# The Calkin algebra, Kazhdan’s property (T), and strongly self-absorbing
$\mathrm{C}^{*}$-algebras

Ilijas Farah  Department of Mathematics and Statistics, York University, 4700
Keele Street, Toronto, Ontario, Canada, M3J 1P3  Matematički Institut SANU
Kneza Mihaila 36
11 000 Beograd, p.p. 367
Serbia  email<EMAIL_ADDRESS>https://ifarah.mathstats.yorku.ca/

###### Abstract.

It is well-known that the relative commutant of every separable nuclear
$\mathrm{C}^{*}$-subalgebra of the Calkin algebra has a unital copy of Cuntz
algebra $\mathcal{O}_{\infty}$. We prove that the Calkin algebra has a
separable $\mathrm{C}^{*}$-subalgebra whose relative commutant has no simple,
unital, and noncommutative $\mathrm{C}^{*}$-subalgebra. On the other hand, the
corona of every stable, separable $\mathrm{C}^{*}$-algebra that tensorially
absorbs the Jiang–Su algebra ${\mathcal{Z}}$ has the property that the
relative commutant of every separable $\mathrm{C}^{*}$-subalgebra contains a
unital copy of ${\mathcal{Z}}$. Analogous result holds for other strongly
self-absorbing $\mathrm{C}^{*}$-algebras. As an application, the Calkin
algebra is not isomorphic to the corona of the stabilization of the Cuntz
algebra $\mathcal{O}_{\infty}$, any other Kirchberg algebra, or even the
corona of the stabilization of any unital, ${\mathcal{Z}}$-stable
$\mathrm{C}^{*}$-algebra.

###### Key words and phrases:

Calkin algebra, Kazhdan’s property (T), and strongly self-absorbing
$\mathrm{C}^{*}$-algebras.

###### 1991 Mathematics Subject Classification:

22D55, 46L05, 19K33

Partially supported by NSERC.

ORCID iD https://orcid.org/0000-0001-7703-6931

The most intriguing open problem about the Calkin algebra $\mathcal{Q}(H)$ is
whether it can have a K-theory reversing automorphism ([8], [9]). A possible
line of attack on this problem was suggested by Shuang Zhang. The corona
$\mathcal{Q}(\mathcal{O}_{\infty}\otimes\mathcal{K})$ of the stabilization of
the Cuntz algebra $\mathcal{O}_{\infty}$ is K-theoretically indistiguishable
from $\mathcal{Q}(H)$, but it is known that the Kirchberg–Phillips
classification theorem (see [43], [46], [27]) implies that it has a $K$-theory
reversing automorphism, and an isomorphism between $\mathcal{Q}(H)$ and
$\mathcal{Q}(\mathcal{O}_{\infty}\otimes\mathcal{K})$ would give a K-theory
reversing automorphism of $\mathcal{Q}(H)$.

Remarkably, many of the defining properties of $\mathcal{Q}(H)$ are shared by
$\mathcal{Q}(\mathcal{O}_{\infty}\otimes\mathcal{K})$ and coronas of other
separable, purely infinite, non-unital $\mathrm{C}^{*}$-algebra. These
algebras have real rank zero and satisfy a (generalized) Weyl–von Neumann
theorem ([59]; for other coronas with real rank zero see [41, Theorem 3.8 and
definitions on pp. 245–246]). They are simple and purely infinite: By [45,
Theorem 3.2] or by [35, Definition 2.5 and Theorem 2.8] and [36], if $A$ is
unital then $\mathcal{Q}(A\otimes\mathcal{K})$ is simple if and only if
$A=M_{n}({\mathbb{C}})$ or $A$ is simple and purely infinite. Every separable,
unital $\mathrm{C}^{*}$-subalgebra of $\mathcal{Q}(H)$ is equal to its double
commutant (this is a consequence of Voiculescu’s theorem for $\mathcal{Q}(H)$,
and proved in [28, Theorem B] using the Elliott–Kucerovsky theory of absorbing
extensions, [20] for $\mathcal{Q}(\mathcal{O}_{\infty}\otimes\mathcal{K})$;
see also [40]). Many general properties of the Ext functor resemble the ones
familiar from the BDF theory ([8], [9], also [15]) by [42]. Nevertheless…

######  Theorem A.

The Calkin algebra is not isomorphic to the corona of the stabilization of
$\mathcal{O}_{\infty}$.

To put this theorem in proper context, see the beginning of §6. Its proof does
not use set theory at all, and model theory is making only a cameo appearance
in Corollary 5.4 (not used in the proof of our main results).

The proof of Theorem A breaks down as follows. Theorem B below is based on an
extension of S. Wasserman’s proof that a certain quotient of the full group
algebra associated with a property (T) group with infinitely many inequivalent
irreducible representations on finite-dimensional Hilbert spaces has the
property that its ${\rm Ext}$ is not a group ([54], rediscovered in [31,
Theorem 3.7.10]). More precisely, we strengthen the variants of Wassermann’s
result developed in [11, Corollary 6.5.11] and [10, §17]. The following is
proved at the end of §2, as a consequence of Proposition 2.3.

######  Theorem B.

There is a separable $\mathrm{C}^{*}$-subalgebra $B$ of $\mathcal{Q}(H)$ whose
relative commutant $\mathcal{Q}(H)\cap B^{\prime}$ has no simple,
noncommutative $\mathrm{C}^{*}$-subalgebra that contains the unit of
$\mathcal{Q}(H)$.

The emphasis in the conclusion is of course on the fact that the simple
$\mathrm{C}^{*}$-subalgebra is assumed to be a unital subalgebra of
$\mathcal{Q}(H)$. This is because every projection $p$ in $\mathcal{Q}(H)\cap
B^{\prime}$ corresponds (after identifying $\mathcal{Q}(H)$ with its corner
corresponding to $p$) to an extension111As customary, we identify extensions
with their Busby invariants, that is, unital ∗-homomorphisms into the Calkin
algebra. See e.g., [31]. of $B$, defined by $\Phi(b):=pb$, and equivalent
extensions correspond to projections that are Murray–von Neumann equivalent in
$\mathcal{Q}(H)\cap B^{\prime}$. Finally, by Voiculescu’s theorem (see e.g.,
[1] or [10]) every extension of $B$ absorbs the trivial extension. Therefore
some $p_{0}\in\mathcal{Q}(H)\cap B’$ corresponds to the trivial extension. By
repeatedly applying Voiculescu’s theorem, one finds orthogonal projections
$p_{n}$, for $n\geq 1$, in $p_{0}(\mathcal{Q}(H)\cap B’)$, each one of them
Murray—von Neumann equivalent to $p_{0}$. If $v_{n}$ is a partial isometry in
$\mathcal{Q}(H)\cap B’$ such that $v_{n}^{*}v_{n}=p_{0}$ and
$v_{n}v_{n}^{*}=p_{n}$, then $\mathrm{C}^{*}(v_{n}:n\in{\mathbb{N}})$ is a
$\mathrm{C}^{*}$-subalgebra of $\mathcal{Q}(H)\cap B’$ isomorphic to
$\mathcal{O}_{\infty}$.

The other component of the proof of Theorem A applies to coronas of
stablizations of strongly self-absorbing $\mathrm{C}^{*}$-algebras (see §1
below). The following is Theorem 4.1.

######  Theorem C.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable, unital, and tensorially
$D$-absorbing then for every separable $\mathrm{C}^{*}$-subalgebra $B$ of
$\mathcal{Q}(A\otimes\mathcal{K})$ there is an isomorphic copy of $D$ in
$\mathcal{Q}(A\otimes\mathcal{K})\cap B^{\prime}$ that contains the unit of
$\mathcal{Q}(A\otimes\mathcal{K})$.

Theorem B and Theorem C immediately imply the following strengthening of
Theorem A. Here ${\mathcal{Z}}$ denotes the Jiang–Su algebra, the initial
object in the category of strongly self-absorbing $\mathrm{C}^{*}$-algebras
([56]), and for Elliott-classifiable $\mathrm{C}^{*}$-algebras see [46].

######  Corollary D.

The Calkin algebra is not isomorphic to the corona of $A\otimes\mathcal{K}$,
for any separable, unital, and tensorially ${\mathcal{Z}}$-absorbing
$\mathrm{C}^{*}$-algebra $A$. In particular, if $A$ is any Elliott-
classifiable $\mathrm{C}^{*}$-algebra then $\mathcal{Q}(A\otimes\mathcal{K})$
is not isomorphic to $\mathcal{Q}(H)$. ∎

It is well-known that many instances of Corollary D can be proven by
K-theoretic means, as follows. To the short exact sequence

$0\to
A\otimes\mathcal{K}\to\mathcal{M}(A\otimes\mathcal{K})\to\mathcal{Q}(A\otimes\mathcal{K})\to
0$

one associates the six-term exact sequence

$K_{0}(A\otimes\mathcal{K})$$K_{0}(\mathcal{M}(A\otimes\mathcal{K}))$$K_{0}(\mathcal{Q}(A\otimes\mathcal{K}))$$K_{1}(\mathcal{Q}(A\otimes\mathcal{K}))$$K_{1}(\mathcal{M}(A\otimes\mathcal{K}))$$K_{1}(A\otimes\mathcal{K})$

.

Since the K-theory of the stable multiplier algebra
$\mathcal{M}(A\otimes\mathcal{K})$ ([4, Definition 12.1.3]) is trivial ([4,
Proposition 12.2.1]), the vertical arrows in this six-term exact sequence are
isomorphisms. Therefore $K_{i}(A)=K_{1-i}(\mathcal{Q}(A\otimes\mathcal{K}))$
(this is [4, Corollary 12.2.3]). This implies many instances of Corollary D,
but not Theorem A because
$K_{0}(\mathcal{O}_{\infty})={\mathbb{Z}}=K_{1}(\mathcal{Q}(H))$ and
$K_{1}(\mathcal{O}_{\infty})=0=K_{0}(\mathcal{Q}(H))$.

For a unital $\mathrm{C}^{*}$-algebra $A$ consider the _path algebra_ 222There
is no universally accepted notation for the path algebra. Some authors use
$A_{\infty}$, a notation largely established for the more common asymptotic
sequence algebra $\ell_{\infty}(A)/c_{0}(A)$. The authors of [29] used
somewhat cumbersome ${\mathfrak{A}}A={\mathfrak{T}}A/{\mathfrak{T}}_{0}A$ and
the original notation in [12], $\mathcal{Q}(A)$, is nowadays commonly used for
coronas. The terminology ‘path algebra’ has been suggested by Gábor Szabó.

$A_{\text{\faForward}}:=C_{b}([0,\infty),A)/C_{0}([0,\infty),A).$

In Theorem 3.1 we prove the analog of Theorem C for $A_{\text{\faForward}}$.
The following is Corollary 5.1.

######  Theorem E.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. Then all unital ∗-homomorphisms of $D$ into
$A_{\text{\faForward}}$ are unitarily equivalent and all unital
∗-homomorphisms of $D$ into $\mathcal{Q}(A\otimes\mathcal{K})$ are unitarily
equivalent.

There is an obvious diagonal embedding of $A$ into $A_{\text{\faForward}}$,
and we identify $A$ with its image. Together with standard methods, Theorem E
and Theorem 3.1 imply the following analog of a vital property of (relative
commutants in) ultrapowers and asymptotic sequence algebras ([46, Theorem
7.2.2]), appearing here as Corollary 5.3.

######  Corollary F.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is a separable and unital
$\mathrm{C}^{*}$-algebra then the following are equivalent.

  1. (1)

$A$ is tensorially $D$-absorbing.

  2. (2)

$D$ unitally embeds into $A_{\text{\faForward}}\cap A^{\prime}$ .

It is not known whether the analogous statement holds for
$\mathcal{Q}(A\otimes\mathcal{K})$ (see Conjecture 6.2).

### Acknowledgments

I would like to thank Ping Wong Ng for communicating Zhang’s question to me
and to Bruce Blackadar, Jamie Gabe, Ilan Hirshberg, Narutaka Ozawa, Gábor
Szabó, and Mikael Rørdam for helpful remarks on the earlier versions of this
paper. Thanks to Gábor Szabó for pointing out to an error in proofs of §3 and
§4 that was as embarrassing as it was inconsequential and to Jamie Gabe for
improving the conclusion of Theorem E by replacing approximate unitary
equivalence with unitary equivalence. Last, but not least, I am indebted to
the anonymous referee for an extremely useful referee report.

