$.

A useful technical fact is that any $A$-definable $f:X\to\mathbb{R}$ is the
uniform limit of a sequence of $A$-definable functions $(f_{n})$ on $X$ such
that for all $n$, there is a finite tuple $a_{n}$ in $A$ and a function
$g_{n}(x,y)$ definable by a term in the language $\mathcal{L}$ such that
$f_{n}(x)=g_{n}(x,a_{n})$. To see this, note that such $g_{n}(x,a_{n})$ form a
lattice of functions on $S_{x}(A)$ which separate points. Therefore the Stone-
Weierstrass theorem applies. One consequence of this is that any $A$-definable
$f:X\to\mathbb{R}$ is definable over a countable subset of $A$.

When working with definable functions, we will often write down formulas which
contain symbols for these functions, e.g $M\models f(a)=0$. This is a slight
abuse of notation, especially when the functions are only defined on a type-
definable set $p$. _These expressions are meant as shorthand for type-
definable sets in $\mathcal{L}$._

As a final comment on definable functions, let $M\models T$, $A\subseteq N$,
let $p$ be a type-definable set over $A$, and let $f:p\to Y$ be $A$-definable,
where $Y$ is any finite product of sorts of $T$. We have seen that $f$ is
defined at the level of the type-space or equivalently at the level of an
elementary extension of $M$. Nevertheless, if $p(M)\neq\emptyset$ then
$f:p(M)\to Y(M)$ is a total function. This is because
$f(a)\in\mathrm{dcl}(Aa)$ for all $a\in M$ and we have seen that
$\mathrm{dcl}(Aa)$ does not depend on $M$.

####  A.2.3. Canonical parameters and imaginaries

In this paper, we make a slightly non-standard use of the notion of canonical
parameter.

######  Definition A.7.

Let $M\models T$, let $A\subseteq M$ and let $f:X\to Y$ be $A$-definable. We
say that a single element $c\in M$ is a canonical parameter for $f$ if for any
elementary extension $N$ of $M$, any automorphism of $N$ preserves $f$ if and
only if it fixes $c$.

This definition is slightly non-standard for the reason that we do not allow
tuples of elements as canonical parameters. This is because canonical
parameters consisting of exactly one element play an important role in this
paper. We obtain canonical parameters by adding imaginary sorts whenever we
need them. Imaginary sorts in continuous logic are a special case of
hyperimaginary sorts in classical logic. See [BYU10] Section 5 for a clear
comparison.

######  Definition A.8.

An imaginary sort $S$ of $T$ is a Cartesian product of at most countably many
sorts $(S_{n})$ of $T$ endowed with a pseudo-metric $d$ such that there is an
increasing sequence $(n_{k})$ in $\mathbb{N}$ and definable pseudo-metrics
$d_{k}$ on $\prod_{i=0}^{n_{k}}S_{i}$ such that the pseudo-metrics $(d_{k})$
converge uniformly to $d$ on $S$.

This means that for any $\epsilon>0$, there is $K\geq 0$ such that for all
$k\geq K$, in any $M\models T$, for any $(a_{n}),(b_{n})\in\prod_{n\geq
0}S_{n}$, $|d((a_{n}),(b_{n}))-d_{k}((a_{n})_{n\leq n_{k}},(b_{n})_{n\leq
n_{k}})|<\epsilon$.

If $S$ is an imaginary sort of $M$ with metric $d$, expressed as the product
of the sorts $(M_{n})$, we can add the sort $S$ to the language with the
metric $d$ and projection maps $\pi_{n}:S\to M_{n}$. This construction is
carried out in detail in [BYU10].

We use imaginary sorts in two ways. Firstly, we can use an imaginary sort to
add a countable Cartesian product of metric spaces to the language: if $d_{n}$
is a metric on $M_{n}$ with diameter $1$ and $S$ is the product of the sorts
$(M_{n})$, then we can define $d(x,y)=\sum d_{n}(x_{n},y_{n})/2^{n}$.

Secondly, we can use imaginary sorts to add canonical parameters for arbitrary
definable functions. If $f$ is a definable function over a countable set
$A\subseteq M$, we have seen that we can express $f$ as the uniform limit of
functions $g_{n}(x,a_{n})$ where $a_{n}\subseteq A$ is finite and $g$ is a
term in the language. For every $n$, let $S_{n}$ be the product of the sorts
corresponding to the tuple $a_{n}$ and let $d_{n}$ be the definable
pseudometric $d_{n}(y,z)=\sup_{x}d^{\prime}(g_{n}(x,y),g_{n}(x,z))$ where
$d^{\prime}$ is the metric on the sort of $x$. The notion of forced limit in
[BYU10] shows how to ensure that the metrics $d_{n}$ are uniformly convergent.
Let $S$ be the Cartesian product of the sorts $S_{n}$ and let $d$ be the limit
of the pseudo-metrics $d_{n}$. Quotienting out $S_{n}$ and $S$ to obtain
metric spaces, $S$ is an imaginary sort of canonical parameters for $f$. See
[BYU10] for details.

