2.1 Notation

We use capital letters $A,B,C,D$ to denote C*-algebras and usually a calligraphic $\mathcal {D}$ to denote a strongly self-absorbing C*-algebra. Generally, small letters $a,b,c,d,\dots ,x,y,z$ will denote operators in C*-algebras. $A_+$ will denote cone of positive elements in a C*-algebra A. If $\varepsilon> 0$ and $a,b$ are elements in a C*-algebra, we will write

(2.1) $$ \begin{align} a \approx_\varepsilon b \end{align} $$

to mean that $\|a - b\| < \varepsilon $ . This will make some approximations more legible.

The symbol $\otimes $ will denote the minimal tensor product of C*-algebras, while $\odot $ will mean the algebraic tensor product. We use the minimal tensor product throughout, and it is common for us to deal with nuclear C*-algebras so there should not be any ambiguity. The symbol $\overline {\otimes }$ will denote the von Neumann tensor product.

We will denote by $M_n$ the C*-algebra of $n\times n$ matrices, and $M_{n^{\infty }}$ the uniformly hyperfinite (UHF) C*-algebra associated with the supernatural number $n^{\infty }$ . We will write $\mathcal {Q}$ for the universal UHF algebra $\mathcal {Q} = \bigotimes _{n \in \mathbb {N}} M_n$ .

By $G \curvearrowright ^\alpha A$ , we will mean that the (discrete) group G acts on A by automorphisms, i.e., $\alpha : G \to \operatorname {\mathrm {Aut}}(A)$ is a homomorphism. $A \rtimes _{r,\alpha }G$ will denote the reduced crossed product, which we will just write as $A \rtimes _\alpha G$ if it is clear from context that the group is amenable and A is nuclear (e.g., if G is finite). We will denote by $A^\alpha $ the fixed point subalgebra of the action (or $A^G$ if the action is clear from context).

For a map $f: X \to Y$ between sets X and Y, we will write $f: X \hookrightarrow Y$ to mean that f is injective and $f: X \twoheadrightarrow Y$ to mean that f is surjective. This will usually be done in the context of *-homomorphisms.

2.2 Ultrapowers, central sequences, and central sequence algebras

Fix a free ultrafilter $\omega \in \beta \mathbb {N}$ . Throughout, we will use ultrapowers to describe asymptotic behavior. Alternatively, one can use sequence algebras, although this comes down to a matter of taste and one can swap between the two if desired, as we will provide local characterizations. This also means that all of what we do will be independent of the specific ultrafilter $\omega $ .

For a C*-algebra A, the ultrapower of A is the C*-algebra

(2.2) $$ \begin{align} A_\omega := \ell^{\infty}(A)/c_{0,\omega}(A), \end{align} $$

where $c_{0,\omega } := \{(a_n)_{n \in \mathbb {N}} \in \ell ^{\infty }(A) \mid \lim _{n \to \omega } \|a_n\| = 0\}$ is the ideal of $\omega $ -null sequences. We can embed A into $A_\omega $ canonically by means of constant sequences: we identify $a \in A$ with the equivalence class of the constant sequence $(a)_{n \in \mathbb {N}}$ .

To ease notation, we will usually write elements of $A_\omega $ as sequences $(a_n)_{n \in \mathbb {N}}$ , keeping in mind that these are equivalence classes without explicitly stating it every time. We note that the norm on $A_\omega $ is given by $\|(a_n)_{n \in \mathbb {N}}\| = \lim _{n \to \omega } \|a_n\|$ .

Kirchberg’s $\varepsilon $ -test [Reference Kirchberg35, Lemma A.1] is essentially the operator algebraists’ Łoś’ theorem without having to turn to (continuous) model theory. Heuristically, it says that if certain things can be done approximately in an ultrapower, then certain things can be done exactly in an ultrapower.

Lemma 2.1 (Kirchberg’s $\varepsilon $ -test)

Let $(X_n)_n$ be a sequence of sets and suppose that for each n, there is a sequence $(f_n^{(k)})_{k \in \mathbb {N}}$ of functions $f_n^{(k)}: X_n \to [0,\infty )$ . For $k \in \mathbb {N}$ , let

(2.3) $$ \begin{align} f_\omega^k(s_1,s_2,\dots) := \lim_{n \to \omega} f_n^{(k)}(s_n). \end{align} $$

Suppose that for every $m \in \mathbb {N}$ and $\varepsilon> 0$ , there is $s \in \prod _n X_n$ with $f_\omega ^{(k)}(s) < \varepsilon $ for $k=1,\dots ,m$ . Then there exists $t \in \prod _nX_n$ with $f_\omega ^{(k)}(t) = 0$ for all $k \in \mathbb {N}$ .

The above is useful, although if one so wishes, one can usually construct exact objects from approximate objects by using standard diagonalization arguments (under some separability assumptions). These sorts of arguments work in both the ultrapower setting and the sequence algebra setting.

Finally, if $\alpha \in \operatorname {\mathrm {Aut}}(A)$ is an automorphism, there is an induced automorphism on $A_\omega $ , which we will denote by $\alpha _\omega $ , given by

(2.4) $$ \begin{align} \alpha_\omega((a_n)_{n \in \mathbb{N}}) := (\alpha(a_n))_{n \in \mathbb{N}}. \end{align} $$

2.3 Central sequences and central sequence subalgebras

For a unital C*-algebra A, the C*-algebra of $\omega $ -central sequences is

(2.5) $$ \begin{align} A_\omega \cap A' = \{x \in A_\omega \mid [x,a] = 0 \text{ for all } a \in A\}, \end{align} $$

where we are identifying $A \subseteq A_\omega $ with the constant sequences. If $B \subseteq A$ is a unital C*-subalgebra and $S \subseteq A_\omega $ is a subset, we can associate the relative commutant of S in $B_\omega $ :

(2.6) $$ \begin{align} B_\omega \cap S' = \{b \in B_\omega \mid [b,s] = 0 \text{ for all } s \in S\}. \end{align} $$

Of particular interest will be when $S = A$ , and $B \subseteq A$ is a unital inclusion of separable C*-algebras.