##  1\. Terminology, notation, and background on property (T)

###  1.1. Terminology and notation

We largely follow [5]. Consider the Calkin algebra
$\mathcal{Q}(H):=\mathcal{B}(H)/\mathcal{K}(H)$. For a non-unital
$\mathrm{C}^{*}$-algebra $A$ consider the multiplier algebra $\mathcal{M}(A)$
and the corona $\mathcal{Q}(A):=\mathcal{M}(A)/A$ (also called outer
multiplier algebra). Thus $\mathcal{Q}(H)$ is the same as
$\mathcal{Q}(\mathcal{K})$, where $\mathcal{K}$ is the algebra of compact
operators on a separable, infinite-dimensional Hilbert space, alternatively
denoted $\mathcal{K}(H)$, as needed. In all of these situations we let
$\pi\colon\mathcal{M}(A)\to\mathcal{Q}(A)$ denote the quotient map.

By $F\Subset X$ we denote the fact that $F$ is a finite subset of a set $X$.

If $A$ is a $\mathrm{C}^{*}$-algebra or a von Neumann algebra, by $Z(A)$ we
denote its center.

###  1.2. Strongly self-absorbing $\mathrm{C}^{*}$-algebras

A unital $\mathrm{C}^{*}$-algebra $D$ is said to be _strongly self-absorbing_
if $D$ is isomorphic to $D\otimes D$ via an isomorphism approximately
unitarily equivalent to the ∗-homomorphism $d\mapsto d\otimes 1_{D}$ and $D$
is not isomorphic to ${\mathbb{C}}$. All strongly self-absorbing
$\mathrm{C}^{*}$-algebras are nuclear and simple, and they satisfy $D\cong
D^{\bigotimes{\mathbb{N}}}$ (see [49]). Known strongly self-absorbing
$\mathrm{C}^{*}$-algebras are the Jiang–Su algebra ${\mathcal{Z}}$, UHF
algebras of infinite type, $\mathcal{O}_{\infty}$, tensor products of UHF
algebras of infinite type with $\mathcal{O}_{\infty}$, and $\mathcal{O}_{2}$.
If all nuclear $\mathrm{C}^{*}$-algebras satisfy the Universal Coefficient
Theorem, UCT (see e.g., [4]) then there are no other separable strongly self-
absorbing $\mathrm{C}^{*}$-algebras (this is Corollary 6.7 of [48]). There
exist exact $\mathrm{C}^{*}$-algebras without the UCT ([47]) but the question
whether there are nuclear examples, or even strongly self-absorbing ones, is
wide open.

###  1.3. Kazhdan’s property (T)

This subsection and §1.4 contain more background than the experts may want to
see. A discrete group $\Gamma$ has _Kazhdan’s property (T)_ if there are
$F\Subset\Gamma$ and $\varepsilon>0$ such that for every unitary
representation $\rho$ of $\Gamma$ on a Hilbert space $H$, if there is a vector
$\xi$ in $H$ such that333When dealing only with a Hilbert space $H$, its norm
will be denoted by $\|\cdot\|$. Later on, when we consider the Hilbert space
of Hilbert–Schmidt operators, we will use $\|\cdot\|$ for the operator norm on
this space and (variously embellished variations on) $\|\cdot\|_{\textrm{HS}}$
for the Hilbert–Schmidt norm.

(1.1) $\max_{g\in F}\|\rho(g)\xi-\xi\|<\varepsilon\|\xi\|.$

for all $g\in F$, (such $\xi\in H$ is called _$F,\varepsilon$ -invariant_),
then there is a unit vector in $H$ that is invariant for $\rho$. The pair
$F,\varepsilon$ is called a _Kazhdan pair_ for $\Gamma$. Suppose $\rho$ is an
irreducible representation of $\Gamma$. Then $\rho$ extends to a
representation of the full group $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\Gamma)$ (see [10] for information on group
$\mathrm{C}^{*}$-algebras). Let $p_{\rho}$ denote the central cover of $\rho$
([10, Definition 1.4.2]) in the second dual, $\mathrm{C}^{*}(\Gamma)^{**}$. If
$\Gamma$ has Kazhdan’s property (T) then $p_{\rho}$ belongs to
$\mathrm{C}^{*}(\Gamma)$ ([10, Definition 17.2.3 and Theorem 17.2.4]; as
pointed out in [10] in the paragraph preceding Definition 17.2.3, this is a
consequence of Schur’s Lemma for property (T) groups, a relative to Lemma 1.2
below). In this case $p_{\rho}$ is called the _Kazhdan projection_ of $\rho$.
Kazhdan projections associated to inequivalent irreducible representations are
orthogonal. See [2], [10] for more on property (T) groups. For more
information on Kazhdan projections see [10, §17], also [31, 3.7.6].

###  1.4. Schur’s Lemma for property (T) groups

This subsection contains a rehashing of ([31, Lemma 3.7.8]), Lemma 1.2, which
is included for convenience of readers not familiar with property (T) groups.

Suppose that $\rho$ is an action of a discrete group $\Gamma$ on a Hilbert
space $H$. For us it will be difficult to keep track on the norm of $\xi$, and
we therefore say that $\xi\in H$ is _scaled
$F,\varepsilon$-invariant_444Arguably, a better terminology would be non-
scaled $F,\varepsilon$-invariant but this sounds rather excessive and after
all the scaling can go either way. if

(1.2) $\max_{g\in F}\|\rho(g)\xi-\xi\|<\varepsilon.$

Clearly every $\xi$ is scaled $F,\varepsilon$ invariant for any $F$ and
$\varepsilon>2\|\xi\|$, but in this case the estimate given in Lemma 1.1 below
is vacuous.

Let $q^{\rho}$ denote the orthogonal projection to the space of invariant
vectors,

$H^{\rho}:=\\{\eta\in H\mid\rho(g)\eta=\eta\text{ for all }g\in\Gamma\\}.$

The following is [10, Proposition 12.1.6] (with $\Gamma=\Lambda$) or [2,
Proposition 1.1.9] (the case when $F$ is compact, and modulo rescaling).

######  Lemma 1.1.

Suppose $\Gamma$ is a group with Kazhdan’s property (T) and $F,\varepsilon$ is
a Kazhdan pair for $\Gamma$. If $\Gamma$ acts on a Hilbert space $H$ then
every scaled $F,\delta\varepsilon$-invariant vector $\xi$ satisfies
$\|\xi-q^{\rho}\xi\|<\delta$. ∎

Suppose that $K$ and $H$ are Hilbert spaces. If $H$ is finite-dimensional then
let $\tau_{H}$ denote the normalized tracial state on $\mathcal{B}(H)$ and
consider the normalized Hilbert—Schmidt norm on $\mathcal{B}(K,H)$,

$\|T\|_{\textrm{HS},H}:=\tau_{H}(TT^{*})^{1/2}.$

This differs from the non-normalized Hilbert–Schmidt norm usually used in this
context ([31], [10]) by the scaling factor $\dim(H)^{-1}$.

Similarly, if $K$ is finite-dimensional let $\tau_{K}$ denote the normalized
tracial state on $\mathcal{B}(K)$ and consider the normalized Hilbert—Schmidt
norm on $\mathcal{B}(K,H)$,

$\|T\|_{\textrm{HS},K}:=\tau_{K}(TT^{*})^{1/2}.$

Note also that we allow for the possibility that only one of $H$ and $K$ is
finite-dimensional. In this case the normalized Hilbert–Schmidt norm
associated with the infinite-dimensional Hilbert space is of course not
defined. We will encounter this situation later on.

It is a good moment to get a few easy facts out of the way. If
$T\in\mathcal{B}(K,H)$ then
$\max(\|T\|_{\textrm{HS},K},\|T\|_{\textrm{HS},H})\leq\|T\|$ (the right-hand
side being the operator norm). Also, if $\dim K\leq\dim H$, then
$\|T\|_{\textrm{HS},K}\geq\|T\|_{\textrm{HS},H}$, more precisely

$\|T\|_{\textrm{HS},K}=\frac{\dim(H)}{\dim(K)}\|T\|_{\textrm{HS},H}.$

We will contract nested subscripts and write $\|\cdot\|_{\textrm{HS},n}$ for
both $\|\cdot\|_{\textrm{HS},H_{n}}$ and $\|\cdot\|_{\textrm{HS},K_{n}}$,
similarly $\tau_{n}$ for both $\tau_{H_{n}}$ and $\tau_{K_{n}}$ (spaces
$H_{n}$ and $K_{n}$ will never appear in the same context).

Lemma 1.2 below is a minor variation on Schur’s Lemma for property (T) groups
([31, Lemma 3.7.8]).

######  Lemma 1.2.

Suppose $\Gamma$ is a property (T) group with Kazhdan pair $F,\varepsilon$ and
$\rho_{j}$ is a representation of $\Gamma$ on $K_{j}$ for $j=1,2$ such that
$K_{1}$ is finite-dimensional and some $T\in\mathcal{B}(K_{1},K_{2})$
satisfies

$\max_{g\in F}\|T\rho_{1}(g)-\rho_{2}(g)T\|<\varepsilon\delta.$

  1. (1)

If $K_{1}=K_{2}$ and $\rho_{1}=\rho_{2}$ is irreducible, then
$\|T-\tau_{1}(T)\|_{\textrm{HS},1}<\delta$.

  2. (2)

If $\rho_{1}$, $\rho_{2}$ have no isomorphic subrepresentations, then
$\|T\|_{\textrm{HS},1}<\delta$.

###### Proof.

Since $\|\cdot\|_{\textrm{HS},1}\leq\|\cdot\|$, we have

(1.3) $\max_{g\in
F}\|T\rho_{1}(g)-\rho_{2}(g)T\|_{\textrm{HS},1}<\varepsilon\delta$

and we will show that this inequality implies both conclusions. Define an
action $\sigma=\sigma_{\rho_{1},\rho_{2}}$ of $\Gamma$ on
$\mathcal{B}(K_{1},K_{2})$ by letting, for $T\in\mathcal{B}(K_{1},K_{2})$,

$\sigma(g)(T):=\rho_{2}(g)T\rho_{1}(g)^{*}.$

A calculation shows that for all $g$ and $h$ in $\Gamma$ we have
$\sigma(g)\in\mathcal{B}(K_{1},K_{2})$ and
$\sigma(gh)=\sigma(g)\circ\sigma(h)$. Thus $\sigma$ is a representation of
$\Gamma$ on $\mathcal{B}(K_{1},K_{2})$. Consider the space
$\mathcal{B}(K_{1},K_{2})^{\sigma}$ of vectors invariant for $\sigma$, and let
$Q$ be the projection onto this space.

Since $F,\varepsilon$ is a Kazhdan pair for $\Gamma$, by (1.3) and Lemma 1.1
we have

(1.4) $\|T-Q(T)\|_{\textrm{HS},1}<\delta.$

Now consider the two specific cases from the statement of the lemma.

(1) If $\rho_{1}=\rho_{2}$ and it is irreducible, then (identifying scalars
with scalar matrices), Schur’s Lemma ([31, Lemma 3.7.7]) implies
$\mathcal{B}(K_{1},K_{1})^{\sigma}={\mathbb{C}}$ and $Q(T)=\tau_{1}(T)$.
Therefore (1.4) reduces to $\|T-\tau_{1}(T)\|_{\textrm{HS},1}<\delta$.

(2) Suppose for a moment that $\rho_{1}$ is irreducible and not isomorphic to
a subrepresentation of $\rho_{2}$. Then Schur’s Lemma ([31, Lemma 3.7.7])
implies that there are no nontrivial intertwiners for $\rho_{1}$ and
$\rho_{2}$, in other words $\mathcal{B}(K_{1},K_{2})^{\sigma}=\\{0\\}$. Hence
$Q(T)=0$ and (1.4) reduces to $\|T\|_{\textrm{HS},1}<\delta$.

Suppose $\rho_{1}$ is not irreducible. Since $K_{1}$ is finite-dimensional,
$\rho_{1}$ is a direct sum of irreducible representations. By applying the
previous paragraph to each one of these irreducible representations, we have
$\mathcal{B}(K_{1},K_{2})^{\sigma}=\\{0\\}$ and $Q(T)=0$ and therefore
$\|T\|_{\textrm{HS},1}<\delta$. ∎

##  2\. Wassermann’s $\mathrm{C}^{*}$-algebra ${\mathcal{W}}(\Gamma)$

Suppose $\Gamma$ is a property (T) group with infinitely many inequivalent
irreducible representations on finite-dimensional Hilbert spaces (e.g., a
residually finite property (T) group such as
$\operatorname{SL}_{3}({\mathbb{Z}})$). Fix a Kazhdan pair, $F\Subset\Gamma$
and $\varepsilon>0$. Let $\rho_{n}$, for $n\in{\mathbb{N}}$, be an enumeration
of irreducible representations of $\Gamma$ on finite-dimensional Hilbert
spaces. Since there are only finitely many inequivalent representations of
$\Gamma$ on every fixed finite-dimensional Hilbert space $H$ ([54, Corollary
3]), we may choose the enumeration so that $\dim(H_{n})\leq\dim(H_{n+1})$ for
all $n$ and $\dim(H_{n})\to\infty$ as $n\to\infty$. For each $n$ let $p_{n}$
be the Kazhdan projection associated with $\rho_{n}$ in
$\mathrm{C}^{*}(\Gamma)$. Each representation $\rho_{n}$ of $\Gamma$ on
$H_{n}$ uniquely extends to a representation of the full group algebra
$\mathrm{C}^{*}(\Gamma)$ on the same space. We will slightly abuse the
notation and denote the latter by $\rho_{n}$. Let
$\rho=\bigoplus_{n}\rho_{n}$, and $H:=\bigoplus_{n}H_{n}$. Then $H$ is an
infinite-dimensional, separable Hilbert space and
$\rho\colon\mathrm{C}^{*}(\Gamma)\to\mathcal{B}(H)$ is a unital ∗-homomorphism
such that $\rho^{-1}(\mathcal{K}(H))$ includes all $p_{n}$.