####  A.2.4. Stability and definable types

######  Definition A.9.

Let $M\models T$. Take $A\subseteq M$ and let $f:X\times Y\to\mathbb{R}$ be an
$A$-definable function where $X$ and $Y$ are finite Cartesian products of
sorts of $M$ and $f$ takes values in $\mathbb{R}$. We say $f$ is unstable if
there is some elementary extension $N$ of $M$, indiscernible sequences
$(a_{n}),(b_{n})$ in $N$ and $\epsilon>0$ such that
$|f(a_{n},b_{m})-f(a_{m},b_{n})|\geq\epsilon$ for all $n\neq m$. We say $f$ is
stable if $f$ is not unstable.

######  Definition A.10.

Let $M\models T$ and let $A,B\subseteq M$. Let $p\in S_{x}(A)$ and let
$f(x,y)$ be an $A$-definable function into $\mathbb{R}$. We say that $p$ is
definable over $B$ with respect to $f$ if there is a $B$-definable function
$g(y)$ such that for any tuple $a$ in $A$ in the sort of $y$, $p(x)$ entails
that $f(x,a)=g(a)$. In that case we write $g(y)=d_{p}f(y)$.

In this paper we work with local stable independence. First developed in
[She78] and [Pil86] for classical logic, local stable independence for
continuous logic has roots in [She75] and was studied in [BYU10]. We only
recall the main definition:

######  Definition A.11.

Let $M\models T$, let $C\subseteq B\subseteq M$ and $p(x)\in S(B)$. Let
$\Delta$ be a set of stable functions definable over $A$. We say that $p(x)$
does not fork over $C$ with respect to $\Delta$ if we can add imaginary sorts
to $M$ and extend $p$ to $\mathrm{bdd}(B)$ so that $p$ is definable over
$\mathrm{bdd}(C)$ with respect to $\Delta$.

Let $A\subseteq M$. We write $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{\Delta}B$ if
$\mathrm{tp}(A/BC)$ does not $\Delta$-fork over $C$.

When $\Delta$ is the set of all stable functions, we simply say that $p$ does
not fork over $C$ and we write $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}B$.

We refer the reader to [Pil86] and [BYU10] for an exposition of the theory of
stable independence. In this paper we make use of Morley sequences in the
context of local stability.

######  Definition A.12.

Let $M\models T$ and let $\Delta$ be a set of stable formulas. We say that a
sequence $(a_{n})$ in $M$ is a Morley sequence over $A$ with respect to
$\Delta$ if for all $n$ we have $a_{n+1}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}^{\Delta}_{A}a_{0}\ldots a_{n}$ and
$\mathrm{tp}(a_{n}/\mathrm{bdd}(A))=\mathrm{tp}(a_{0}/\mathrm{bdd}(A))$.

####  A.2.5. Continuous logic and classical logic

Continuous logic is a direct generalisation of classical logic, and there is a
canonical way of taking a classical logic theory $T$ and viewing it as a
continuous logic theory $T^{cont}$. The construction of $T^{cont}$ is as
follows.

Every sort of $T$ is now viewed as a metric space with the discrete metric
with diameter $1$. We remove the equality symbol from the sorts of $T$.
Observe that there is no loss of information in doing this, since $x=y$ is
equivalent to $d(x,y)\leq 1/2$ and $x\neq y$ is equivalent to $d(x,y)\geq
1/2$, both of which are approximations of positive formulas. We add the usual
value sorts to $T$. Function symbols in the language of $T$ are unchanged. For
each relation symbol $R$ in the language of $T$, we substitute a function
symbol $f_{R}$ which we view as the indicator function of $R$ in the
corresponding continuous logic sort. It is then clear how to axiomatise a
continuous logic theory $T^{cont}$ so that there is an exact correspondence
between models of $T$ and models of $T^{cont}$.

In this paper, when we construct a theory $T^{cont}$ from a classical logic
theory $T$, we distinguish two kinds of imaginary sorts in $T^{cont}$. Firstly
we have the _classical imaginary sorts of $T^{cont}$_ which come from the
imaginary sorts of $T$ defined in the usual way (see [TZ12]). Secondly we have
the _continuous logic imaginary sorts of $T^{cont}$_ which are imaginary sorts
obtained by the constructions sketched in A.8. These correspond to certain
hyperimaginary sorts of $T$. Unless specified otherwise, ‘imaginary sort’ and
‘imaginary element’ always refer to imaginaries in continuous logic, as
defined in A.8.

####  A.2.6. Stable embeddedness

We show that the classical notion of stable embeddedness adapts easily from
classical logic to continuous logic. An account of stable embeddedness in the
context of classical logic can be found in the appendix of [CH99]. Here we
take $T$ to be a complete continuous logic theory.

######  Definition A.13.

Let $\mathcal{D}$ be a collection of type-definable sets of $T$. We say that
$\bigcup\mathcal{D}$ is stably embedded in $T$ if for any $\kappa$-saturated
$M\models T$ with $|M|=\kappa$, any $M$-definable function from a finite
Cartesian products of sets in $\mathcal{D}$ to $\mathbb{R}$ is definable over
$\bigcup\mathcal{D}(M)$.

In the above definition, saturation is not essential, but it is convenient to
include it. Saturation can be eliminated by considering imaginaries as in
[CH99]. We will replicate that argument in Proposition 1.8 in a more
restricted setting so we only consider saturated structures for now.

######  Lemma A.14.

Let $\mathcal{D}$ be a collection of distance-definable sets in $T$ (see
Definition 1.6). The following are equivalent:

  1. (1)

$\bigcup\mathcal{D}$ is stably embedded in $T$.