2.4 Strongly self-absorbing C*-algebras

A unital separable C*-algebra $\mathcal {D}$ is strongly self-absorbing if $\mathcal {D} \not \simeq \mathbb {C}$ and there is an isomorphism $\phi : \mathcal {D} \to \mathcal {D} \otimes \mathcal {D}$ which is approximately unitarily equivalent to the first factor embedding $d \mapsto d \otimes 1_{\mathcal {D}}$ (see [Reference Toms and Winter66]). All known strongly self-absorbing C*-algebras are: the Jiang–Su algebra $\mathcal {Z}$ [Reference Jiang and Su32], the Cuntz algebras $\mathcal {O}_2$ and $\mathcal {O}_{\infty }$ [Reference Cuntz15], UHF algebras of infinite type, and $\mathcal {O}_{\infty }$ tensor a UHF algebra of infinite type. Strongly self-absorbing C*-algebras have approximately inner flip and therefore have K-theoretic restrictions (see [Reference Enders, Schemaitat and Tikuisis20, Reference Tikuisis61]). They are also nuclear, simple, and have at most one tracial state [Reference Toms and Winter66],

Tensorial absorption with strongly self-absorbing C*-algebras gives rise to many regular properties, for example, in terms of K-theory, traces, and the Cuntz semigroup [Reference Jiang and Su32, Reference Rørdam46, Reference Rørdam47, Reference Rørdam50]. Of paramount interest is the Jiang–Su algebra $\mathcal {Z}$ . An accumulation of work has successfully classified all (unital) separable, simple, nuclear, infinite-dimensional, $\mathcal {Z}$ -stable C*-algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet [Reference Rosenberg and Schochet53] by means of K-theory and traces (see [Reference Carrión, Gabe, Schafhauser, Tikuisis and White9] and the references therein). We describe how one might work with $\mathcal {Z}$ -stability in terms of its standard building blocks. Recall that, for $n,m \geq 2$ , the dimension drop algebras are

(2.7) $$ \begin{align} \mathcal{Z}_{n,m} := \{f \in C([0,1],M_n \otimes M_m) \mid f(0) \in M_n \otimes 1_{M_m}, f(1) \in 1_{M_n} \otimes M_m\}. \end{align} $$

Such an algebra is a called a prime dimension drop algebra when n and m are coprime. The Jiang–Su algebra $\mathcal {Z}$ is the unique separable simple C*-algebra with unique tracial state which is an inductive limit of prime dimension drop algebras with unital connecting maps [Reference Jiang and Su32] (in fact, the dimension drop algebras can be chosen to have the form $\mathcal {Z}_{n,n+1}$ ). It is $KK$ -equivalent to $\mathbb {C}$ and $\mathcal {Z}$ -stability is a often necessary condition for K-theoretic classification.

By [Reference Rørdam and Winter52, Proposition 5.1] (or [Reference Sato54, Proposition 2.1] for our desired formulation), $\mathcal {Z}_{n,n+1}$ is the universal C*-algebra generated by elements $c_1,\dots ,c_n$ and s such that:

• • $c_1 \geq 0$ ;

• • $c_ic_j^* = \delta _{ij}c_1^2$ ;

• • $s^*s + \sum _{i=1}^n c_i^*c_i = 1$ ;

• • $c_1s = s$ .

If there are uniformly tracially large (in the sense of [Reference Toms, White and Winter65, Definition 2.2]) order zeroFootnote ¹ c.p.c. maps $M_n \to A_\omega \cap A'$ , these give rise to elements $c_1,\dots ,c_n \in A_\omega \cap A'$ with $c_1 \geq 0$ and $c_ic_j^* = \delta _{ij}c_1^2$ , along with certain tracial information. If A has strict comparison, Matui and Sato used this tracial information to show that A has property (SI) [Reference Matui and Sato39], from which one can get an element $s \in A_\omega \cap A'$ such that $s^*s + \sum _{i=1}^n c_i^*c_i = 1$ and $c_1s = s$ . This gives a *-homomorphism $\mathcal {Z}_{n,n+1} \to A_\omega \cap A'$ , which if can be done for each $n \in \mathbb {N}$ , is enough to conclude that $\mathcal {Z} \hookrightarrow A_\omega \cap A'$ unitally and hence $A \simeq A \otimes \mathcal {Z}$ (see [Reference Toms and Winter67, Reference Winter71]). In fact, it suffices to show that $\mathcal {Z}_{2,3} \hookrightarrow A_\omega \cap A'$ (or $\mathcal {Z}_{n,n+1}$ for some $n \geq 2$ ) (see [Reference Rørdam and Winter52, Theorem 3.4(ii)] and [Reference Schemaitat56, Theorem 5.15]).