######  Definition 2.1.

If $\Gamma$ is a property (T) group with infinitely many irreducible
representations $\rho_{n}\colon\Gamma\to\mathcal{B}(H_{n})$, for
$n\in{\mathbb{N}}$, on finite-dimensional Hilbert spaces and
$\rho=\bigoplus_{n}\rho_{n}$ is considered as a representation of the full
group algebra $\mathrm{C}^{*}(\Gamma)$ as in the previous paragraph, then with
$\pi\colon\mathcal{B}(H)\to\mathcal{Q}(H)$ denoting the quotient map let

${\mathcal{W}}(\Gamma):=\pi\circ\rho[\mathrm{C}^{*}(\Gamma)].$

With $p_{n}$ denoting the Kazhdan projection associated with $\rho_{n}$, let

$\mathcal{D}:=\\{T\in\mathcal{B}(H)\mid p_{m}Tp_{n}\neq 0\text{ implies
}m=n\\}.$

In other words,
$\mathcal{D}=\prod_{n}p_{n}\mathcal{B}(H)p_{n}=\prod_{n}\mathcal{B}(H_{n})$.

Had we chosen an orthonormal basis of $H$ whose subsets span the $H_{n}$’s,
then $\mathcal{D}$ would be a special case of a von Neumann algebra of the
form $\mathcal{D}[{\mathbf{E}}]$ studied in [22, §9.7.1]. These von Neumann
algebras are of central importance both in author’s construction of outer
automorphisms of $\mathcal{Q}(H)$ ([22, §17.1]—the original construction of
outer automorphisms given in [44] uses separable $\mathrm{C}^{*}$-subalgebras
of $\mathcal{Q}(H)$ instead, and one of the key lemmas used in this proof
plays a role in the proof of Corollary 5.1 below) and in the proof that all
automorphisms of $\mathcal{Q}(H)$ are inner ([21], [22, §17.2–17.8]).

Note that the center $Z(\mathcal{D})$ of $\mathcal{D}$ is equal to

(2.1) $\textstyle
Z(\mathcal{D})=\\{\sum_{n}\lambda_{n}p_{n}\mid(\lambda_{n})\in\ell_{\infty}\\}$

and that
$\pi[Z(\mathcal{D})]\subseteq\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$
because each $p_{n}$ is invariant for $\rho(T)$ for all
$T\in\mathrm{C}^{*}(\Gamma)$.

Since for a scalar $\lambda$ we have $\|\lambda p_{n}\|=|\lambda|=\|\lambda
p_{n}\|_{\textrm{HS},n}$,555A clarification for the readers who skipped §1.4:
this is the normalized Hilbert–Schmidt norm. every $T\in Z(\mathcal{D})$
satisfies

$\|T\|=\sup_{n}\|Tp_{n}\|_{\textrm{HS},n}.$

The following convention, going back at least to [55], will considerably
simplify the notation, and it will not lead to confusion if the reader keeps
it in mind.

######  Convention 2.2.

In Proposition 2.3 and elsewhere we denote the elements of the Calkin algebra
by dotted letters, with understanding that if $\dot{a}\in\mathcal{Q}(H)$ then
$a\in\mathcal{B}(H)$ is a lift of $\dot{a}$.

######  Proposition 2.3.

Suppose that $\Gamma$, ${\mathcal{W}}(\Gamma)$, and $\mathcal{D}$ are as in
Definition 2.1. Then, with $A:=\pi[Z(\mathcal{D})]$, there is a ∗-homomorphism

$\Psi\colon\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}\to A$

whose restriction to $A$ is the identity map. Thus, if $I:=\ker(\Psi)$ then
the short exact sequence

$0\to
I\to\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}\buildrel\Psi\over{\to}A\to
0$

splits.

Prior to proving this proposition, we show that Theorem B is its immediate
consequence.

###### Proof of Theorem B.

Let $\Gamma$ and ${\mathcal{W}}(\Gamma)$ be as in the first paragraph of §2.
We will prove that $B:={\mathcal{W}}(\Gamma)$ is as required. Let $C$ be a
unital and simple $\mathrm{C}^{*}$-algebra, and suppose that $\Phi$ is a
unital ∗-homomorphism from $C$ into
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$. If $\Psi$ is as
constructed in Proposition 2.3, then (since $C$ is simple), $\Psi\circ\Phi$ is
a unital, and therefore injective, *-homomorphism from $C$ into the range of
$\Psi$. However, the range of $\Psi$ is abelian, and therefore
$C\cong{\mathbb{C}}$. ∎

The proof of Proposition 2.3 commences with a sequence of statements, some of
them self-explanatory, phrased as a proposition.

######  Proposition 2.4.

Suppose that $\Gamma$, $\rho\colon\mathrm{C}^{*}(\Gamma)\to\mathcal{B}(H)$,
${\mathcal{W}}(\Gamma)$, and $\mathcal{D}$ are as in Definition 2.1. Then we
have the following.

  1. (1)

$\rho[\mathrm{C}^{*}(\Gamma)]^{\prime\prime}=\mathcal{D}$.

  2. (2)

$\rho[\mathrm{C}^{*}(\Gamma)]^{\prime}=\mathcal{D}’=\mathrm{W}^{*}(p_{n}:n\in{\mathbb{N}})=Z(\mathcal{D})$.

  3. (3)

A conditional expectation $E\colon\mathcal{B}(H)\to\mathcal{D}^{\prime}$ is
defined by ($\tau_{n}$ is the normalized trace on $\mathcal{B}(H_{n})$)

(2.2) $E(T)=\sum_{n}\tau_{n}(p_{n}Tp_{n})p_{n}.$

  4. (4)

$\pi[\mathcal{D}^{\prime}]=\mathcal{Q}(H)\cap\pi[\mathcal{D}]^{\prime}\subseteq\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$.

  5. (5)

There is a conditional expectation
$\dot{E}\colon\mathcal{Q}(H)\to\mathcal{Q}(H)\cap\pi[\mathcal{D}]^{\prime}$
such that $\dot{E}(\dot{T})=\pi(E(T))$ for all $T\in\mathcal{B}(H)$.

  6. (6)

Every $T$ such that
$\dot{T}\in\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$ satisfies

$\displaystyle\lim_{n}\|Tp_{n}-\tau_{n}(p_{n}Tp_{n})p_{n}\|_{\textrm{HS},n}$
$\displaystyle=0,$ $\displaystyle\lim_{n}\|Tp_{n}-E(T)p_{n}\|_{\textrm{HS},n}$
$\displaystyle=0,\text{ and}$
$\displaystyle\lim_{n}\|p_{n}T-E(T)p_{n}\|_{\textrm{HS},n}$ $\displaystyle=0.$

###### Proof.

(1) For every $n$, $\rho_{n}$ is irreducible and $H_{n}$ is finite-
dimensional, and therefore
$p_{n}\rho[\mathrm{C}^{*}(\Gamma)]p_{n}=\rho_{n}[\mathrm{C}^{*}(\Gamma)]=\mathcal{B}(H_{n})$
for all $n$. Since the $\rho_{n}$’s are pairwise inequivalent, there are no
nontrivial intertwiners and by a consequence of the Glimm–Kadison transitivity
theorem ([22, Proposition 3.5.2]) we have

$\rho[\mathrm{C}^{*}(\Gamma)]^{\prime\prime}=\prod_{n}\mathcal{B}(H_{n})=\mathcal{D}.$

(2) By (1),
$\rho[\mathrm{C}^{*}(\Gamma)]^{\prime}=\mathcal{D}^{\prime}=\mathrm{W}^{*}(p_{n}:n\in{\mathbb{N}})$
and $Z(\mathcal{D})=\mathcal{D}\cap\mathcal{D}’=\mathcal{D}’$.

(3) Follows from (2), since $\mathcal{D}=\prod_{n}\mathcal{B}(H_{n})$.

(4) We first prove that
$\pi[\mathcal{D}^{\prime}]=\mathcal{Q}(H)\cap\pi[\mathcal{D}]^{\prime}$
follows from the results in [32]. Let $H$ be a Hilbert space and let
$\mathcal{D}$ be a von Neumann algebra acting on $H$. Property $P_{1}$ ([32,
p. 39]) asserts that if $b\in\mathcal{B}(H)$ and $ab-ba\in\mathcal{K}(H)$ for
all $a\in\mathcal{D}$, then $b\in\mathcal{D}+\mathcal{K}(H)$. We need to
verify that $\mathcal{D}$ has property $P_{1}$.

For each $n$, $p_{n}$ belongs to the center of $\mathcal{D}$ and
$\mathcal{D}p_{n}=\mathcal{B}(H_{n})$ is a finite type I factor. By [32, Lemma
3.1], every derivation of $\mathcal{D}p_{n}$ into $\mathcal{K}(H)$ is inner,
and (in the terminology of [32, p. 39]), $\mathcal{D}p_{n}$ has property
$P_{2}$. [32, Theorem 3.4] now implies that every derivation of $\mathcal{D}$
is inner, and by [32, Lemma 1.4] $\mathcal{D}$ has property $P_{1}$ as well,
as required.

We proceed to prove
$\pi[\mathcal{D}^{\prime}]\subseteq\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$.
Since $\mathcal{D}=\rho[\mathrm{C}^{*}(\Gamma)]^{\prime\prime}$, we have
$\pi[\mathcal{D}]\supseteq\pi[\mathrm{C}^{*}(\Gamma)]={\mathcal{W}}(\Gamma)$.
Therefore
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}\supseteq\mathcal{Q}(H)\cap\pi[\mathcal{D}]^{\prime}$.

(5) Since $E$ as defined in (2.2) sends compact operators to compact
operators, it defines a conditional expectation
$\dot{E}\colon\mathcal{Q}(H)\to\mathcal{Q}(H)\cap\pi[\mathcal{D}]^{\prime}$ by
$\dot{E}(\dot{T})=\pi(E(T))$, as required.

(6) For $Y\subseteq{\mathbb{N}}$ it will be convenient to write

$H_{Y}:=\bigoplus_{m\in Y}H_{m}$

and let $\rho_{Y}:=\bigoplus_{m\in Y}\rho_{m}$, so that $\rho_{Y}$ is a
representation of $\mathrm{C}^{*}(\Gamma)$ on $H_{Y}$. Finally let (the right-
hand side is strongly convergent)

$p_{Y}:=\sum_{m\in Y}p_{m}.$

Fix $T\in\mathcal{B}(H)$ such that
$\dot{T}\in\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$. Then
$\rho(g)T-T\rho(g)$ is compact for all $g\in\Gamma$. We can therefore fix
$0<\delta<1/5$ and $n_{0}$ such that for all $g\in F$ we have

$\textstyle\|(\rho(g)T-T\rho(g))(1-\sum_{m<n_{0}}p_{m})\|<\delta\varepsilon$

and this implies

(2.3) $\|p_{X}(\rho(g)T\rho(g)^{*}-T)p_{Y}\|<\delta\varepsilon$

for all $g\in F$ and all $X$ and $Y$ included in $[n_{0},\infty)$. Write
$T\sim_{\varepsilon}S$ for $\|T-S\|<\varepsilon$ (this is the operator norm).
By (2.3), every $Y\subseteq[n_{0},\infty)$ and every $n\geq n_{0}$ satisfy

(2.4)
$\rho_{Y}(g)p_{Y}Tp_{n}\rho_{n}(g)^{*}=p_{Y}\rho(g)T\rho(g)^{*}p_{n}\sim_{\delta\varepsilon}p_{Y}Tp_{n}.$

Since all $\rho_{j}$ are inequivalent and irreducible, if $n\notin Y$ then
$\rho_{n}$ and $\rho_{Y}$ have no isomorphic subrepresentations. By (2.4) and
Lemma 1.2 (2),