  2. (2)

For any $\kappa$-saturated $M\models T$ with $|M|=\kappa>|\mathcal{L}|$, for
every finite tuple $a\in M$, $\mathrm{tp}(a/\bigcup\mathcal{D}(M))$ is
definable over a subset of $\bigcup\mathcal{D}(M)$ of size at most
$|\mathcal{L}|$.

  3. (3)

For any $\kappa$-saturated $M\models T$ with $|M|=\kappa>|\mathcal{L}|$, for
every finite tuple $a\in M$, there is a subset $C$ of $\bigcup\mathcal{D}(M)$
of size at most $|\mathcal{L}|$ such that $\mathrm{tp}(a/C)$ extends uniquely
to $\bigcup\mathcal{D}(M)$.

  4. (4)

For any $\kappa$-saturated $M\models T$ with $|M|=\kappa>|\mathcal{L}|$, every
automorphism of $\bigcup\mathcal{D}(M)$ extends to an automorphism of $M$.

###### Proof.

To make notation lighter we can assume that $\mathcal{D}$ is closed under
finite Cartesian products. We fix $M\models T$ $\kappa$-saturated with
$|M|=\kappa$.

$(1)\Rightarrow(2)$ Let $a\in M$. Let $f(x,y)$ be a definable function into
$\mathbb{R}$ with $y$ in the sort of $D\in\mathcal{D}$ and $x$ in the sort of
$a$. By $(1)$ there is a $\bigcup\mathcal{D}(M)$-definable function $g(y)$
such that $g(y)=f(a,y)$ on $D$. $g$ defines $\mathrm{tp}(a/D(M))$ for $f$.
Moreover, $g$ is definable over a countable $A\subseteq D(M)$, so
$\mathrm{tp}(a/\bigcup\mathcal{D}(M))$ is definable over a subset $B$ of
$\bigcup\mathcal{D}(M)$ with $|B|\leq|\mathcal{L}|$.

$(2)\Rightarrow(1)$: Let $f(x)$ be an $M$-definable function into $\mathbb{R}$
where $x$ is a finite tuple in the sort of $D\in\mathcal{D}$. We can assume
that $f(x)=g(x,b)$ where $b$ is a finite tuple in $M$ and $g$ is
$0$-definable. $q=\mathrm{tp}(b/\bigcup\mathcal{D}(M))$ is definable over some
small $C\subseteq\mathcal{D}(M)$ so we have a $C$-definable function
$d_{q}g(x)=f(x)$.

$(2)\Rightarrow(3)$: Let $p(x)=\mathrm{tp}(a/\bigcup\mathcal{D}(M))$, $a\in
M$. Suppose that $p$ is definable over $C\subseteq\bigcup\mathcal{D}(M)$. Let
$f(x)$ be a $\bigcup\mathcal{D}(M)$-definable function into $\mathbb{R}$. We
show that the restriction of $p(x)$ to $C$ determines the value of $f(x)$. We
can assume that $f(x)=g(x,b)$ where $b$ is a finite tuple in
$\bigcup\mathcal{D}(M)$ and $g(x,y)$ is $0$-definable.

By (2), $p$ is definable over $C$ with respect to $g$ and we write $d_{p}g(y)$
for its definition. Write $d(y,D)$ for the definable function which gives the
distance to $D$. An easy compactness argument shows that for every
$\epsilon>0$ there is $\delta>0$ such that $p(x)$ contains the positive
formula over $C$:

$\forall y(d(y,D)<\delta\rightarrow|g(x,y)-d_{p}(g(y))|\leq\epsilon)$

Therefore $p\upharpoonright C$ has a unique extension to
$\bigcup\mathcal{D}(M)$.

$(3)\Rightarrow(2)$: Let $a\in M$, write
$p(x)=\mathrm{tp}(a/\bigcup\mathcal{D}(M))$ and let $f(x,y)$ be a definable
function with $y$ in the sort of $D\in\mathcal{D}$ and $x$ a tuple in the sort
of $a$. By $(3)$, there is $C\subseteq\bigcup\mathcal{D}(M)$ such that $p(x)$
is the unique extension of $p\upharpoonright C$ to $\bigcup\mathcal{D}(M)$.
Let $N$ be an elementary extension of $M$. Suppose there are $c,c^{\prime}\in
N$ both realising $p\upharpoonright C$ and $b\in D(N)$ such that
$|f(c,b)-f(c^{\prime},b)|\geq\delta$ for some $\delta>0$. Then

$N\models\exists x,x^{\prime},y\big{(}(p\upharpoonright
C)(x)\wedge(p\upharpoonright C)(x^{\prime})\wedge
D(y)\wedge|f(x,y)-f(x^{\prime},y)|\geq\delta\big{)}$

The above is a type-definable set. Since $M\prec N$ and $M$ is saturated over
$C$, $M$ satisfies the same type-definable set. This contradicts (3) and this
proves that $p$ has a unique extension to $\bigcup\mathcal{D}(N)$. Hence for
any $M\prec N$, $p(x)$ is $C$-invariant in $N$. It follows that $p(x)$ is
$C$-definable.