4.1 Relative intertwinings

It is well known that a strongly self-absorbing C*-algebra $\mathcal {D}$ embeds unitally into the central sequence algebra $(\mathcal {M}(A))_\omega \cap A'$ of a separable C*-algebra A if and only if $A \simeq A \otimes \mathcal {D}$ , where $\mathcal {M}(A)$ is the multiplier algebra of A (see, for example, [Reference Rørdam49, Theorem 7.2.2(i)]). We alter the proof to keep track of a subalgebra in order to show that for a unital inclusion $B \subseteq A$ of separable C*-algebras, $\mathcal {D} \hookrightarrow B_\omega \cap A'$ unitally if and only if there is an isomorphism $\Phi : A \to A \otimes \mathcal {D}$ which is approximately unitarily equivalent to the first factor embedding and satisfies $\Phi (B) = B \otimes \mathcal {D}$ . This was initially done for (irreducible) inclusions of $\text {II}_1$ factors in [Reference Bisch2] and commented on in [Reference Izumi31] for $\mathcal {D}$ being $M_{n^{\infty }},\mathcal {O}_2,\mathcal {O}_{\infty }$ . The proof we alter is Elliott’s intertwining argument, which can be found as a combination of Propositions 2.3.5 and 7.2.1 and Theorem 7.2.2 of [Reference Rørdam49].

Proposition 4.2 (Countable relative intertwining)

Let $A,B_m,C$ be unital, separable C*-algebras, $m \in \mathbb {N}$ , and $\phi : A \hookrightarrow C,\theta _m: B_m \to A,\psi _m: B_m \to C$ be such that $\phi \circ \theta _m(B_m) \subseteq \psi _m(B_m)$ and $\psi _1(B_1) \subseteq \psi _m(B_m)$ . Suppose there is a sequence $(v_n)_{n \in \mathbb {N}} \subseteq \psi _1(B_1)_\omega \cap \phi (A)'$ of unitaries such that:

• • $\text {dist}(v_n^*cv_n,\phi (A)_\omega ) \to 0$ for all $c \in C$ ;

• • $\text {dist}(v_n^*\psi _m(b)v_n,\phi \circ \theta _m(B_m)_\omega ) \to 0$ for all $b \in B_m$ .

Then $\phi $ is approximately unitarily equivalent to an isomorphism $\Phi :A \simeq C$ such that $\Phi \circ \theta _m (B_m) = \psi _m(B_m)$ for all $m \in \mathbb {N}$ .

Proof We show that if there are unitaries $(v_n)_{n \in \mathbb {N}} \subseteq \psi _1(B_1)$ satisfying:

• • $[v_n,\phi (a)] \to 0$ for all $a \in A$ ;

• • $\text {dist}(v_n^*cv_n,\phi (A)) \to 0$ for all $c \in C$ ;

• • $\text {dist}(v_n^*\psi _m(b)v_n,\phi \circ \theta _m(B_m)) \to 0$ for all $b \in B_m$ ,

then the conclusion holds. Such unitaries can be found using Kirchberg’s $\varepsilon $ -test (Lemma 2.1).

Let $(a_n)_{n \in \mathbb {N}},(b_n^{(m)})_{n \in \mathbb {N}},(c_n)_{n \in \mathbb {N}}$ be dense sequences of $A,B_m,C$ , respectively, $m \in \mathbb {N}$ . We can inductively choose $v_n$ , forming a subsequence $(v_n)_{n \in \mathbb {N}}$ of the unitaries above (after re-indexing, we are still calling them $v_n$ ), such that there are $a_{jn} \in A, b_{jn}^{(m)} \in B_m$ with:

• • $v_n^*\dots v_1^*c_jv_1\dots v_n \approx _{\frac {1}{n}} \phi (a_{jn})$ ;

• • $v_n^*\dots v_1^*\psi (b_j^{(m)})v_1\dots v_n \approx _{\frac {1}{n}} \phi \circ \theta _m(b_{jn}^{(m)})$ ;

• • $[v_n,\phi (a_j)] \approx _{\frac {1}{2^n}} 0$ ;

• • $[v_n,\phi (a_{jl})] \approx _{\frac {1}{2^n}} 0$ ;

• • $[v_n,\phi \circ \theta _m(b_j^{(m)})] \approx _{\frac {1}{2^n}} 0$ ;

• • $[v_n,\phi \circ \theta _m(b_{jl}^{(m)})] \approx _{\frac {1}{2^n}} 0$ ,

where $j,m=1,\dots ,n$ and $l=1,\dots ,n-1$ . Define, for $a \in \{a_n \mid n \in \mathbb {N}\}$ ,

(4.1) $$ \begin{align} \Phi(a) = \lim_n v_1\dots v_n\phi(a) v_n^*\dots v_1^* \end{align} $$

which extends to a *-isomorphism $\Phi : A \simeq C$ , as in [Reference Rørdam49, Proposition 2.3.5]. The proof also yields the following useful approximation:

(4.2) $$ \begin{align} \Phi \circ \theta_m(b_{jn}^{(m)}) \approx_{\frac{1}{2^n}} v_1 \dots v_n \phi \circ \theta_m(b_{jn}^{(m)})v_n^* \dots v_1^* \end{align} $$

for appropriate $n \geq j,m$ .

We now need to check that $\Phi \circ \theta _m(B_m) = \psi _m(B_m)$ . Approximate

(4.3) $$ \begin{align} \psi_m(b_j^{(m)}) \approx_{\frac{1}{n}} v_1\dots v_n \phi \circ \theta_m(b_{jn}^{(m)})v_n^*\dots v_1^* \approx_{\frac{1}{2^n}} \Phi \circ \theta_m(b_{jn}^{(m)}). \end{align} $$

As $n \in \mathbb {N}$ can be made arbitrarily large, this yields $\psi _m(B_m) \subseteq \overline {\Phi \circ \theta _m(B_m)} = \Phi \circ \theta _m(B_m)$ . On the other hand, for any $\varepsilon> 0$ and $b \in B_m$ , we can find n such that

(4.4) $$ \begin{align} \Phi \circ \theta_m(b) \approx_\varepsilon v_1\dots v_n \phi \circ \theta_m(b)v_n^*\dots v_1^* \in \psi_m(B_m) \end{align} $$

since $v_i \in \psi _1(B_1) \subseteq \psi _m(B_m)$ and $\phi \circ \theta _m(B_m) \subseteq \psi _m(B_m)$ . Hence $\Phi \circ \theta _m(B_m) \subseteq \overline {\psi _m(B_m)} = \psi _m(B_m)$ .