(2.5) $\|p_{Y}Tp_{n}\|_{\textrm{HS},n}<\delta.$

On the other hand, since $p_{[0,n_{0})}T$ has finite rank, there is
$n_{1}>n_{0}$ such that all $n\geq n_{1}$ satisfy

(2.6) $\|p_{[0,n_{0})}Tp_{n}\|_{\textrm{HS},n}<\delta.$

By (2.6) and (2.5) applied to $Y:=[n_{1},\infty)\setminus\\{n\\}$, every
$n\geq n_{1}$ satisfies
$\|p_{{\mathbb{N}}\setminus\\{n\\}}Tp_{n}\|_{\textrm{HS},n}<2\delta$.
Equivalently,

(2.7) $\|Tp_{n}-p_{n}Tp_{n}\|_{\textrm{HS},n}<2\delta.$

On the other hand, since $\rho_{n}$ is irreducible, Lemma 1.2 (1) implies

$\|p_{n}Tp_{n}-\tau_{n}(p_{n}Tp_{n})p_{n}\|_{\textrm{HS},n}<\delta$

which together with (2.7) implies

(2.8) $\|Tp_{n}-\tau_{n}(p_{n}Tp_{n})p_{n}\|_{\textrm{HS},n}<3\delta.$

Since $\delta>0$ was arbitrary and (2.8) holds for every large enough $n$,
this completes the proof of
$\lim_{n}\|Tp_{n}-\tau_{n}(p_{n}Tp_{n})p_{n}\|_{\textrm{HS},n}=0$, which is
the first equation in (6). The second equation,
$\lim_{n}\|Tp_{n}-E(T)p_{n}\|_{\textrm{HS},n}=0$, follows since
$E(T)=\sum_{n}\tau_{n}(p_{n}Tp_{n})p_{n}$. The third equation is obtained by
replacing $T$ with $T^{*}$ in the second. ∎

###### Proof of Proposition 2.3.

With $\Gamma$, ${\mathcal{W}}(\Gamma)$ as in Definition 2.1, let $\Psi$ be the
restriction of $\dot{E}$ as in Proposition 2.4 (5) to
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$ and let

$A:=\pi[Z(\mathcal{D})].$

Since (2) of Proposition 2.4 implies that
$Z(\mathcal{D})=\mathcal{D}^{\prime}=\mathrm{W}^{*}(p_{n}:n\in{\mathbb{N}})$,
it is isomorphic to $\ell_{\infty}({\mathbb{N}})$, hence $A$ is isomorphic to
$\ell_{\infty}({\mathbb{N}})/c_{0}({\mathbb{N}})$. In particular, $A$ is
abelian, and, by (4) of Proposition 2.4, included in
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$.

Fix $T\in\mathcal{B}(H)$ such that $\dot{T}$ is in the domain of $\Psi$. Being
a restriction of $\dot{E}$, $\Psi$ satisfies

$\textstyle\Psi(\dot{T}):=\pi(\sum_{n}\tau_{n}(p_{n}Tp_{n})p_{n}).$

It remains to prove that $\Psi$ is a ∗-homomorphism. Since it is a conditional
expectation, it suffices to check that it is multiplicative. This will be
proven by multiple uses of Proposition 2.4 (6).

If $\dot{S}$ and $\dot{T}$ belong to
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$, then for every $n$ and
scalars $\eta$ and $\lambda$ we have (using
$\|ab\|_{\textrm{HS},n}\leq\|a\|\|b\|_{\textrm{HS},n}$)

$\displaystyle\|STp_{n}-\eta\lambda p_{n}\|_{\textrm{HS},n}$
$\displaystyle\leq\|STp_{n}-S\lambda p_{n}\|_{\textrm{HS},n}+\|S\lambda
p_{n}-\eta\lambda p_{n}\|_{\textrm{HS},n}$
$\displaystyle\leq\|S\|\|Tp_{n}-\lambda
p_{n}\|_{\textrm{HS},n}+|\lambda|\|Sp_{n}-\eta p_{n}\|_{\textrm{HS},n}.$

Thus if $E(T)=\sum_{n}\lambda_{n}p_{n}$ and $E(S)=\sum_{n}\eta_{n}p_{n}$, then
the previous formula together with the first equation of Proposition 2.4 (6)
applied to $S$ and $T$ implies
$\|STp_{n}-\eta_{n}\lambda_{n}p_{n}\|_{\textrm{HS},n}\to 0$, and the same
equation applied to $ST$ implies
$\lim_{n}\|STp_{n}-\tau_{n}(p_{n}STp_{n})p_{n}\|_{\textrm{HS},n}=0$. Therefore

$\lim_{n}\tau_{n}(p_{n}STp_{n})-\tau_{n}(p_{n}Sp_{n})\tau_{n}(p_{n}Tp_{n})=0.$

After passing to the quotient, this gives
$\dot{E}(\dot{S}\dot{T})=\pi(\sum_{n}\eta_{n}\lambda_{n}p_{n})=\dot{E}(\dot{S})\dot{E}(\dot{T})$,
as required.

Thus $\Psi$ is multiplicative, hence it is a ∗-homomorphism whose restriction
to its range $A$ is the identity map. Thus, if $I:=\ker(\Psi)$ then the short
exact sequence

$0\to
I\to\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}\buildrel\Psi\over{\to}A\to
0$

splits with $A\ni a\mapsto a+I$ being the right inverse of $\Psi$. ∎

Although it is not used elsewhere in the present paper, I find it that the
following proposition provides sufficient insight into the structure of the
relative commutant of ${\mathcal{W}}(\Gamma)$ in $\mathcal{Q}(H)$ and the
structure of ${\rm Ext}({\mathcal{W}}(\Gamma))$ to merit its inclusion.

######  Proposition 2.5.

Suppose that $\Gamma$, ${\mathcal{W}}(\Gamma)$, and $p_{n}$, for
$n\in{\mathbb{N}}$, are as in Definition 2.1.

  1. (1)

For every projection $p\in\mathcal{B}(H)$ such that
$\dot{p}\in\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$ there is
$X=X_{p}\subseteq{\mathbb{N}}$ which satisfies (considering $pp_{n}$ as an
element of $\mathcal{B}(H_{n},H)$)

$\lim_{n\in X,\\\ n\to\infty}\|pp_{n}\|_{\textrm{HS},n}=1\text{ and
}\lim_{n\notin X,\\\ n\to\infty}\|pp_{n}\|_{\textrm{HS},n}=0.$

  2. (2)

If $\dot{p}$ and $\dot{q}$ are Murray–von Neumann equivalent projections
belonging to $\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$, then the
symmetric difference $X_{p}\Delta X_{q}$ is finite.

###### Proof.

With the ∗-homomorphism
$\Psi\colon\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}\to\pi[Z(\mathcal{D})]$
as in Proposition 2.3, $\Psi(\dot{p})$ is a projection in
$\pi[Z(\mathcal{D})]$. Since $Z(\mathcal{D})$, being a von Neumann algebra,
has real rank zero, by [22, Lemma 3.1.13] $\Psi(\dot{p})$ can be lifted to a
projection $q$ in $Z(\mathcal{D})$. This projection is of the form
$q=\sum_{n\in X_{p}}p_{n}$ for some $X_{p}\subseteq{\mathbb{N}}$. Writing
$\tau_{n}$ for the normalized trace on $\mathcal{B}(H_{n})$ as in the proof of
Proposition 2.3, we have
$\dot{E}(\dot{p})=\dot{E}(\dot{q})=\pi(\sum_{n}\tau_{n}(p_{n}pp_{n})p_{n})$.
Since $E(p)-\sum_{n\in X_{p}}p_{n}$ is compact, we have666If one of $X_{p}$ or
${\mathbb{N}}\setminus X_{p}$ is finite, then only one of the two displayed
limits is well-defined and relevant, and we ignore the other one.

$\lim_{n\to\infty,n\in X_{p}}\tau_{n}(p_{n}pp_{n})=1\text{ and
}\lim_{n\to\infty,n\notin X_{p}}\tau_{n}(p_{n}pp_{n})=0.$

Since $\|pp_{n}\|_{\textrm{HS},n}^{2}=\tau_{n}(p_{n}pp_{n})$ the set $X_{p}$
is as required.

(2) Suppose that $\dot{p}$ and $\dot{q}$ are Murray–von Neumann equivalent
projections in $\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$ and let
$\dot{v}$ be a partial isometry in
$\mathcal{Q}(H)\cap{\mathcal{W}}(\Gamma)^{\prime}$ such that
$\dot{v}^{*}\dot{v}=\dot{v}\dot{v}^{*}$.

Since the range of $\Psi$ as provided by Proposition 2.3 is abelian, we have
$\Psi(\dot{p})=\Psi(\dot{v}^{*}\dot{v})=\Psi(\dot{v}\dot{v}^{*})=\Psi(\dot{q})$
and therefore $X_{p}$ and $X_{q}$ correspond to the same projection in
$\pi[\mathcal{D}]$. This is equivalent to $X_{p}\Delta X_{q}$ being finite. ∎

Perhaps it should be noted that the converse of Proposition 2.5 (2) is not
true. For example, if $\dot{p}$ corresponds to the trivial extension of
${\mathcal{W}}(\Gamma)$ then $X_{p}$ is finite, but $\dot{p}$ is not equal to
$0$.

##  3\. $D$-saturation of the path algebra $A_{\text{\faForward}}$

The main result of this section, Theorem 3.1, is a warmup for Theorem 4.1 that
will be used in the proof of Theorem A. Theorem 3.1 will not be used in this
proof and readers not interested in path algebras (as well as those not
requiring a warmup) can proceed directly to §4.

Suppose that $D$ is a separable and unital $\mathrm{C}^{*}$-algebra. Following
[25], we say that a (nonseparable) unital $\mathrm{C}^{*}$-algebra $C$ is _$D$
-saturated_ if for every separable $\mathrm{C}^{*}$-subalgebra of $C$ there is
an injective, unital ∗-homomorphism from $D$ into $C\cap A^{\prime}$. Prime
examples of $D$-saturated algebras are the ultrapower $A_{\mathcal{U}}$ and an
asymptotic sequence algebra $\ell_{\infty}(A)/c_{0}(A)$ of a tensorially
$D$-absorbing $\mathrm{C}^{*}$-algebra $A$ (see [17], [46, Theorem 7.2.2], and
[25]).

For a $\mathrm{C}^{*}$-algebra $A$ consider the path algebra

$A_{\text{\faForward}}:=C_{b}([0,\infty),A)/C_{0}([0,\infty),A).$

######  Theorem 3.1.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable, unital, and tensorially
$D$-absorbing then $A_{\text{\faForward}}$ is $D$-saturated.

###### Proof.

Since $D\cong D^{\bigotimes{\mathbb{N}}}$, we can identify $A$ with $A\otimes
D^{\otimes{\mathbb{N}}}$. Letting $D_{j}$ denote the $j$-th copy777We adopt
the convention that $0\in{\mathbb{N}}$. of $D$ in
$D^{\bigotimes{\mathbb{N}}}$, we have that $A$ is the inductive limit of
$\mathrm{C}^{*}$-algebras $A_{0}\cong A$ and
$A_{n}:=A\otimes\bigotimes_{j<n}D_{j}$, for $n\geq 1$. Let $\Theta_{n}\colon
D\to A$ be the embedding that identifies $D$ with $D_{n}$, so that

$\Theta_{n}(d):=1_{A}\otimes\bigotimes_{j<n}1_{D_{j}}\otimes
d\otimes\bigotimes_{j>n}1_{D_{j}}.$

By [56, Remark 3.3] $D$ is $K_{1}$-injective hence [14, Theorem 2.2] implies
that the flip automorphism of $D\otimes D$ (the one that interchanges
$a\otimes b$ with $b\otimes a$) is _strongly asymptotically equivalent to the
identity_. This means that there is a continuous path of unitaries $u_{t}$,
for $0\leq t<1$, in $D\otimes D$ such that $u_{0}=1$ and $\lim_{t\to
1}u_{t}(a\otimes b)u_{t}^{*}=b\otimes a$. Fix $m<n$ and identify $D_{m}\otimes
D_{n}$ with $D\otimes D$. Then the previous sentence gives a continuous path
of unitaries $w^{m,n}_{t}$, for $0\leq t<1$, in $A$ such that $w^{m,n}_{0}=1$
and for all $d\in D$ we have

$\lim_{t\to 1}w^{m,n}_{t}\Theta_{m}(d)(w^{m,n}_{t})^{*}=\Theta_{n}(d).$

In addition, $w^{m,n}_{t}\in A\cap A_{m}^{\prime}$ for all $t$. If $m=n$ then
for definiteness let $w^{m,n}_{t}=1$ for all $t$.