$(3)\Rightarrow(4):$ Let $\sigma$ be an automorphism of
$\bigcup\mathcal{D}(M)$ and suppose that we have extended it to an
automorphism $\sigma:\bigcup\mathcal{D}(M)\cup A\to\bigcup\mathcal{D}(M)\cup
B$ where $|A|<\kappa$. Let $a\in M$. There is
$C\subseteq\bigcup\mathcal{D}(M)$ with $|C|<\kappa$ such that
$\mathrm{tp}(a/A,C)$ extends uniquely to $A\cup\bigcup\mathcal{D}(M)$. By
saturation, we can find $b\in M$ such that $\sigma$ extends to an automorphism
$\bigcup\mathcal{D}(M)\cup Aa\to\bigcup\mathcal{D}(M)\cup Bb$. The result
follows by a back-and-forth argument.

$(4)\Rightarrow(3)$: Suppose $(3)$ fails for $M\models T$, where $M$ is
$\kappa$-saturated with cardinality $\kappa$. Fix $a\in M$ which witnesses the
failure of $(3)$. Let $(a_{i})_{i<\kappa}$ be an enumeration of the
realisations of $\mathrm{tp}(a)$ in $M$. Suppose we have constructed an
isomorphism $\sigma:C\to\sigma(C)$ where $C$ is a subset of
$\bigcup\mathcal{D}(M)$ such that for some $\alpha<\kappa$ and all $i<\alpha$,
the maps $\sigma:aC\to a_{i}\sigma(C)$ are not isomorphisms.

Suppose that $\sigma_{1}:aC\to a_{\alpha}\sigma(C)$ is an isomorphism. By the
failure of $(3)$ there is $a^{\prime}$ in $M$ and $b\in\bigcup\mathcal{D}(M)$
such that $\mathrm{tp}(a/C)=\mathrm{tp}(a^{\prime}/C)$ and
$\mathrm{tp}(a/bC)\neq\mathrm{tp}(a^{\prime}/bC)$. Then we can find
$b^{\prime}$ such that $\sigma_{1}:a^{\prime}bC\to
a_{\alpha}b^{\prime}\sigma(C)$ is an isomorphism. Then we extend $\sigma$ by
putting $\sigma(b)=b^{\prime}$. Note that now we cannot extend $\sigma$ to $a$
by sending $a$ to $a_{\alpha}$. By enumerating $\bigcup\mathcal{D}(M)$, we can
also make sure that after $\kappa$ iterations of this procedure $\sigma$ is
defined on all of $\bigcup\mathcal{D}(M)$. This contradicts $(4)$. ∎

###  A.3. Hilbert spaces in continuous logic

We recall here basic facts about the model theory of Hilbert spaces which we
use in this paper. We refer the reader to [BYBHU08] for a more complete
summary. In this paper we work with Hilbert spaces over $\mathbb{R}$, but all
results can be transposed to complex Hilbert spaces without modification.

On one presentation of the model theory of Hilbert spaces, the language of
Hilbert spaces in continuous logic consists of countably many metric sorts,
which stand for balls with radius $n$ around $0$. We add appropriate inclusion
maps between the metric sorts. The language consists of the usual vector space
structure over $\mathbb{R}$ and a function $\langle\cdot,\cdot\rangle$ on each
metric sort into an appropriate value sort which stands for the inner product.
The axiomatisation of the theory of infinite dimensional Hilbert spaces
$T^{Hilb}$ is as expected. In this paper, we usually don’t disinguish between
a Hilbert space and a model of $T^{Hilb}$.

$T^{Hilb}$ is complete, has quantifier-elimination, is stable, and is totally
categorical. The theory of Hilbert spaces does not have elimination of
imaginaries, but it has weak elimination of imaginaries:

######  Lemma A.15 ([BYB04] 1.2).

Let $M\models T^{Hilb}$. Let $\alpha$ be a canonical parameter for an
$M$-definable function $f$ in an arbitrary imaginary sort of $T^{Hilb}$. Then
there is a closed subspace $H$ of $M$ such that each point of $H$ is in
$\mathrm{bdd}(\alpha)$ and $\alpha$ is definable over $H$.

######  Definition A.16.

If $H$ is a Hilbert space and $V\leq H$ is a closed subspace, write $P_{V}$
for the orthogonal projection onto $V$.

We will make much use of the characterisation of forking independence in
Hilbert spaces. See [BYBHU08] for a proof:

######  Lemma A.17.

Let $M\models T^{Hilb}$. Let $A,B,C\subseteq M$ with $B\subseteq C$. Then
$\mathrm{bdd}(B)$ is the closed subspace of $M$ generated by $B$ and
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}C$ if and only
if for all $a\in A$, $P_{\mathrm{bdd}(C)}(a)=P_{\mathrm{bdd}(B)}(a)$.

Finally, we recall some elementary facts about the weak topology in Hilbert
spaces which we use in our proofs. If $H$ is a Hilbert space, recall that the
weak topology has a sub-basis consisting of the sets

$\\{v\in H\mid\langle v,w\rangle\in U,w\in H,U\subseteq\mathbb{R}\text{
open}\\}.$

Recall also that the unit ball is compact in the weak topology and that every
bounded sequence in $H$ has a weakly convergent subsequence.

######  Lemma A.18.

Let $H$ be a Hilbert space and $A\subseteq H$ a bounded subset of $H$. Then
the closure of $A$ in the weak topology is the set of limit points of
sequences in $A$ in the weak topology.