4.2 $\mathcal {D}$ -stable embeddings

We will mostly have interest in the case where $\iota $ corresponds to the inclusion map and $B \subseteq A$ is a subalgebra. In this form, we will say $B \subseteq A$ is $\mathcal {D}$ -stable (or $\mathcal {D}$ -absorbing). Clearly, $\iota $ being $\mathcal {D}$ -stable is the same as $\iota (B) \subseteq A$ being $\mathcal {D}$ -stable. We note that we can define the above for any *-homomorphism. Namely, a *-homomorphism $\phi :B \to A$ is $\mathcal {D}$ -stable if $\phi (B) \subseteq A$ is.

Lemma 4.3 If $\iota : B \hookrightarrow A$ is an embedding, then $\iota \otimes \text {id}_D: B \otimes \mathcal {D} \hookrightarrow A \otimes \mathcal {D}$ is $\mathcal {D}$ -stable.

Proof Let $\phi : D \simeq D \otimes \mathcal {D}$ be an isomorphism. Then

(4.5) $$ \begin{align} \Phi := \text{id}_A \otimes \phi: A \otimes \mathcal{D} \to A \otimes \mathcal{D} \otimes \mathcal{D} \end{align} $$

is an isomorphism with

(4.6) $$ \begin{align} \Phi(\iota \otimes \text{id}_{\mathcal{D}}(B \otimes \mathcal{D})) = \left(\iota \otimes \text{id}_{\mathcal{D}}(B \otimes \mathcal{D})\right) \otimes \mathcal{D}.\\[-35pt]\nonumber \end{align} $$

We note that this is a strengthening of the notion of $\mathcal {D}$ -stability for C*-algebras because if $\iota := \text {id}_A: A \to A$ , then $\iota $ is $\mathcal {D}$ -stable if and only if A is $\mathcal {D}$ -stable. This condition is different from the notion of $\mathcal {O}_2$ or $\mathcal {O}_{\infty }$ -absorbing morphisms discussed in [Reference Bosa, Gabe, Sims and White4, Reference Gabe21, Reference Gabe22] – they require sequences from a larger algebra to commute with a smaller algebra, while we require sequences from a smaller algebra to commute with the larger algebra. In the former, neither of the algebras are required to be $\mathcal {D}$ -stable, while the latter necessitates both to be $\mathcal {D}$ -stable.

The following adapts [Reference Rørdam49, Theorem 7.2.2].

Theorem 4.4 Suppose that $B \subseteq A$ is a unital inclusion of separable C*-algebras. If $\mathcal {D}$ is strongly self-absorbing, then $B \subseteq A$ is $\mathcal {D}$ -stable if and only if there is a unital inclusion $\mathcal {D} \hookrightarrow B_\omega \cap A'$ .

Proof Let $\phi : A \to A \otimes \mathcal {D}$ be the first factor embedding $\phi (a) := a \otimes 1_{\mathcal {D}}$ . First, suppose that $\xi : \mathcal {D} \hookrightarrow B_\omega \cap A' \simeq (B \otimes 1_{\mathcal {D}})_\omega \cap (A \otimes 1_{\mathcal {D}})'$ is an embedding (so that $\phi (a)\xi (d) \in \phi (A)_\omega $ and $\phi (b)\xi (d) \in \phi (B)_\omega $ ). Let $\eta : \mathcal {D} \hookrightarrow (B \otimes \mathcal {D})_\omega \cap (A \otimes 1_{\mathcal {D}})'$ be given by $\eta (d) := (1 \otimes d)_n$ and notice that $\xi ,\eta $ have commuting ranges. As all endomorphisms of $\mathcal {D}$ are approximately unitarily equivalent by [Reference Toms and Winter66, Corollary 1.12], let $(v_n)_{n \in \mathbb {N}} \subseteq C^*(\xi (\mathcal {D}),\eta (\mathcal {D})) \simeq \mathcal {D} \otimes \mathcal {D}$ be such that $v_n^*\eta (d)v_n \to \xi (d)$ for $d \in \mathcal {D}$ . For $b \in B$ and $d \in \mathcal {D}$ , we have

(4.7) $$ \begin{align} \nonumber v_n^*(b \otimes d)v_n &= v_n^*(b \otimes 1_{\mathcal{D}})(1_A \otimes d)v_n^* \\ &= v_n^*\phi(b)\eta(d)v_n \\ \nonumber &= \phi(b)v_n^*\eta(d)v_n \\ \nonumber &\to \phi(b)\xi(d) \in \phi(B)_\omega. \end{align} $$

Moreover, the same argument shows that, for $a \in A$ , we have

(4.8) $$ \begin{align} v_n^*(a \otimes d)v_n \to \phi(a)\xi(d) \in \phi(A)_\omega. \end{align} $$

Now $(v_n)_{n \in \mathbb {N}}$ satisfy the hypothesis of Proposition 4.1 with $C := A \otimes D$ , $\phi $ being the first factor embedding, $\theta : B \to A$ being the inclusion and $\psi : B \simeq B \otimes \mathcal {D} \subseteq A \otimes \mathcal {D} = C$ (where this isomorphism exists since if $\mathcal {D} \hookrightarrow B_\omega \cap A'$ , then clearly $\mathcal {D} \hookrightarrow B_\omega \cap B'$ ). From this, we see that $\phi $ is approximately unitarily equivalent to an isomorphism $\Phi :A \simeq A \otimes \mathcal {D}$ such that $\Phi (B) = B \otimes \mathcal {D}$ .