We are now ready to prove that $A_{\text{\faForward}}$ is $D$-saturated. Fix a
separable $\mathrm{C}^{*}$-subalgebra $B$ of $A_{\text{\faForward}}$, and let
$\\{a_{n}\mid n\in{\mathbb{N}}\\}$, be a subset of $C_{b}([0,\infty),A)$ such
that (with $\pi$ denoting the quotient map) $\pi(a_{n})$, for
$n\in{\mathbb{N}}$, is dense in $B$. For each $m$ and $\varepsilon>0$ define
$f_{m,\varepsilon}\colon[0,\infty)\to{\mathbb{N}}$ by

$f_{m,\varepsilon}(t)=\min\\{j\mid\operatorname{dist}(a_{m}(t),A_{j})<\varepsilon\\}.$

This function is clearly upper semicontinuous.

If $d\in D$, $\|d\|\leq 1$, $n>f_{m,\varepsilon}(t)$, and $b\in A_{n-1}$ is
such that $\|b-a_{m}(t)\|<\varepsilon$, we then have

(3.1) $\|[\Theta_{n}(d),a_{m}(t)]\|=\|[\Theta_{n}(d),a_{m}(t)-b]\|\leq
2\|d\|\|a_{m}(t)-b\|<2\varepsilon.$

For all $n\in{\mathbb{N}}$, the value

$g(n):=\max\\{f_{m,1/(n+1)}(t)\mid m\leq n+1\text{ and }t\leq n+1\\}$

is finite by upper semicontinuity of $f_{m,\varepsilon}$ and compactness of
$[0,n+1]$. Clearly $g$ is nondecreasing.

Define $\Theta\colon D\to C_{b}([0,\infty),A)$ by first letting, for
$n\in{\mathbb{N}}$,

$\Theta(d)(n):=\Theta_{g(n)}(d).$

For $n\in{\mathbb{N}}$ and $0\leq t<1$ let

$\Theta(d)(n+t):=w^{g(n),g(n+1)}_{t}\Theta_{g(n)}(d)(w^{g(n),g(n+1)}_{t})^{*}.$

By the choice of the unitaries $w^{g(n),g(n+1)}_{t}$ we have

$\lim_{t\to
1}w^{g(n),g(n+1)}_{t}\Theta_{g(n)}(d)(w^{g(n),g(n+1)}_{t})^{*}=\Theta_{g(n+1)}(d),$

and therefore $\Theta(d)$ is continuous, $\|\Theta(d)(n+t)\|=\|d\|$ for all
$n$ and $t$, hence $\Theta(d)\in C_{b}([0,\infty),A)$. Also, (3.1) and
$w_{t}^{g(n),g(n+1)}\in A\cap A_{g}(n)^{\prime}$ for all $n$ and $t$ together
imply that for every $m$ and every $d\in D$ we have
$\lim_{t\to\infty}\|[a_{m}(t),\Theta(d)(t)]\|=0$, and therefore
$\pi\circ\Theta[D]\subseteq A_{\text{\faForward}}\cap B^{\prime}$. It is clear
that $\pi\circ\Theta\colon D\to A_{\text{\faForward}}$ is unital and
injective. Since $B$ was an arbitrary separable $\mathrm{C}^{*}$-subalgebra of
$A_{\text{\faForward}}$, we conclude that $A_{\text{\faForward}}$ is
$D$-saturated. ∎

##  4\. $D$-saturation of the corona $\mathcal{Q}(A\otimes\mathcal{K})$

In this section we prove the folloqing analog of Theorem 3.1.

######  Theorem 4.1.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable, unital, and tensorially
$D$-absorbing then $\mathcal{Q}(A\otimes\mathcal{K})$ is $D$-saturated.

The proof of Theorem 4.1 is an elaboration of the proof of Theorem 3.1, and as
such it also depends on [14, Theorem 2.2] in an essential way. The additional
ingredient in this proof is Lemma 4.3 below. The origins of this lemma can be
traced back at least to Voiculescu’s tri-diagonal argument ([52], more
recently [53]) and more directly to [18, Theorem 3.1] (see [19]) and [21,
Lemma 1.2]. The simplest form of these lemmas shows that every element of the
corona of a $\sigma$-unital $\mathrm{C}^{*}$-algebra that has an approximate
unit consisting of projections is the sum of two block-diagonal elements. The
appropriately uniform version of these lemmas for countable sets of operators
is indispensable in the study of automorphisms of coronas, both in
constructing outer automorphisms (when combined with the Continuum Hypothesis
or its weakenings, see [13, Lemma 4.3], [22, §17.1]) and in proving that all
automorphisms of a given corona are inner, or of a very simple form (when
combined with forcing axioms, see [22, §17], [39], and [51]).

The original construction of outer automorphisms of the Calkin algebra ([44])
uses a different approach that is more directly related to Voiculescu’s tri-
diagonal trick. This is the unfolding argument of [44, Proposition 1.4] and
[37, §3], used in the proof of Corollary 5.1 below. The two approaches appear
to be interchangeable for many purposes.

In the context of (strongly) self-absorbing $\mathrm{C}^{*}$-algebras similar
methods had been used in [33, Theorem 4.5] and [49, §4], and the variant of
these results proven in [14] is used in the proof of Lemma 4.2 below. As the
referee pointed out, Lemma 4.3 can also be considered as an instance of the
‘2-coloured equivalence’ used in Elliott’s classification program. The first
instance of this was the seminal 2-coloured equivalence of projections ([38,
Lemma 2.1]) that has been perfected further to obtain a method for presenting
a *-homomorphism into a massive ${\mathcal{Z}}$-stable
$\mathrm{C}^{*}$-algebra as the sum of two order zero completely positive
contractions in [7, §6]. At the hindsight, this is very similar to what Lemma
4.3 and its predecessors (including the tri-diagonal-type arguments) do for
*-homomorphisms of separable $\mathrm{C}^{*}$-algebras, with ultraproducts in
their codomain replaced by coronas of $\sigma$-unital
$\mathrm{C}^{*}$-algebras, with the difference that ${\mathcal{Z}}$-stability
is in our situation not needed.

The remainder of this section is devoted to the proof of Theorem 4.1. Suppose
that $D$ and $A$ are as in Theorem 4.1. As in the first paragraph of the proof
of Theorem 3.1, we write $A=\lim_{n}A_{n}$, where $A_{0}\cong A$ and
$A_{n+1}=A_{n}\otimes D_{n}$, with $D_{n}$ being an isomorphic copy of $D$. As
before, we identify $D_{n}$ with a unital $\mathrm{C}^{*}$-subalgebra of $A$
and let $\Theta_{n}\colon D\to A$ be the embedding that identifies $D$ with
$D_{n}$. We will need the following discretization of [14, Theorem 2.2].

######  Lemma 4.2.

Suppose that $D$ and $A$ are as in Theorem 4.1 and that $A_{j}$, $D_{j}$, and
$\Theta_{j}$ are as in the previous paragraph. Then for every $\varepsilon>0$
and every $F\Subset D$ there is $k(F,\varepsilon)\in{\mathbb{N}}$ such that
for all $m<n$ there are unitaries $w_{j}=w^{m,n,\varepsilon}_{j}(F)$, for
$j<k(F,\varepsilon)$, in $A$ which satisfy the following.

  1. (1)

$w_{0}=1$.

  2. (2)

$\|w_{j}-w_{j+1}\|<\varepsilon$ for all $j<k(F,\varepsilon)-1$.

  3. (3)

$w_{j}\in A\cap A_{m}^{\prime}$ for all $j<k(F,\varepsilon)$.

  4. (4)

$\|w_{k(F,\varepsilon)-1}\Theta_{m}(d)w_{k(F,\varepsilon)-1}^{*}-\Theta_{n}(d)\|<\varepsilon$
for all $d\in F$.

Clearly the unitaries $w^{m,n,\varepsilon}_{j}(F)$ depend on the presentation
of $A$ as $\lim_{n}A_{n}$, but this presentation is fixed once and for all. It
is important that $k(F,\varepsilon)$ does not depend on the choice of $m$ and
$n$.

###### Proof.

We may let $w^{m,m,\varepsilon}_{j}(F):=1$ for all $F,m=n$ and
$j<k(F,\varepsilon)$ (with $k(F,\varepsilon)$ to be determined shortly) and
assume that $m<n$. By [56, Remark 3.3] $D$ is $K_{1}$ injective hence [14,
Theorem 2.2] implies that in $D\otimes D$ there is a continuous path of
unitaries $u_{t}$, for $0\leq t<1$ such that $u_{0}=1$ and $\lim_{t\to
1}u_{t}(a\otimes b)u_{t}^{*}=b\otimes a$ for all $a$ and $b$ in $D$. For
$\varepsilon>0$, fix $t^{\prime}<1$ large enough to have
$\|u_{t^{\prime}}\Theta_{m}(d)u_{t^{\prime}}^{*}-\Theta_{n}(d)\|<\varepsilon$
for all $d\in F$. Then fix $k(F,\varepsilon)$ large enough to have
$0=t(0)<t(1)<\dots<t(k(F,\varepsilon)-1)=t^{\prime}$ which satisfy
$\|u_{t(j)}-u_{t(j+1)}\|<\varepsilon$ for all $j<k(F,\varepsilon)-2$ and
$\|u_{k(F,\varepsilon)-1}\Theta_{m}(d)u_{k(F,\varepsilon)-1}^{*}-\Theta_{n}(d)\|<\varepsilon$
for all $d\in F$. Transferring the unitaries $u_{t(j)}$ to $A$, by identifying
$D\otimes D$ with $D_{m}\otimes D_{n}$ as in the proof of Theorem 3.1, we
obtain $w^{m,n,\varepsilon}_{j}(F):=u_{t(j)}$ as required. ∎

Part (1a) of the following lemma is closely related to [22, Lemma 9.6.7], [18,
Theorem 3.1], and [39, Lemma 2.6] (see also the introduction to the present
section for a discussion of the origins of this lemma). Although the latter
result is fairly general, it does not literally imply the statement that we
need and we sketch a proof for reader’s convenience.

######  Lemma 4.3.

Suppose that $B$ is a non-unital $\mathrm{C}^{*}$-algebra that has an
approximate unit $(e_{n})$ consisting of projections.

  1. (1)

If $a_{n}$, for $n\in{\mathbb{N}}$, belong to $\mathcal{M}(B)$ then there are
an increasing sequence $j(n)$ in ${\mathbb{N}}$ and $a_{n}^{0}$, $a_{n}^{1}$,
for $n\in{\mathbb{N}}$, in $\mathcal{M}(B)$ with the following property. With
the projections (letting $j(0)=0$ and $e_{0}=0$)

$p_{n}:=e_{j(n+1)}-e_{j(n)}$

the following holds.

    1. (a)

$(p_{2i}+p_{2i+1})a_{n}^{0}(p_{2j}+p_{2j+1})=0$ for all $n$ and $i\neq j$.

    2. (b)

$(p_{2i+1}+p_{2i+2})a_{n}^{1}(p_{2j+1}+p_{2j+2})=0$ for all $n$ and $i\neq j$.

    3. (c)

$a_{n}-a_{n}^{0}-a_{n}^{1}\in B$ for all $n$.

  2. (2)

If in addition $B=A\otimes\mathcal{K}$, for $A$ unital and such that
$A=\lim_{n}A_{n}$, then $p_{n}$ can be chosen of the form $1_{A}\otimes r_{n}$
for orthogonal projections $r_{n}\in\mathcal{K}$, so that for some
$f\colon{\mathbb{N}}\to{\mathbb{N}}$ and all $i$ in addition to (1a)–(1c) we
have

    1. (a)

$(p_{2i}+p_{2i+1})a_{n}^{0}(p_{2i}+p_{2i+1})$ belongs to

$(p_{2i}+p_{2i+1})(A_{f(i)}\otimes\mathcal{K})(p_{2i}+p_{2i+1}),$

    2. (b)

$(p_{2i+1}+p_{2i+2})a_{n}^{1}(p_{2i+1}+p_{2i+2})$ belongs to

$(p_{2i+1}+p_{2i})(A_{f(i)}\otimes\mathcal{K})(p_{2i+1}+p_{2i+2}),$

and

    3. (c)

$\sum_{n\leq m}r_{n}$, for $m\in{\mathbb{N}}$, is an approximate unit in
$\mathcal{K}$.

###### Proof.

(1) Find an increasing sequence $j(n)$, for $n\in{\mathbb{N}}$, such that
$j(0):=0$ and for all $m\leq n$ we have

(4.1) $\displaystyle\begin{split}\|(1-e_{j(n+1)})a_{m}e_{j(n)}\|<2^{-n-2},\\\
\|(1-e_{j(n+1)})a_{m}^{*}e_{j(n)}\|<2^{-n-2}.\end{split}$

This is possible because both $a_{m}e_{j(n)}$ and $a_{m}^{*}e_{j(n)}$ belong
to the ideal $B$. Figure 1 adapted from [22, p. 263] may be helpful in
visualizing the construction to follow.