###### Proof.

Suppose $w$ is in the closure of $A$ in the weak topology. Let $v_{0}\in A$ be
any vector such and construct $(v_{n})$ inductively as follows: given
$v_{0},\ldots,v_{n}$, find $v_{n+1}\in A$ such that $|\langle
v_{n+1},v_{i}\rangle-\langle w,v_{i}\rangle|<1/(n+1)$ and $|\langle
v_{n+1},w\rangle-\langle w,w\rangle|<1/(n+1)$. Since $A$ is bounded, we can
assume that $(v_{n})$ converges weakly to some $z\in H$. We check that $z=w$.
Firstly we have

$\langle z,w\rangle=\lim_{n}\langle v_{n},w\rangle=\lim_{n}\lim_{m}\langle
v_{n},v_{m}\rangle=\lim_{n}\langle v_{n},z\rangle=\langle z,z\rangle.$

Similarly,

$\langle w,w\rangle=\lim_{n}\lim_{m}\langle v_{n},v_{m}\rangle=\langle
z,z\rangle.$

Hence $z=w$. We also deduce that the original sequence $(v_{n})$ converges
weakly to $w$ and that it is not necessary to take a subsequence. ∎

The next lemma is proved in a similar way to Lemma A.18:

######  Lemma A.19.

Let $M\models T^{Hilb}$ and $A\subseteq M$ and $v\in M$. Let $(v_{n})$ be an
indiscernible sequence in $\mathrm{tp}(v/A)$ in $M$. Then there is an
orthogonal sequence $(w_{n})$ in $M$ and $w\in M$ such that for all $n$,
$v_{n}=w_{n}+w$, $w_{n}\perp A$ and $w_{n}\perp w$. It follows that $(v_{n})$
converges weakly to $w$ (we write $v_{n}\rightharpoonup w$).

Moreover, $w$ is the unique element of $M$ such that $\langle
w,w\rangle=\langle v_{1},v_{0}\rangle=\langle w,v_{n}\rangle$ for all $n$.

######  Lemma A.20.

Let $M\models T^{Hilb}$. Let $(v_{n})$ be a Morley sequence over $A\subseteq
M$. Then $(v_{n})$ converges weakly to $P_{\mathrm{bdd}{A}}(v_{0})$.

###### Proof.

It is enough to check that $\lim_{n}\langle v_{n},v_{m}\rangle=\langle
P_{\mathrm{bdd}(A)}(v_{0}),v_{m}\rangle$ for all $m$. For $n>m$ we have
$\langle v_{n},v_{m}\rangle=\langle P_{\mathrm{bdd}(Av_{0}\ldots
v_{m})}(v_{n}),v_{m}\rangle=\langle
P_{\mathrm{bdd}(A)}(v_{n}),v_{m}\rangle=\langle
P_{\mathrm{bdd}(A)}(v_{0}),v_{m}\rangle$. ∎

##  Appendix B More theory of interpretable Hilbert spaces

_In this appendix, we elaborate on some technical notions which we only
briefly touched upon in Section 1.1. _

Let $T$ be a continuous logic theory with an interpretable Hilbert space
$\mathcal{H}$. Let $M\models T$.

######  Definition B.1.

Let $H(M),H^{\prime}(M)$ be interpretable Hilbert spaces in $M$. Suppose
$H(M)$ is the direct limit of $(M_{j})_{j\in J}$ and $H^{\prime}(M)$ is the
direct limit of $(M_{j^{\prime}})_{J^{\prime}}$. Write $h_{j}:M_{j}\to H(M)$
and $h_{j}^{\prime}:M_{j^{\prime}}\to H^{\prime}(M)$ for the direct limit
maps.

We say that a map $F:H(M)\to H^{\prime}(M)$ is an embedding of piecewise
interpretable Hilbert spaces if $F$ is a unitary map and for all $j\in J$ and
$j^{\prime}\in J^{\prime}$ and all $\epsilon\geq 0$, the set $\\{(x,y)\in
M_{j}\times
M^{\prime}_{j^{\prime}}\mid\|F(h_{j}x)-h_{j^{\prime}}^{\prime}y\|\leq\epsilon\\}$
is type-definable.

If $F$ is also surjective, we say that $F$ is an isomorphism of piecewise
interpretable Hilbert spaces.

######  Lemma B.2.

Let $H(M),H^{\prime}(M)$ be interpretable Hilbert spaces in $M$. Let
$(M_{i})_{i\in I}$ and $(M_{i^{\prime}})_{i^{\prime}\in I^{\prime}}$ be pieces
of $H(M)$ and $H^{\prime}(M)$ respectively which generate $H(M)$ and
$H^{\prime}(M)$. Write $h_{i}:M_{i}\to H(M)$ and
$h_{i^{\prime}}^{\prime}:M_{i^{\prime}}\to H^{\prime}(M)$ for the direct limit
maps.

A unitary map $F:H(M)\to H^{\prime}(M)$ is an embedding of interpretable
Hilbert spaces if and only if for all $i\in I$ and $i^{\prime}\in I^{\prime}$
the map $M_{i}\times M_{i^{\prime}}\to\mathbb{R},(x,y)\mapsto\langle
F(h_{i}x),h_{i^{\prime}}^{\prime}y\rangle$ is definable.