Conversely, if $B \subseteq A$ is $\mathcal {D}$ -stable, let $\Phi : A \simeq A \otimes \mathcal {D}$ be an isomorphism such that $\Phi (B) = B \otimes \mathcal {D}$ . By [Reference Toms and Winter66, Proposition 1.10(iv)], we can identify $\mathcal {D} \simeq \mathcal {D}^{\otimes \infty }$ and take $\xi : \mathcal {D} \hookrightarrow B_\omega \cap A'$ to be given by

(4.9) $$ \begin{align} \xi(d) = (\Phi^{-1}(1_A \otimes 1_{\mathcal{D}}^{\otimes n-1} \otimes d \otimes 1_{\mathcal{D}}^{\otimes \infty}))_n. \end{align} $$

Corollary 4.5 Let $\iota : B \hookrightarrow A$ be a unital embedding between separable C*-algebras. If $\mathcal {D}$ is strongly self-absorbing and $\iota $ is $\mathcal {D}$ -stable, then for every intermediate unital C*-algebra C with $\iota (B) \subseteq C \subseteq A$ , we have that $\iota (B) \subseteq C$ and $C \subseteq A$ are $\mathcal {D}$ -stable. In particular, $C \simeq C \otimes \mathcal {D}$ for all such C.

Proof We have

(4.10) $$ \begin{align} \mathcal{D} \hookrightarrow B_\omega \cap A' \subseteq B_\omega \cap C' \end{align} $$

and

(4.11) $$ \begin{align} \mathcal{D} \hookrightarrow B_\omega \cap A' \subseteq C_\omega \cap A'. \end{align} $$

Now apply Theorem 4.4.

It is not, however, the case that any isomorphism $\Phi : A \simeq A \otimes \mathcal {D}$ with $\Phi (B) = B \otimes \mathcal {D}$ maps C to $C \otimes \mathcal {D}$ .

Example 4.6 Let $\mathcal {D}$ be strongly self-absorbing and consider

(4.12) $$ \begin{align} \nonumber A &:= \mathcal{D} \otimes \mathcal{D} \otimes \mathcal{D}, \\ C_1 &:= \mathcal{D} \otimes 1_{\mathcal{D}} \otimes \mathcal{D},\\ \nonumber C_2 &:= 1_{\mathcal{D}} \otimes \mathcal{D} \otimes \mathcal{D},\\ \nonumber B &:= 1_{\mathcal{D}} \otimes 1_{\mathcal{D}} \otimes \mathcal{D}. \end{align} $$

If $f: \mathcal {D} \otimes \mathcal {D} \to \mathcal {D} \otimes \mathcal {D}$ is the tensor flip and $\phi : \mathcal {D} \simeq \mathcal {D} \otimes \mathcal {D}$ is an isomorphism, let

(4.13) $$ \begin{align} \Phi := f \otimes \phi: A \simeq A \otimes \mathcal{D} \end{align} $$

which satisfies $\Phi (B) = B \otimes \mathcal {D}$ (in particular, $B \subseteq A$ is $\mathcal {D}$ -stable). However,

(4.14) $$ \begin{align} \Phi(C_1) = C_2 \otimes \mathcal{D} \text{ and } \Phi(C_2) = C_1 \otimes \mathcal{D}. \end{align} $$

In fact, the above example can be generalized to show that for any $\mathcal {D}$ -stable inclusion $B \subseteq A$ , there are an isomorphism $\Phi : A \simeq A \otimes \mathcal {D}$ such that $\Phi (B) = B \otimes \mathcal {D}$ and an intermediate algebra $B \subseteq C \subseteq A$ with $\Phi (C) \neq C \otimes \mathcal {D}$ (obviously, we may still have that $\Phi (C) \simeq C \otimes \mathcal {D}$ , but equality may not happen).

Corollary 4.7 Let $B \subseteq A$ be a $\mathcal {D}$ -stable inclusion. There exist a C*-algebra C with $B \subseteq C \subseteq A$ and an isomorphism $\Phi : A \simeq A \otimes \mathcal {D}$ such that $\Phi (B) = B \otimes \mathcal {D}$ but $\Phi (C) \neq C \otimes \mathcal {D}$ .

Proof We first claim that if $B \subseteq A$ is $\mathcal {D}$ -stable, then we can identify $B \subseteq A$ with $B \otimes 1_{\mathcal {D}} \subseteq A \otimes \mathcal {D}$ . If $\Psi : A \simeq A \otimes \mathcal {D}$ is such that $\Psi (B) = B \otimes \mathcal {D}$ and $f: \mathcal {D} \otimes \mathcal {D} \simeq \mathcal {D} \otimes \mathcal {D}$ is the tensor flip, we have

(4.15) $$ \begin{align} \Xi:= (\text{id}_A \otimes f) \circ (\Psi \otimes\text{id}_{\mathcal{D}}): A \otimes \mathcal{D} \simeq A \otimes \mathcal{D} \otimes \mathcal{D} \end{align} $$

is such that $\Xi (B \otimes 1_{\mathcal {D}}) = B \otimes 1_{\mathcal {D}} \otimes \mathcal {D}$ . This proves the claim.