$p_{0}$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$$p_{6}$$\cdot$$\cdot$$\cdot$ Figure
1.  The supports of $a^{0}_{n}$ (with respect to any orthonormal basis
diagonalizing all $p_{n}$) are included in the solid square regions and the
supports of $a^{1}_{n}$ are included in the dashed line regions.

Let $p_{n}:=e_{j(n+1)}-e_{j(n)}$ and let (the series on the right-hand side of
each of the following formulas is strictly convergent because $p_{n}$ are
orthogonal projections in $B$)

$\displaystyle a_{m}^{0}$
$\displaystyle:=\sum_{n}(p_{2n}a_{m}p_{2n}+p_{2n+1}a_{m}p_{2n}+p_{2n}a_{m}p_{2n+1}),$
$\displaystyle a_{m}^{1}$
$\displaystyle:=\sum_{n}(p_{2n+1}a_{m}p_{2n+1}+p_{2n+2}a_{m}p_{2n+1}+p_{2n+1}a_{m}p_{2n+2}).$

Clearly the conditions (1a) and (1b) are satisfied. In order to verify (1c),
fix $2k>m$. Then $a^{0}_{m}p_{2k}=(p_{2k}+p_{2k+1})a_{m}p_{2k}$ and
$a^{1}_{m}p_{2k}=p_{2k-1}a_{m}p_{2k}$. Therefore

$\displaystyle(a-a^{0}_{m}-a^{1}_{m})p_{2k}$
$\displaystyle=\left(\sum_{i<2k-1}p_{i}+\sum_{i>2k+1}p_{i}\right)a_{m}p_{2k}$
$\displaystyle=(e_{j(2k-1)}+(1-e_{j(2k+2)}))a_{m}p_{2k}.$

Similarly,
$(a-a^{0}_{m}-a^{1}_{m})p_{2k+1}=(e_{j(2k)}+(1-e_{j(2k+3)}))a_{m}p_{2k+1}$ for
$2k+1>m$, and therefore
$(a-a^{0}_{m}-a^{1}_{m})p_{k}=(e_{j(k-1)}+(1-e_{j(k+2)}))a_{m}p_{k}$ for all
$k>m$. Since $p_{k}=e_{j(k+1)}-e_{j(k)}$, for $k>m$ we have, using (4.1),

$\displaystyle\|(a-a^{0}_{m}-a^{1}_{m})p_{k}\|$
$\displaystyle=\|(e_{j(k-1)}+(1-e_{j(k+2)}))a_{m}p_{k}\|$
$\displaystyle\leq\|e_{j(k-1)}a_{m}(1-e_{j(k)})\|+\|(1-e_{j(k+2)})a_{m}e_{j(k+1)}\|$
$\displaystyle<2^{-k-2}+2^{-k-1}<2^{-k}.$

Therefore all $n>m$ satisfy

$\|(a_{m}-a^{0}_{m}-a^{1}_{m})(1-e_{j(n)})\|\leq\sum_{k\geq
n}\|(a_{m}-a^{0}_{m}-a^{1}_{m})p_{k}\|<\sum_{k=n}^{\infty}2^{-k}=2^{-n+1}.$

Since $n$ was arbitrary, we have $a_{m}-a_{m}^{0}-a_{m}^{1}\in B$ for all $m$,
and (1c) follows.

For (2), assume $B=A\otimes\mathcal{K}$ for a unital $\mathrm{C}^{*}$-algebra
$A$. Apply (1) with $e_{n}=1_{A}\otimes q_{n}$, for an increasing approximate
unit $q_{n}$ of $\mathcal{K}(H)$ consisting of projections to produce first
approximations to $a^{0}_{n}$, $a^{1}_{n}$, and $p_{n}$ satisfying (1a)–(1c).
By the choice of the $e_{n}$’s, each $p_{n}$ is of the form $1_{A}\otimes
r_{n}$ for a projection $r_{n}$ in $\mathcal{K}(H)$ such that finite partial
sums of $r_{n}$’s form an approximate identity in $\mathcal{K}(H)$, which
takes care of (2c).

Before the notation gets out of control, we define the _cutdown of $x$ by a
projection $q$_ to be $qxq$. By replacing $a^{l}_{n}$ with
$(1-e_{j(n)})a^{l}_{n}(1-e_{j(n)})$ for $l=0,1$ (note that the image of
$a^{l}_{n}$ under the quotient map is unaffected by this change), we may
assume that $p_{i}a^{l}_{n}=a^{l}_{n}p_{i}=0$ for all $i<j(n)$ and $l=0,1$.
Then in each block of the form
$(p_{2i}+p_{2i+1})(A\otimes\mathcal{K})(p_{2i}+p_{2i+1})$ only finitely many
of the $a_{n}^{0}$ have nonzero cutdowns, and in each block of the form
$(p_{2i+1}+p_{2i+2})(A\otimes\mathcal{K})(p_{2i+1}+p_{2i+2})$ only finitely
many of the $a_{n}^{1}$ have nonzero cutdowns. Since $A=\lim_{l}A_{l}$, the
cutdown of each $a^{0}_{n}$ is within $1/(i+1)$ from some

$b(n,i)\in(p_{2i}+p_{2i+1})(A_{l}\otimes\mathcal{K})(p_{2i}+p_{2i+1})$

for a large enough $l$. The norm of $b(n,i)$ can be assumed to be no greater
than $\|a^{0}_{n}\|$. Similarly, for every $i\geq j(n)$ there are
$l\in{\mathbb{N}}$ and

$c(n,i)\in(p_{2i+1}+p_{2i+2})(A_{l}\otimes\mathcal{K})(p_{2i+1}+p_{2i+2})$

which is within $1/(i+1)$ of the cutdown of $a^{1}_{n}$ by $p_{2i+1}+p_{2i+2}$
and satisfies $\|c(n,i)\|\leq\|a^{1}_{n}\|$.

Let $f(i)$ be the maximum of the $l$’s obtained in this way for all $n$ such
that $i\geq j(n)$. For $i<j(n)$ let $b(n,i)=0$ and $c(n,i)=0$ (to be precise,
this is $0$ in the corresponding cutdown of $A_{f(i)}\otimes\mathcal{K}$).
Thus in each block of the form
$(p_{2i}+p_{2i+1})(A\otimes\mathcal{K})(p_{2i}+p_{2i+1})$, every $a^{0}_{n}$
has a $1/(i+1)$-approximation $b(n,i)$ in the corresponding cutdown of
$A_{f(i)}\otimes\mathcal{K}$ and in each block of the form
$(p_{2i+1}+p_{2i+2})(A_{j}\otimes\mathcal{K})(p_{2i+1}+p_{2i+2})$ every
$a^{1}_{n}$ has a $1/(i+1)$-approximation $c(n,i)$ in the corresponding
cutdown of $A_{f(i)}\otimes\mathcal{K}$.

A special case of [22, Lemma 13.1.10] when $\\{e_{n}\\}$ is an increasing
sequence of projections implies that for every $n$ the series
$b_{n}:=\sum_{i}b(n,i)$ is strictly convergent and therefore
$b_{n}\in\mathcal{M}(A\otimes\mathcal{K})$. Since $b(n,i)$ is within $1/i$ of
the cutdown of $a^{0}_{n}$ by $p_{2i}+p_{2i+1}$, we have
$\|(a^{0}_{n}-b_{n})(1-e_{j(2i)})\|\leq 1/(i+1)$ for all $i$, and therefore
$a^{0}_{n}-b_{n}\in A\otimes\mathcal{K}$. Moreover, with $a^{0}_{n}$ replaced
with $b_{n}$, (1a) continues to hold, and (2a) holds by the choice of the
function $f$.

Similarly, $c_{n}:=\sum_{i}c(n,i)$ defines an element of
$\mathcal{M}(A\otimes\mathcal{K})$ such that $a^{1}_{n}-c_{n}\in
A\otimes\mathcal{K}$ and $c_{n}$ satisfies (1b) and (2b), completing the
proof. ∎

A notational convention is in order. Suppose that $p_{n}$ is a sequence of
orthogonal projections in $A$ such that
$\sum_{n}p_{n}=1_{\mathcal{M}(A\otimes\mathcal{K})}$ (such series is
convergent in the strict topology by [22, Lemma 13.1.11]). Suppose in addition
that $p_{n}=1_{A}\otimes r_{n}$, for $r_{n}\in\mathcal{K}$. Then
$p_{n}(A\otimes\mathcal{K})p_{n}$ is isomorphic to $M_{m}(A)$ (where
$m=\operatorname{rank}(r_{n})$, but the exact value of $m$ is of no importance
for us) and we let

(4.2) $\Upsilon_{n}\colon A\to p_{n}(A\otimes\mathcal{K})p_{n}$

be the diagonal embedding, $a\mapsto a\otimes 1_{m}$.

###### Proof of Theorem 4.1.

Suppose that $A$ and $D$ are separable and unital, $D$ is strongly self-
absorbing, and $A$ is tensorially $D$-absorbing. Fix a separable
$\mathrm{C}^{*}$-subalgebra $B$ of $\mathcal{Q}(A\otimes\mathcal{K})$, and let
$\\{a_{n}\mid n\in{\mathbb{N}}\\}$ be a subset of
$\mathcal{M}(A\otimes\mathcal{K})$ such that $\pi(a_{n})$, for
$n\in{\mathbb{N}}$, is dense in $B$. Write $A=\lim_{n}A_{n}$ as in the
paragraph preceding Lemma 4.2, so that $A_{n+1}=A_{n}\otimes D_{n}$ with
$D_{n}\cong D$ for all $n$. By Lemma 4.3 (2), there are projections $p_{i}$,
for $i\in{\mathbb{N}}$, in $A\otimes\mathcal{K}$ such that
$\sum_{i}p_{i}=1_{\mathcal{M}(A\otimes\mathcal{K})}$ (the sum is convergent in
the strict topology), and there are $f\colon{\mathbb{N}}\to{\mathbb{N}}$ and
$a_{n}^{0}$, $a_{n}^{1}$ in $\mathcal{M}(A\otimes\mathcal{K})$ such that the
following holds for all $n$ and all $i$ (all infinite sums are strictly
convergent).

  1. (1)

$a_{n}-(a_{n}^{0}+a_{n}^{1})\in\mathcal{M}(A\otimes\mathcal{K})$.

  2. (2)

$a_{n}^{0}=\sum_{i}(p_{2i}+p_{2i+1})a_{n}^{0}(p_{2i}+p_{2i+1})$.

  3. (3)

$a_{n}^{1}=\sum_{i}(p_{2i+1}+p_{2i+2})a_{n}^{1}(p_{2i+1}+p_{2i+2})$.

  4. (4)

$(p_{2i}+p_{2i+1})a_{n}^{0}(p_{2i}+p_{2i+1})$ belongs to

$(p_{2i}+p_{2i+1})(A_{f(i)}\otimes\mathcal{K})(p_{2i}+p_{2i+1})$

  5. (5)

$(p_{2i+1}+p_{2i+2})a_{n}^{1}(p_{2i+1}+p_{2i+2})$ belongs to

$(p_{2i+1}+p_{2i+2})(A_{f(i)}\otimes\mathcal{K})(p_{2i+1}+p_{2i+2}).$

Let $F_{i}$, for $i\in{\mathbb{N}}$, be an increasing sequence of finite
subsets of the unit ball of $D$ with dense union. Let
$\varepsilon(j)=1/(j+1)$. Let $l(0)=0$ and (with $k(F_{j},\varepsilon(j))$ as
provided by Lemma 4.2) let $l(j+1):=l(j)+k(F_{j},\varepsilon(j))$ for $j\geq
0$. Also let $f^{+}(j):=f(l(j))$ for $j\in{\mathbb{N}}$. Define $\Psi\colon
D\to\mathcal{M}(A\otimes\mathcal{K})$ as follows.