###### Proof.

Suppose that $F$ is an embedding of piecewise interpretable Hilbert spaces.
Fix $i\in I$, $i^{\prime}\in I^{\prime}$ and $D$ a closed bounded subset of
$\mathbb{R}$. For every $\epsilon>0$, let $D_{\epsilon}$ be the closed set
$\\{x\in\mathbb{R}\mid\exists y\in D,|y-x|\leq\epsilon\\}$. Let $B$ be an
upper bound on $\\{\|h_{i^{\prime}}^{\prime}y\|\mid y\in M_{i^{\prime}}\\}$.

By compactness, for every $\epsilon>0$ we can find $i^{\prime}_{\epsilon}\in
I^{\prime}$ such that for every $x\in M_{i}$, there is $x^{\prime}\in
M_{i^{\prime}_{\epsilon}}$ such that
$\|h_{i^{\prime}_{\epsilon}}^{\prime}x^{\prime}-F(h_{i}x)\|<\epsilon/B$. The
set $\\{(x,y)\in M_{i}\times M_{i^{\prime}}\mid\langle
F(h_{i}x),h_{i^{\prime}}^{\prime}y\rangle\in D\\}$ is equal to the
intersection over $\epsilon>0$ of the sets

$\\{(x,y)\in M_{i}\times M_{i^{\prime}}\mid\exists x^{\prime}\in
M_{i_{\epsilon}},\|F(h_{i}x)-h_{i^{\prime}_{\epsilon}}^{\prime}x^{\prime}\|\leq\epsilon/B\text{
and }\langle
h_{i^{\prime}_{\epsilon}}^{\prime}x^{\prime},h_{i^{\prime}}y\rangle\in
D_{\epsilon}\\}.$

Hence this set is type-definable.

Conversely, suppose that $H(M)$ and $H^{\prime}(M)$ respectively are the
direct limits of the sorts $(M_{j})_{J}$ and $(M_{j^{\prime}})_{J^{\prime}}$.
We write $(h_{j})$ and $(h_{j^{\prime}}^{\prime})$ for the direct limit maps.
Fix $j\in J$, $j^{\prime}\in J^{\prime}$. For any $x\in M_{j}$, the element
$F(h_{j}x)$ is uniquely determined by the collection of maps
$M_{i^{\prime}}\to\mathbb{R},y\mapsto\langle
F(h_{j}x),h_{i^{\prime}}^{\prime}y\rangle$, for $i^{\prime}\in I^{\prime}$.

Moreover, by a standard compactness argument, for arbitrary $\epsilon>0$ there
is $n_{\epsilon}\geq 1$ such that for any $x\in M_{j}$, $h_{j}x$ is within
distance $\epsilon$ of a vector of the form
$\sum_{k=1}^{n_{\epsilon}}\lambda_{k}h_{i_{k}}x_{i_{k}}$ where $i_{k}\in I$
for all $k\leq n_{\epsilon}$, $x_{i_{k}}\in M_{i_{k}}$ and $|\lambda_{k}|\leq
n_{\epsilon}$. By considering small enough $\delta$ and quantifying over
elements $\sum_{k=1}^{n_{\delta}}\lambda_{k}h_{i_{k}}x_{i_{k}}$, we find that
$\\{(x,y)\in M_{j}\times
M_{j^{\prime}}\mid\|F(h_{j}x)-h_{j^{\prime}}^{\prime}y\|\leq\epsilon\\}$ is
type-definable if the maps $\langle F(h_{i}x),h^{\prime}_{i^{\prime}}y\rangle$
are definable.

∎

The following sharpening of Proposition 1.4 follows easily:

######  Lemma B.3.

Let $M$ be a continuous logic structure. As in Proposition 1.4, suppose we are
given a Hilbert space $H$, a collection of distinct imaginary sorts
$(M_{i})_{i\in I}$ of $M$ and maps $F_{i}:M_{i}\to H$ such that for all
$i,j\in I$, the map $M_{i}\times M_{j}\to\mathbb{R},(x,y)\mapsto\langle
F_{i}x,F_{j}y\rangle$ is definable.

Then there is an interpretable Hilbert space $H(M)$ which is unique up to
isomorphism such that the sorts $(M_{i})_{I}$ are pieces of $H(M)$, $\bigcup
h_{i}(M_{i})$ has dense span in $H(M)$, and for $x\in M_{i}$, $y\in M_{j}$,
$\langle h_{i}x,h_{j}y\rangle_{H(M)}=\langle F_{i}x,F_{j}y\rangle_{H}$.

###### Proof.

Let $H(M)$, $H^{\prime}(M)$ be two piecewise interpretable Hilbert spaces
satisfying the existence claim. Since the inner product maps on the pieces
$M_{i}$ are identical, the uniqueness statement of the GNS theorem applies and
we find an isomorphism of Hilbert spaces $F:H(M)\to H^{\prime}(M)$. Lemma B.2
applies directly so $F$ is an isomorphism of interpretable Hilbert spaces. ∎

Recall that we did not require direct limits of sorts of $M$ to have injective
transition maps. We now show that this does not present any significant
disadvantage.