Now, by applying the claim twice, we can identify $B \subseteq A$ with the inclusion

(4.16) $$ \begin{align} B \otimes 1_{\mathcal{D}} \otimes 1_{\mathcal{D}} \otimes \mathcal{D} \subseteq A \otimes \mathcal{D} \otimes \mathcal{D} \otimes \mathcal{D}. \end{align} $$

If $\phi : \mathcal {D} \simeq \mathcal {D} \otimes \mathcal {D}$ is any *-isomorphism,

(4.17) $$ \begin{align} \Phi := \text{id}_A \otimes f \otimes \phi: A \otimes \mathcal{D} \otimes \mathcal{D} \otimes \mathcal{D} \simeq A \otimes \mathcal{D} \otimes \mathcal{D} \otimes \mathcal{D} \otimes \mathcal{D} \end{align} $$

is such that

(4.18) $$ \begin{align} \Phi(B \otimes 1_{\mathcal{D}} \otimes 1_{\mathcal{D}} \otimes \mathcal{D}) = B \otimes 1_{\mathcal{D}} \otimes 1_{\mathcal{D}} \otimes \mathcal{D} \otimes \mathcal{D}. \end{align} $$

Taking $C_1$ and $C_2$ as in Example 4.6, we have that

(4.19) $$ \begin{align} \Phi(B \otimes C_1) = B \otimes C_2 \otimes \mathcal{D} \text{ and } \Phi(B \otimes C_2) = B \otimes C_1 \otimes \mathcal{D}.\\[-36pt]\nonumber \end{align} $$

However, we can always realize $\mathcal {D}$ -stability for countably many intermediate C*-algebras at once using some isomorphism $A \simeq A \otimes \mathcal {D}$ .

The above works since norm ultrapowers have the property that unitaries lift to sequences of unitaries.Footnote ² Tracial ultrapowers of $\text {II}_1$ von Neumann algebras also have this property.Footnote ³ Consequently, if we work with the 2-norm $\|x\|_2 = \tau (x^*x)^{\frac {1}{2}}$ , where $\tau $ is the unique trace on a $\text {II}_1$ factor, all of the above arguments with the C*-norm replaced by $\|\cdot \|_2$ will allow us to recover Bisch’s result [Reference Bisch2, Theorem 3.1], provided we have the appropriate separability conditions.

4.3 Existence of $\mathcal {D}$ -stable embeddings

We move to discuss the existence of $\mathcal {D}$ -stable embeddings. First, we show that each unital embedding of unital, separable $\mathcal {D}$ -stable C*-algebras is approximately unitarily equivalent to a $\mathcal {D}$ -stable embedding. From this, it will follow that there are many $\mathcal {D}$ -stable embeddings.

Proposition 4.11 Let $\mathcal {D}$ be strongly self-absorbing, $A,B$ be unital separable $\mathcal {D}$ -stable C*-algebras, and let $\iota : B \hookrightarrow A$ be a unital embedding. Then $\iota $ is approximately unitarily equivalent to a unital $\mathcal {D}$ -stable embedding $B \hookrightarrow A$ .

Proof As $A,B$ are $\mathcal {D}$ -stable, there are isomorphisms

(4.20) $$ \begin{align} \phi: B \simeq B \otimes \mathcal{D} \text{ and }\psi: A \simeq A \otimes \mathcal{D,} \end{align} $$

which are approximately unitarily equivalent to the first factor embeddings $b \mapsto b \otimes 1_{\mathcal {D}}, b \in B$ and $a \mapsto a \otimes 1_{\mathcal {D}}, a \in A,$ respectively. As $\iota \otimes \text {id}_{\mathcal {D}}: B \otimes \mathcal {D} \hookrightarrow A \otimes \mathcal {D}$ is $\mathcal {D}$ -stable by Lemma 4.3,

(4.21) $$ \begin{align} \sigma := \psi^{-1} \circ (\iota \otimes \text{id}_{\mathcal{D}}) \circ \phi: B \hookrightarrow A \end{align} $$

is $\mathcal {D}$ -stable by Lemma 3.8. Now we show that $\sigma $ is approximately unitarily equivalent to $\iota $ . Let $\mathcal {F} \subseteq B$ be finite and $\varepsilon> 0$ . Let $u \in U(B \otimes \mathcal {D})$ be such that $u^*(b \otimes 1_{\mathcal {D}})u \approx _{\frac {\varepsilon }{2}} \phi (b)$ for $b \in \mathcal {F}$ and $v \in U(A \otimes \mathcal {D})$ be such that $v^*(\iota (b) \otimes 1_{\mathcal {D}})v \approx _{\frac {\varepsilon }{2}} \psi \circ \iota (b)$ for $b \in \mathcal {F}$ . Set $w = \psi ^{-1}(\iota \otimes \text {id}_{\mathcal {D}}(u))^*\psi ^{-1}(v) \in U(A)$ . Then for $b \in \mathcal {F}$ ,

(4.22) $$ \begin{align} \nonumber w^*\sigma(b)w &= \psi^{-1}(v)^*\psi^{-1}(\iota \otimes \text{id}_{\mathcal{D}}(u\phi(b)u^*))\psi^{-1}(v) \\ &\approx_{\frac{\varepsilon}{2}} \psi^{-1}(v)^*\psi^{-1}(\iota \otimes \text{id}_{\mathcal{D}}(b \otimes 1_{\mathcal{D}}))\psi^{-1}(v) \\ \nonumber &= \psi^{-1}(v)^*\psi^{-1}(\iota(b) \otimes 1_{\mathcal{D}})\psi^{-1}(v) \\ \nonumber &\approx_{\frac{\varepsilon}{2}} \psi^{-1}(\psi(\iota(b))) \\ \nonumber &= \iota(b).\\[-37pt]\nonumber \end{align} $$

Later on, there will be some examples of non- $\mathcal {D}$ -stable embeddings between $\mathcal {D}$ -stable C*-algebras. Consequently, despite the fact $\mathcal {D}$ -stable embeddings are point-norm dense, the set of unital $\mathcal {D}$ -stable embeddings need not coincide with the set of all unital embeddings $B \hookrightarrow A$ . Another clear consequence is that despite $\mathcal {D}$ -stability of an embedding being closed under conjugation by a unitary, it is not true that it is preserved under approximate unitary equivalence (in fact, the examples in question show that $\mathcal {D}$ -stability is not even preserved under asymptotic unitary equivalence). We finish with a corollary about embeddings into the Cuntz algebra $\mathcal {O}_2$ [Reference Cuntz15].