Fix $d\in D$. We will have $\Psi(d)=\sum_{n}x_{n}p_{n}$, for some $x_{n}$ in
$p_{n}(A\otimes\mathcal{K})p_{n}$ to be determined shortly. Let
$\Upsilon_{n}\colon A\to p_{n}(A\otimes\mathcal{K})p_{n}$ be the diagonal
embedding defined in (4.2). Thus for every $j\in{\mathbb{N}}$ the embedding
$\Theta_{j}$ of $D$ into $A$ defined by identifying it with $D_{j}$ (the
$j$-th copy of $D$ in $D^{\otimes{\mathbb{N}}}$) used earlier determines a
unital embedding

$\Upsilon_{n}\circ\Theta_{j}\colon D\to p_{n}(A\otimes\mathcal{K})p_{n}.$

We will need to twist this embedding a little. For $i\in{\mathbb{N}}$ and
$j<k(F_{i},\varepsilon(i))$ let, using $k(F_{i},\varepsilon(i))$ and the
unitaries $w^{m,n,\varepsilon(i)}_{j}(F_{i})$ produced by Lemma 4.2,

(4.3) $v_{l(i)+j}:=w^{f^{+}(i),f^{+}(i+1),\varepsilon(i)}_{j}(F_{i}).$

Since every natural number can be uniquely represented in the form $l(i)+j$
for some $i$ and $j<k(F_{i},\varepsilon(i))$, this defines $v_{n}$ for all
$n\in{\mathbb{N}}$ (analogous remark applies to the definition of $\Psi_{n}$
below). Then for all $i$ and $j<k(F_{i},\varepsilon(i))-1$ we have

(4.4) $\|v_{l(i)+j}-v_{l(i)+j+1}\|<\varepsilon(i).$

Also, by (4) of Lemma 4.2, for all $i$ we have

(4.5)
$\|(\operatorname{Ad}v_{l(i)+k(F_{i},\varepsilon(i))-1})\circ\Theta_{f^{+}(i)}(d)-\Theta_{f^{+}(i+1)}(d)\|<\varepsilon(i)\text{
for all }d\in F_{i}.$

For $n$ in ${\mathbb{N}}$, with $n=l(i)+j$ for unique $i$ and
$j<k(F_{i},\varepsilon(i))$,

(4.6)
$\Psi_{l(i)+j}(d):=(\operatorname{Ad}v_{l(i)+j})\circ\Theta_{f^{+}(i)}(d)$

defines a ∗-homomorphism $\Psi_{n}\colon D\to A$ for all $n$.

######  Claim 4.4.

  1. (1)

For all $i$ and $j<k(F_{i},\varepsilon(i))$, the range of $\Psi_{l(i)+j}$ is
included in $A\cap A_{f(i)}’$.

  2. (2)

For all $n$ we have $\|\Psi_{n}(d)-\Psi_{n+1}(d)\|\leq 2\varepsilon(i)$.

###### Proof.

The first part follows immediately from the fact that both the range of
$\Theta_{f^{+}(i)}$ and all $v_{l(i)+j}$ are included in $A\cap
A_{f^{+}(i)}’$.

Fix $n$. By (4.3), if $n=l(i)+j$ for some $j$ such that $0\leq
j<k(F_{i},\varepsilon(i))-1$, then
$\|v_{l(i)+j}-v_{l(i)+j+1}\|<\varepsilon(i)$ and therefore
$\|\Psi_{n}(d)-\Psi_{n+1}(d)\|<2\varepsilon(i)$ for all $d$. If
$n=l(i)+k(F_{i},\varepsilon(i))-1$, then $v_{n+1}=v_{l(i+1)}=1$ and
$v_{n}=v_{l(i)+k(F_{i},\varepsilon(i))-1}=w^{f^{+}(i),f^{+}(i+1),\varepsilon(i)}_{k(F_{i},\varepsilon(i)-1}(F_{i})$
satisfies, by (4) of Lemma 4.2,

$\|(\operatorname{Ad}v_{n}\circ\Theta_{f^{+}(i)})(d)-\Theta_{f^{+}(i+1)}(d)\|<\varepsilon(i),$

and the left-hand side of this formula is equal to
$\|\Psi_{n}(d)-\Psi_{n+1}(d)\|$ (see (4.6)), giving the required estimate. ∎

Let (the right-hand side is again strictly convergent for all $d$)

$\Psi(d):=\sum_{n}\Upsilon_{n}\circ\Psi_{n}(d).$

This is a ∗-homomorphism of $D$ into $\mathcal{M}(A\otimes\mathcal{K})$ such
that

$\Psi(1_{D})=\sum_{n}p_{n}=1_{\mathcal{M}(A\otimes\mathcal{K})}.$

Since each $\Psi_{n}$ is an isometry we have
$\Psi[D]\cap(A\otimes\mathcal{K})=\\{0\\}$, and therefore $\pi\circ\Psi$
defines an injective, unital ∗-homomorphism of $D$ into
$\mathcal{Q}(A\otimes\mathcal{K})$.

######  Claim 4.5.

For every $n=l(i)+j$ with $j<k(\varepsilon(i))$ and $d\in F_{i}$ we have

  1. (1)

$\Psi(d)p_{n}\in
p_{n}(A\otimes\mathcal{K})p_{n}\cap(p_{n}(A_{f^{+}(i)}\otimes\mathcal{K})p_{n})^{\prime}$.

  2. (2)

If $a\in p_{n+1}(A_{f^{+}(i)}\otimes\mathcal{K})p_{n}$ then

$\|a(\Upsilon_{n}\circ\Psi_{n}(d))-(\Upsilon_{n+1}\circ\Psi_{n+1}(d))a\|<2\varepsilon(i)\|a\|.$

  3. (3)

For all $m\in{\mathbb{N}}$ and $d\in D$, $[a_{m},\Psi(d)]\in
A\otimes\mathcal{K}$.

###### Proof.

(1) follows from the first part of Claim 4.4 and the fact that $\Upsilon_{n}$
is an isomorphic embedding of $A$ into a corner of $A\otimes\mathcal{K}$.

(2) Considering $a$ as a matrix over $A_{f^{+}(i)}$, by the choice of $v_{n}$
and $\Theta_{f^{+}(i)}$ all of its entries commute with
$\Psi_{n}(d)=(\operatorname{Ad}v_{n})\circ\Theta_{f^{+}(i)}(d)$. Therefore

$a(\Upsilon_{n}\circ\Psi_{n}(d))=(\Upsilon_{n+1}\circ\Psi_{n}(d))a.$

The second part of Claim 4.4 implies that
$\|\Psi_{n}(d)-\Psi_{n+1}(d)\|<2\varepsilon(i)$, therefore
$\|(\Upsilon_{n+1}\circ\Psi_{n+1}(d))a-(\Upsilon_{n+1}\circ\Psi_{n}(d))a\|<2\varepsilon(i)$,
and the claim follows.

(3) Since $a_{m}-a^{0}_{m}-a^{1}_{m}\in A\otimes\mathcal{K}$, it suffices to
prove that for all $m$ and $d\in D$, both $[a_{m}^{0},\Psi(d)]$ and
$[a_{m}^{1},\Psi(d)]$ belong to $A\otimes\mathcal{K}$. By density it suffices
to prove this for all $d\in\bigcup_{i}F_{i}$. Fix $i$, $d\in F_{i}$, and $m$
and consider the commutator $[a_{m}^{0},\Psi(d)]$.

We have $\Psi(d)=\sum_{n}p_{n}\Psi(d)p_{n}$ and
$a^{0}_{m}=\sum_{n}(p_{2n}+p_{2n+1})a^{0}_{m}(p_{2n}+p_{2n+1})$. Thus for all
$n$ and $n’$ such that $|n-n’|\geq 2$ we have
$p_{n}[a^{0}_{m},\Psi(d)]p_{n’}=0$. By (1), $p_{n}\Psi(d)p_{n}$ and
$p_{n}a^{0}_{m}p_{n}$ commute for all $n$, hence
$p_{n}[a^{0}_{m},\Psi(d)]p_{n}=0$ for all $n$. Therefore, after rearranging
the strictly convergent sum we have that $[a^{0}_{m},\Psi(d)]$ is equal to the
following.

(4.7)
$\sum_{n}(p_{2n+1}a_{m}^{0}p_{2n}\Psi(d)-\Psi(d)p_{2n+1}a_{m}^{0}p_{2n})\\\
+\sum_{n}(p_{2n}a_{m}^{0}p_{2n+1}\Psi(d)-\Psi(d)p_{2n}a_{m}^{0}p_{2n+1}).$

Then (2) implies that for $d\in F_{i}$ the $n$th summand of the first infinite
sum has norm smaller than $2\varepsilon(i)\|a_{m}^{0}\|$ (for $i$ such that
$l(i)\leq 2n<l(i+1)$). Since $\varepsilon(i)\to 0$ and the projections $p_{n}$
are orthogonal, this implies that the first sum strictly converges to an
element of $A\otimes\mathcal{K}$ for all $d\in\bigcup_{i}F_{i}$, and therefore
for all $d\in D$. Analogous argument shows that the second sum in (4.7) also
belongs to $A\otimes\mathcal{K}$, and therefore that $[a_{m}^{0},\Psi(d)]\in
A\otimes\mathcal{K}$.

Analogous argument gives $[a_{m}^{1},\Psi(d)]\in A\otimes\mathcal{K}$ for all
$m$ and all $d\in D$, concluding the proof. ∎

Since $B$ is generated by $\pi(a_{m})$, for $m\in{\mathbb{N}}$, the unital
copy of $D$ that is the range of $\pi\circ\Psi$ is in the relative commutant
of $B$, as required. Since $B$ was an arbitrary separable
$\mathrm{C}^{*}$-subalgebra of $\mathcal{Q}(A\otimes\mathcal{K})$, this
completes the proof. ∎

##  5\. Other applications

It is well-known that all embeddings of a strongly self-absorbing
$\mathrm{C}^{*}$-algebra $D$ into an ultrapower of a $D$-absorbing
$\mathrm{C}^{*}$-algebra $A$ are unitarily equivalent ([46, Theorem 7.2.2],
[17]). Analogous results hold for embeddings of $D$ into the asymptotic
sequence algebra of $A$. Corollary 5.1 below shows that this holds for path
algebras and coronas and it appears to be new. In the original version of the
present paper I only claimed approximate unitary equivalence in both (1) and
(2). The proof provided clearly implied asymptotic unitary equivalence, but
Jamie Gabe pointed out that one can even obtain unitary equivalence.888At
first this may appear to be interesting because neither
$A_{\text{\faForward}}$ nor $\mathcal{Q}(A\otimes\mathcal{K})$ is likely to be
fully (or even quantifier-free) countably saturated (see [22, §15 and §16]).
However, as the referee kindly pointed out, the model-theoretic point of view
is misleading because this is proven by a standard reindexing argument similar
to one used below in the proof of Corollary 5.1.  His argument is included
with his kind permission.

######  Corollary 5.1.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable, unital, and tensorially
$D$-absorbing then the following holds.

  1. (1)

Every two unital embeddings of $D$ into $A_{\text{\faForward}}$ are unitarily
equivalent.

  2. (2)

Every two unital embeddings of $D$ into $\mathcal{Q}(A\otimes\mathcal{K})$ are
unitarily equivalent.

The following lemma is a straightforward application of reindexing in coronas
of $\sigma$-unital $\mathrm{C}^{*}$-algebras of Phillips and Weaver, known as
‘(un)folding’, and it is the key point of the proof of Corollary 5.1.

######  Lemma 5.2.

Suppose that $E$ is a $\sigma$-unital, non-unital $\mathrm{C}^{*}$-algebra and
that $D$ is a separable, unital $\mathrm{C}^{*}$-algebra. Two unital copies of
$D$ in $\mathcal{Q}(E)$ are asymptotically unitarily equivalent if and only if
they are unitarily equivalent.

###### Proof.

Only the direct implication requires a proof. Let $C$ denote the
$\mathrm{C}^{*}$-subalgebra of $\mathcal{Q}(E)$ generated by two
asymptotically unitarily equivalent unital copies of $D$. We have a continuous
path of unitaries $u_{t}$, for $0\leq t<1$, in $\mathcal{Q}(E)$ which
implements the asymptotic unitary equivalence. Embed $\mathcal{Q}(E)$ into
$\mathcal{Q}(E)_{\text{\faForward}}$ diagonally and consider the diagonal
image of $C$. The path $u_{t}$ gives a unitary $u$ in
$\mathcal{Q}(E)_{\text{\faForward}}$ that implements an isomorphism between
the two copies of $D$ comprising $C$. We thus have an embedding of the
$\mathrm{C}^{*}$-algebra generated by $C$ and $u$ into the path algebra
$\mathcal{Q}(E)_{\text{\faForward}}$ which factors through a ∗-homomorphism
from $C$ into $\mathcal{Q}(E)$. The unitary $u$ can be lifted to
$\mathcal{Q}(E)$ by [44, Proposition 1.4] and this lift implements the
required isomorphism between the two copies of $D$. ∎

###### Proof of Corollary 5.1.

Let $E$ be $A\otimes\mathcal{K}$ or $C_{b}([0,\infty),A)$. Then
$\mathcal{Q}(E)$ is $A_{\text{\faForward}}$ or
$\mathcal{Q}(A\otimes\mathcal{K})$ and it is $D$-saturated by Theorem 3.1 or
Theorem 4.1. By Lemma 5.2 we need to prove that every two unital copies of $D$
in $\mathcal{Q}(E)$ are asymptotically unitarily equivalent. By $D$-saturation
it will suffice to prove that two commuting unital copies of $D$ in
$\mathcal{Q}(E)$ are asymptotically unitarily equivalent. Since $D$ is nuclear
and simple, any two commuting copies of $D$ generate an isomorphic copy of
$D\otimes D$. By [56, Remark 3.3] and [14, Theorem 2.2] the flip automorphism
of $D\otimes D$ is strongly asymptotically inner, and the desired conclusion
follows. ∎

Here are Corollary F and its proof (see also Conjecture 6.2).