######  Lemma B.4.

Let $H(M)$ be an interpretable Hilbert space in $M$. Then $H(M)$ is isomorphic
to an interpretable Hilbert space $H^{\prime}(M)$ which is a direct limit of
imaginary sorts $(M_{j^{\prime}})_{J^{\prime}}$ with isometric direct limit
maps.

###### Proof.

Suppose that $H(M)$ is the direct limit of the sorts $(M_{j})$ with inner
product maps $(f_{ij})$. Write $h_{ij}:M_{i}\to M_{j}$ for the transition maps
between the sorts $M_{i},M_{j}$ when $i\leq j$ and $h_{j}:M_{j}\to H(M)$ for
the direct limit maps. For every $j\in J$, let $M^{\prime}_{j}$ be the
imaginary sort of canonical parameters of the inner product map $f_{jj}$,
where we view $f_{jj}(x,y)$ as a function in $y$ with a parameter $x$. For
$i\leq j\in J$, define the transition map $h^{\prime}_{ij}:M^{\prime}_{i}\to
M^{\prime}_{j}$ as the map which takes the canonical parameter for the map
$f_{ii}(a,.)$ to the canonical parameter for the map $f_{jj}(h_{ij}a,.)$.
Observe $h^{\prime}_{ij}$ is well-defined because $f_{ii}(a,.)=f_{ii}(b,.)$ if
and only if $h_{i}a=h_{i}b$. It is clear that the direct limit $H^{\prime}(M)$
of $(M^{\prime}_{j})_{J}$ satisfies the lemma. ∎

As a direct application of the construction in Proposition 1.4 and Lemma B.4,
we have:

######  Lemma B.5.

Let $H(M)$ be an interpretable Hilbert space in $M$. Then $H(M)$ is isomorphic
to an interpretable Hilbert space $H^{\prime}(M)$ in $M$ which is a direct
limit of $(M_{j^{\prime}})_{j^{\prime}\in J^{\prime}}$ with isometric direct
limit maps $h^{\prime}_{j^{\prime}}$ such that the Hilbert space operations on
$H^{\prime}(M)$ are piecewise bounded333 Thanks to Arturo Rodríguez Fanlo for
this terminology. This means that

  1. (1)

for every $i,j\in J^{\prime}$, there is $k\in J^{\prime}$ such that $i,j\leq
k$ and $h_{i}^{\prime}(M_{i})+h_{j}^{\prime}(M_{j})\subseteq
h_{k}^{\prime}(M_{k})$ and the map $M_{i}\times M_{j}\to
M_{k},(x,y)\mapsto(h_{k}^{\prime})^{-1}(x+y)$ is definable

  2. (2)

for every $i\in J^{\prime}$ and $n\geq 0$, there is $k$ such that for $x\in
M_{i}$ and $\lambda\in[-n,n]$, $\lambda h_{i}^{\prime}x\in
h_{i}^{\prime}M_{k}$ and the map $[-n,n]\times M_{i}\to M_{k}$,
$(\lambda,x)\mapsto(h_{i}^{\prime})^{-1}(\lambda h_{i}^{\prime}x)$ is
definable.

###### Proof.

Apply the construction of Proposition 1.4 to all pieces $(M_{j})_{J}$ of
$H(M)$ with the direct limit maps $M_{j}\to H(M)$ to obtain $H_{1}(M)$
isomorphic to $H(M)$. Observe that the construction of Proposition 1.4 implies
that the operations on $H_{1}(M)$ are piecewise bounded. Now apply Lemma B.4
to obtain $H^{\prime}(M)$ with isometric direct limit maps. ∎

Finally we show that an isomorphism of interpretable Hilbert spaces in an
$\omega$-saturated structure induces an isomorphism at the level of the theory
$T$.

######  Lemma B.6.

Let $T$ be a complete theory and let $\mathcal{H}$, $\mathcal{H}^{\prime}$ be
two interpretable Hilbert spaces in $T$. Suppose that for some
$\omega$-saturated $M\models T$, $H(M)$ and $H^{\prime}(M)$ are isomorphic as
interpretable Hilbert spaces. Then for every $N\models T$, $H(N)$ and
$H^{\prime}(N)$ are isomorphic as interpretable Hilbert spaces.

###### Proof.

Suppose that $H(M)$ and $H^{\prime}(M)$ are the direct limits of $(M_{j})_{J}$
and $(M_{j^{\prime}})_{J^{\prime}}$ respectively. We write $h_{j}$ and
$h_{j^{\prime}}^{\prime}$ for the direct limit maps $M_{j}\to H(M)$,
$M_{j^{\prime}}\to H^{\prime}(M)$ respectively. Write $F_{M}:H(M)\to
H^{\prime}(M)$ for the isomorphism of interpretable Hilbert spaces. Fix
$N\models T$, $j\in J$, and take $a\in N_{j}$. Find $x\in M_{j}$ with
$\mathrm{tp}(x)=\mathrm{tp}(a)$ and fix $j^{\prime}\in J^{\prime}$ such that
$F_{M}h_{j}x\in h^{\prime}_{j^{\prime}}M_{j^{\prime}}$.