We include this last result about the classification of morphisms via functors.

4.4 Permanence properties

We now discuss some permanence properties.

Lemma 4.16 Let $\mathcal {D}$ be strongly self-absorbing. Suppose that $\iota _j: B_j \hookrightarrow A_j, j=1,2$ are $\mathcal {D}$ -stable inclusions. Then $\iota _1 \oplus \iota _2: B_1 \oplus B_2 \hookrightarrow A_1 \oplus A_2$ is $\mathcal {D}$ -stable.

Proof Let $\Phi _j: A_j \simeq A_j \otimes \mathcal {D}$ be isomorphisms such that $\Phi _j\circ \iota _j(B_j) = \iota _j(B_j) \otimes \mathcal {D}$ and consider

(4.24) $$ \begin{align} \Phi: A_1 \oplus A_2 \simeq (A_1 \oplus A_2) \otimes \mathcal{D} \end{align} $$

given by the composition

(4.25)

where the last isomorphism follows from (finite) distributivity of the min-tensor. Then we see that

(4.26) $$ \begin{align} \Phi(\iota_1(B_1) \oplus \iota_2(B_2)) = \left(\iota_1(B_1) \oplus \iota_2(B_2)\right) \otimes \mathcal{D}.\\[-34pt]\nonumber \end{align} $$

Lemma 4.17 Let $\mathcal {D}$ be strongly self-absorbing. Suppose that $\iota _j: B_j \hookrightarrow A_j, j=1,2$ are inclusions and that at least one of $\iota _1$ or $\iota _2$ is $\mathcal {D}$ -stable. Then $\iota _1 \otimes \iota _2: B_1 \otimes B_2 \hookrightarrow A_1 \otimes A_2$ is $\mathcal {D}$ -stable.

Proof We prove this if $\iota _2$ is $\mathcal {D}$ -stable, and a symmetric argument will yield the result if $\iota _1$ is. Let $\Phi _2: A_2 \simeq A_2 \otimes \mathcal {D}$ be such that $\Phi _2 \circ \iota _2(B_2) = \iota (B_2) \otimes \mathcal {D}$ . Taking

(4.27) $$ \begin{align} \Phi := \text{id}_{A_1} \otimes \Phi_2: A_1 \otimes A_2 \simeq A_1 \otimes A_2 \otimes \mathcal{D}, \end{align} $$

we have that

(4.28) $$ \begin{align} \Phi(\iota_1(B_1) \otimes \iota_2(B_2)) = \iota_1(B_1) \otimes \iota_2(B_2) \otimes \mathcal{D}.\\[-34pt]\nonumber \end{align} $$

Lastly, we discuss unital inclusions $B \subseteq A$ of $C(X)$ algebras, where X is a compact Hausdorff space. We show that if X has finite covering dimension, then such an inclusion is $\mathcal {D}$ -stable if and only if the inclusion $B_x \subseteq A_x$ along each fiber is $\mathcal {D}$ -stable.

Lemma 4.19 Let $\mathcal {D}$ be strongly self-absorbing. Suppose that $B_i \subseteq A_i$ are unital inclusions, for $i=1,2$ , and $\psi : A_1 \to A_2$ is a surjective *-homomorphism such that $\psi (B_1) = B_2$ . If $B_1 \subseteq A_1$ is $\mathcal {D}$ -stable, then so is $B_2 \subseteq A_2$ .

Proof We note that $\psi $ induces a *-homomorphism

(4.29) $$ \begin{align} \tilde{\psi}: (B_1)_\omega \cap A_1' \to (B_2)_\omega \cap A_2' \end{align} $$

and consequently if $\xi : \mathcal {D} \hookrightarrow (B_1)_\omega \cap A_1'$ , we have a unital *-homomorphism

(4.30) $$ \begin{align} \eta := \tilde{\psi} \circ \xi: \mathcal{D} \to (B_2)_\omega \cap A_2'. \end{align} $$

The homomorphism $\eta $ is automatically injective since $\mathcal {D}$ is simple.

Rephrasing the above in terms of commutative diagrams, it says that if we have a commutative diagram

(4.31)

where the left inclusion is $\mathcal {D}$ -stable, then the right inclusion is $\mathcal {D}$ -stable as well.

Now we consider many of the results discussed in [Reference Hirshberg, Rørdam and Winter27, Section 4], except for inclusions of C*-algebras.

If $Y \subseteq X$ is a closed subset, we set $I_Y := C_0(X \setminus Y)A$ , which is a closed two-sided ideal in A. We denote $A_Y := A/I_Y$ and the quotient map $A \to A_Y$ by $\pi _Y$ . For an element $a \in A$ , we write $a_Y := \pi _Y(a)$ and if Y consists of a single point x, we write $A_x,I_x,\pi _x$ and $a_x$ . We say that $A_x$ is the fiber of A at x. We note that $A_X = A$ .