######  Corollary 5.3.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable and unital
$\mathrm{C}^{*}$-algebra then the following are equivalent.

  1. (1)

$A\otimes D\cong A$.

  2. (2)

$D$ embeds unitally into $A_{\text{\faForward}}\cap A^{\prime}$.

  3. (3)

For every separable $\mathrm{C}^{*}$-subalgebra $S$ of
$A_{\text{\faForward}}$, $D$ embeds unitally into $A_{\text{\faForward}}\cap
S^{\prime}$.

###### Proof.

(1) implies (3) is Theorem 3.1, and (3) trivially implies (2).

It remains to prove that (2) implies (1). Let ${\mathcal{U}}$ be a
nonprincipal ultrafilter on ${\mathbb{N}}$. Then
$C_{b}([0,\infty),A)\to\ell_{\infty}(A):f\mapsto f\restriction{\mathbb{N}}$ is
a quotient map that lifts a ∗-homomorphism $\Psi\colon
A_{\text{\faForward}}\to A_{\mathcal{U}}$ (where $A_{\mathcal{U}}$ is the
ultrapower of $A$ associated with ${\mathcal{U}}$). Clearly $\Psi$ sends the
diagonal copy of $A$ in $A_{\text{\faForward}}$ to the diagonal copy of $A$ in
$A_{\mathcal{U}}$. Since $D$ is nuclear and simple, $\Psi$ is injective on $D$
and we conclude that $D$ embeds into $A_{\mathcal{U}}\cap A^{\prime}$. By [46,
Theorem 7.2.2], $A$ absorbs $D$ tensorially. ∎

In [25, Theorem 2.8] it was proven that, with $A$ and $D$ as in Corollary
5.1999A sufficient assumption on $D$ is that it has approximately inner half-
flip., the Continuum Hypothesis implies that $A_{\mathcal{U}}\cap D^{\prime}$
is isomorphic to $A_{\mathcal{U}}$ and $\ell_{\infty}(A)/c_{0}(A)\cap
D^{\prime}$ is isomorphic to $\ell_{\infty}(A)/c_{0}(A)$. The proof uses
countable saturation (see [22, §16]) of these quotient algebras. Since neither
$\mathcal{Q}(A\otimes\mathcal{K})$ nor $A_{\text{\faForward}}$ is likely to be
countably saturated (for $A_{\text{\faForward}}$ see [22, Exercise 16.8.36],
but beware the following typo: ‘degree-1 saturated’ should be ‘saturated’; the
present formulation clearly contradicts [22, Theorem 15.1.5]), we have only
the following weak analog of [25, Theorem 2.8] (for the definition of an
elementary submodel see [22, Appendix D]).

######  Corollary 5.4.

Suppose that $D$ is a separable, unital, and strongly self-absorbing
$\mathrm{C}^{*}$-algebra. If $A$ is separable, unital, and tensorially
$D$-absorbing then the following holds.

  1. (1)

For any unital embedding of $\Phi\colon D\to A_{\text{\faForward}}$, the
relative commutant $A_{\text{\faForward}}\cap\Phi[D]^{\prime}$ is an
elementary submodel of $A_{\text{\faForward}}$.

  2. (2)

For any unital embedding of $\Phi\colon D\to\mathcal{Q}(A\otimes\mathcal{K})$,
the relative commutant $\mathcal{Q}(A\otimes\mathcal{K})\cap\Phi[D]^{\prime}$
is an elementary submodel of $\mathcal{Q}(A\otimes\mathcal{K})$.

###### Proof.

The proofs are analogous to the proof of [25, Theorem 2.8], but using Lemma
1.5 of the same paper in place of Lemma 1.6. ∎

The possibilities that $A_{\text{\faForward}}$ and
$A_{\text{\faForward}}\cap\Phi[D]’$ are isomorphic or that
$\mathcal{Q}(A\otimes\mathcal{K})$ and
$\mathcal{Q}(A\otimes\mathcal{K})\cap\Phi[D]’$ are isomorphic remain open. By
[22, Corollary 15.1.6] both
$\mathcal{Q}(A\otimes\mathcal{K})\cap\Phi[D]^{\prime}$ and
$A_{\text{\faForward}}\cap\Phi[D]^{\prime}$ are countably degree-1 saturated,
however this does not suffice to prove that $A_{\text{\faForward}}$ and
$A_{\text{\faForward}}\cap\Phi[D]’$ are isomorphic or that
$\mathcal{Q}(A\otimes\mathcal{K})$ and
$\mathcal{Q}(A\otimes\mathcal{K})\cap\Phi[D]’$ are isomorphic. Notably,
$\mathcal{Q}(H)$ is not even countably homogeneous ([26]). This poses a
serious obstruction to using model-theoretic methods for constructing
automorphisms of $\mathcal{Q}(H)$ whose restriction to some separable
subalgebra is not implemented by a unitary. It is presently not known whether
such automorphism of $\mathcal{Q}(H)$ can exist in any model of ZFC. The
logical complexity of the assertion that such automorphism exists is
$\Sigma^{2}_{1}$ and therefore standard set-theoretic arguments (including
Woodin’s $\Sigma^{2}_{1}$ absoluteness theorem, see [34], also [57]) suggest
that if the existence of such automorphism of $\mathcal{Q}(H)$ is relatively
consistent with ZFC, then it follows from the Continuum Hypothesis.

##  6\. Concluding remarks

To put Theorem A in proper context, note that forcing axioms MA and
$\operatorname{OCA_{T}}$ (see [22, §8.7]) together imply that $\mathcal{Q}(H)$
is not isomorphic to the corona of any simple $\mathrm{C}^{*}$-algebra not
stably isomorphic to the algebra $\mathcal{K}$ of compact operators on a
separable, infinite-dimensional Hilbert space. This is a consequence of [51,
Theorem B] and the straightforward fact that in this situation there are no
topologically trivial isomorphisms. The same axioms imply that the Calkin
algebra has no outer automorphisms ([21], [22, §17.2–17.8]) and even that each
one of its endomorphisms is unitarily equivalent to the amplifications by a
matrix algebra ([50]). However, the Continuum Hypothesis implies that the
Calkin algebra has outer automorphisms ([44], see also [22, §17.1]).

It is not known whether there are nonisomorphic separable, simple,
$\mathrm{C}^{*}$-algebras with isomorphic coronas (see [23, Question 3.19]).

######  Conjecture 6.1.

The Calkin algebra is not isomorphic to the corona of any simple, separable,
non-unital $\mathrm{C}^{*}$-algebra other than $\mathcal{K}$.

A few remarks on obtaining a candidate $A$ for a counterexample to Conjecture
6.1. When $A$ is of the form $B\otimes\mathcal{K}$ for a unital $B$, then by
[45, Theorem 3.2] or by [35, Definition 2.5 and Theorem 2.8] and [36], it has
to be purely infinite and simple in order to have a simple and purely infinite
corona. Notably, a relative of the first part of Lemma 4.3 (to be precise,
[18, Theorem 3.1]) is crucial in analyzing ideals of multiplier algebras and
coronas (see e.g., the proof of [45, Theorem 2.8]).

By the main result of [58], a purely infinite and simple
$\mathrm{C}^{*}$-algebra is either unital or stable (i.e., tensorially
$\mathcal{K}$-absorbing). There are examples of $\mathrm{C}^{*}$-algebras $A$
that are neither unital nor stable but have simple coronas (see [45, Remark
3.3]), but I’ll concentrate on the purely infinite, simple case.

By a result of Kirchberg, if $B$ is nuclear in addition to being simple,
purely infinite, and separable, then it necessarily absorbs
$\mathcal{O}_{\infty}$ tensorially (see [46, §7.2]), and therefore Corollary D
implies that $\mathcal{Q}(B\otimes\mathcal{K})$ is not isomorphic to
$\mathcal{Q}(H)$. The non-nuclear examples of purely infinite, simple, and
separable $\mathrm{C}^{*}$-algebras from [43, Theorem 4.3.8 and Theorem
4.3.11] all absorb $\mathcal{O}_{\infty}$ tensorially, and are therefore ruled
out as well. In [6, Example 4.8] and [16, Corollary 1.2], purely infinite and
simple $\mathrm{C}^{*}$-algebras have been constructed inside type III factors
(by Blackadar’s method or by taking elementary submodels, see [24, Proposition
7.3.1]). These examples are not approximately divisible, hence not
$\mathcal{O}_{\infty}$-absorbing and possibly not ${\mathcal{Z}}$-absorbing.
However, they inherit trivial $K$-groups from the ambient factors. This rules
them out as counterexamples to Conjecture 6.1, since standard methods (see the
paragraph following Corollary D) imply that $B$ would have to have the same
$K$-theory as the algebra of compact operators, namely $K_{0}(B)={\mathbb{Z}}$
and $K_{1}(B)=0$. Possibly the reduced free products as in [16, Theorem 3.1]
can produce a purely infinite and simple, but not ${\mathcal{Z}}$-stable,
$\mathrm{C}^{*}$-algebra with the right $K$-theory, but here is another
example (using the methods of [24]).

Here is another example. Let $B_{0}$ be a separable elementary submodel of
$\mathcal{Q}(H)$, and let $B$ be a separable elementary submodel of
$\mathcal{Q}(B_{0}\otimes\mathcal{K})$. Then $B$ is purely infinite and simple
(because this property of $\mathrm{C}^{*}$-algebras is axiomatizable, see [24,
Theorem 2.5.1]), $K_{0}(B)=K_{1}(B_{0})=K_{0}(\mathcal{K})$, and
$K_{1}(B)=K_{0}(B_{0})=K_{1}(\mathcal{K})$, as required. By Theorem B combined
with a clever use of the folding argument (i.e., [44, Proposition 1.4] again)
communicated to me by Jamie Gabe, Chris Schafhauser and Stuart White, $B_{0}$
is not $\mathcal{O}_{\infty}$-absorbing.

It is however not at all clear how would one construct a nontrivial
isomorphism between $\mathcal{Q}(B\otimes\mathcal{K})$ and $\mathcal{Q}(H)$
(or how to assure that $\mathcal{Q}(B\otimes\mathcal{K})$ has a $K$-theory
reversing automorphism). What is clear is that such isomorphism cannot be
constructed in ZFC (by Vignati’s result mentioned earlier) and that the
Continuum Hypothesis could be helpful (see [23, §11] on this and [23, §5 and
§6] for more information on this sort of a rigidity question).

Theorem B gives a rather crude description of a small part of the ${\rm Ext}$
group of Wassermann’s $\mathrm{C}^{*}$-algebra ${\mathcal{W}}(\Gamma)$. I
haven’t been able to prove anything new about the ${\rm Ext}$ of the reduced
group algebra of the free group, $\mathrm{C}^{*}_{r}(F_{2})$, although it is
well-known that its Ext is not a group ([30]). The techniques developed in [3]
may be relevant to this problem. As pointed out by Nigel Higson, there is no
known example of a separable, unital $\mathrm{C}^{*}$-algebra $A$ such that
$\operatorname{Aut}(\mathcal{Q}(H))$ acts on ${\rm Ext}^{w}(A)$ nontrivially
in some model on ZFC (see [23, Question 3.18]; note that we use the weak Ext,
allowing unitaries with nonzero Fredholm index, because otherwise letting $A$
be $M_{2}({\mathbb{C}})$ provides an example), and ${\mathcal{W}}(\Gamma)$
provides an interesting test case. Under forcing axioms this action is
trivial, since all automorphisms of $\mathcal{Q}(H)$ are inner.

The following conjectures that there is an analog of Corollary F (and [46,
Theorem 7.2.2]) for $\mathcal{Q}(A\otimes\mathcal{K})$.

######  Conjecture 6.2.

Suppose that $A$ and $D$ are separable and unital $\mathrm{C}^{*}$-algebras,
and that $D$ is strongly self-absorbing. Then the following are equivalent.

  1. (1)

$A\otimes D\cong A$.

  2. (2)

For every separable $\mathrm{C}^{*}$-subalgebra $S$ of
$\mathcal{Q}(A\otimes\mathcal{K})$ there is a unital embedding of $D$ into
$\mathcal{Q}(A\otimes\mathcal{K})\cap S^{\prime}$.

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