$F_{M}h_{j}x$ is uniquely determined by the value $\lambda=\langle
F_{M}h_{j}x,F_{M}h_{j}x\rangle$ and the $x$-definable function
$f_{x}:M_{j^{\prime}}\to\mathbb{R},y\mapsto\langle
F_{M}h_{j}x,h^{\prime}_{j^{\prime}}y\rangle$. By elementarity, for every $n$,
there is $b_{n}\in N_{j^{\prime}}$ such $|\langle
h^{\prime}_{j^{\prime}}b_{n},h^{\prime}_{j^{\prime}}b_{n}\rangle-\lambda|<2^{-n}$
and for all $y\in N_{j^{\prime}}$ $|f_{a}(y)-\langle
h^{\prime}_{j^{\prime}}b_{n},h^{\prime}_{j^{\prime}}y\rangle|<2^{-n}$. Then
$(h^{\prime}_{j^{\prime}}b_{n})$ is Cauchy and there is $b\in N_{j^{\prime}}$
such that $h^{\prime}_{j^{\prime}}b$ is the limit of
$(h^{\prime}_{j^{\prime}}b_{n})$. Define
$F_{N}h_{j}a=h^{\prime}_{j^{\prime}}b$. It is straightforward to check that
$F_{N}$ is well-defined, definable, and is an isomorphism of interpretable
Hilbert spaces. ∎

Finally, we give a category-theoretic reformulation of Proposition 1.4:

######  Lemma B.7.

Let $M\models T$. The categories $\mathcal{A},\mathcal{B},\mathcal{C}$ defined
below are all equivalent:

Let $\mathcal{A}$ be the category of interpretable Hilbert spaces in $M$ with
embeddings of interpretable Hilbert spaces.

Let $\mathcal{B}$ be the category of pairs $(H,(h_{i})_{I})$ where $H$ is a
Hilbert space and for every $i\in I$, $h_{i}:M_{i}\to H$ is a map on an
imaginary sort of $M$ such that

  1. (1)

the set $\bigcup_{I}h_{i}(M_{i})$ has dense span in $H$

  2. (2)

for all $i_{1},i_{2}\in I$, the map $M_{i_{1}}\times
M_{i_{2}}\to\mathbb{R},(x,y)\mapsto\langle h_{i_{1}}x,h_{i_{2}}y\rangle$ is
definable.

The morphisms between objects $(H,(h_{i})_{I})$ and
$(H^{\prime},(h_{i^{\prime}})_{I^{\prime}})$ of $\mathcal{B}$ are unitary maps
$F:H\to H^{\prime}$ such that for all $i\in I$, $i^{\prime}\in I^{\prime}$ the
map $M_{i}\times M_{i^{\prime}}\to\mathbb{R},(x,y)\mapsto\langle
F(h_{i}x),h_{i^{\prime}}y\rangle$ is definable.

Let $\mathcal{C}$ be the category of pairs $((M_{i})_{I},(f_{ij})_{i,j\in I})$
where $(M_{i})_{I}$ is a set of imaginary sorts of $M$ and the
$f_{ij}:M_{i}\times M_{j}\to\mathbb{R}$ are definable functions such that
their concatenation $f:\bigcup_{I}M_{i}\times\bigcup_{I}M_{i}\to\mathbb{R}$ is
positive semidefinite. The morphisms between objects $((M_{i})_{I},(f_{ij}))$
and $((M_{i^{\prime}})_{I^{\prime}},f_{i^{\prime}j^{\prime}})$ of
$\mathcal{C}$ are piecewise definable functions $G:\bigcup_{k\in I\cup
I^{\prime}}M_{k}\times\bigcup_{k\in I\cup I^{\prime}}M_{k}\to\mathbb{R}$ such
that

  1. (1)

$G$ is positive semidefinite and $G$ extends each function $f_{ij}$ where
$i,j\in I$ or $i,j\in I^{\prime}$

  2. (2)

writing $H_{I}$ and $H_{I\cup I^{\prime}}$ for the Hilbert spaces induced by
$\bigcup_{I}M_{i}$ and $\bigcup_{k\in I\cup I^{\prime}}M_{k}$ respectively as
in the GNS theorem, the resulting Hilbert space embedding $F:H_{I^{\prime}}\to
H_{I\cup I^{\prime}}$ induced from $G$ is surjective 444 The GNS theorem gives
Hilbert spaces $H_{I}$, $H_{I^{\prime}}$ and $H_{I\cup I^{\prime}}$ and maps
$a:\bigcup_{I}M_{i}\to H_{I}$, $b:\bigcup_{I^{\prime}}M_{i^{\prime}}\to
H_{I^{\prime}}$, $c:\bigcup_{k\in I\cup I^{\prime}}M_{k}\to H_{I\cup
I^{\prime}}$ and Hilbert space embeddings $\phi:H_{I}\to H_{I\cup I^{\prime}}$
and $\phi^{\prime}:H_{I^{\prime}}\to H_{I\cup I^{\prime}}$ such that the
corresponding diagram commutes. So we have $G(x,y)=\langle\phi\circ
a(x),\phi^{\prime}\circ b(y)\rangle$ when $x\in\bigcup_{I}M_{i}$ and
$y\in\bigcup_{I^{\prime}}M_{i^{\prime}}$

Composition of morphisms in $\mathcal{C}$ is induced by the GNS theorem555 It
is possible to give a definition of the morphisms of $\mathcal{C}$ and their
composition which does not rely explicitly on the GNS theorem by using the
Gram-Schmidt orthogonalisation process and Bessel’s inequality. The details
are left to the reader.

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