If $B \subseteq A$ is a unital inclusion and $\theta _A: C(X) \to A, \theta _B: C(X) \to B$ are morphisms which witness A and B as $C(X)$ -algebras, respectively, we say that $B \subseteq A$ is an inclusion of $C(X)$ -algebras if

(4.32)

commutes. Note that $\theta _B(C(X)) \subseteq \mathcal {Z}(A)$ , and when discussion an inclusion of fibers $B_Y \subseteq A_Y$ we are considering $B_Y := \pi ^A_Y(B) \subseteq \pi ^A_Y(A) =: A_Y$ , where $\pi _Y^A: A \to A_Y$ is the associated quotient map.

We note that Lemma 4.19 gives that if $B\subseteq A$ is $\mathcal {D}$ -stable and $Y \subseteq X$ is closed, then $B_Y \subseteq A_Y$ is automatically $\mathcal {D}$ -stable as well since we have the commuting diagram

(4.33)

The converse needs a bit of work. This is the embedding analog of the beginning of [Reference Hirshberg, Rørdam and Winter27, Section 4]. We discuss how the proofs can be adapted and often omit approximations that were otherwise done there. We want a version of [Reference Hirshberg, Rørdam and Winter27, Lemma 4.5], which is a result about gluing c.c.p. maps together along fibers. In our setting, we are only interested in u.c.p. maps, and we want to show that if we glue two u.c.p. maps together whose images are contained in some $C(X)$ -subalgebra B, then the glued map also has image contained in B. We borrow their Definition 4.2.

First, we need a lemma that follows as a consequence of $\mathcal {D}$ -stability. It is the embedding analog of [Reference Hirshberg, Rørdam and Winter27, Proposition 4.1].

Lemma 4.21 Let $\mathcal {D}$ be strongly self-absorbing, and $B \subseteq A$ be a unital, $\mathcal {D}$ -stable inclusion of separable C*-algebras. Then for any $\mathcal {G} \subseteq A$ finite and $\varepsilon> 0$ , there exist unital *-homomorphisms $\kappa : A \to A$ and $\mu : \mathcal {D} \to B$ such that:

1. (1) $\kappa (B) \subseteq B$ ,

2. (2) $[\kappa (A),\mu (\mathcal {D})] = 0$ ,

3. (3) $\kappa (a) \approx _\varepsilon a$ for all $a \in \mathcal {G}$ .

Proof The proof is essentially the same as the proof of (a) $\Rightarrow (c)$ in [Reference Hirshberg, Rørdam and Winter27, Proposition 4.1]. As $B \subseteq A$ is $\mathcal {D}$ -stable, let us identify $B \subseteq A$ with $B \otimes \mathcal {D} \subseteq A \otimes \mathcal {D}$ . As $\mathcal {D}$ is strongly self-absorbing, [Reference Toms and Winter66, Theorem 2.3] gives a sequence $(\phi _n)_{n \in \mathbb {N}}$ of *-homomorphisms $\phi _n: \mathcal {D} \otimes \mathcal {D} \to \mathcal {D}$ such that

(4.34) $$ \begin{align} \phi_n(d \otimes 1_{\mathcal{D}}) \to d \text{ for all } d \in \mathcal{D}. \end{align} $$

Define $\kappa _n: A \otimes \mathcal {D} \to A \otimes \mathcal {D}$ by

(4.35) $$ \begin{align} \kappa_n:= (\text{id}_A \otimes \phi) \circ (\text{id}_A \otimes \text{id}_{\mathcal{D}} \otimes 1_{\mathcal{D}}), \end{align} $$

and $\mu _n: \mathcal {D} \to B \otimes \mathcal {D}$ by

(4.36) $$ \begin{align} \mu_n := (\text{id}_B \otimes \phi_n) \circ (1_A \otimes 1_{\mathcal{D}} \otimes \text{id}_{\mathcal{D}}). \end{align} $$

Then taking n large enough and letting $\kappa $ and $\mu $ be $\kappa _n$ and $\mu _n$ , respectively, its clear that $\kappa (B \otimes \mathcal {D}) \subseteq B \otimes \mathcal {D}$ , $[\kappa (A),\mu (\mathcal {D})] = 0$ and that $\kappa (a) \approx _\varepsilon a$ whenever a is in some prescribed finite subset $\mathcal {G} \subseteq A$ and $\varepsilon> 0$ is some prescribed error.

Lemma 4.22 Let $\mathcal {D}$ be strongly self-absorbing and A be a unital, separable $C([0,1])$ -algebra. Suppose $\mathcal {F} \subseteq \mathcal {D}, \mathcal {G} \subseteq A$ are finite self-adjoint subsets of contractions with $1_{\mathcal {D}} \in \mathcal {F}$ . Suppose that we have points $0 \leq r < s < t \leq 1$ and two u.c.p. maps $\rho ,\sigma : \mathcal {D} \to A$ which are $(\mathcal {F},\varepsilon ,\mathcal {G})$ -good for $[r,s],[s,t]$ , respectively. Suppose that $A_s$ is $\mathcal {D}$ -stable.

Then there are u.c.p. maps $\rho ',\sigma ': \mathcal {D} \to A$ which are $(\mathcal {F},\varepsilon ,\mathcal {G})$ -good for $[r,s],[s,t]$ , respectively, and u.c.p. maps $\nu _{\rho '},\nu _{\sigma '}: \mathcal {D} \to A,\mu _{\rho '},\mu _{\sigma '}: \mathcal {D} \otimes \mathcal {D} \to A$ such that $\nu _{\rho '},\nu _{\sigma '}$ are $(\mathcal {F},3\varepsilon ,\mathcal {G})$ -good for some interval $I \subseteq (r,t)$ containing s in its interior, and such that for any $a \in \mathcal {G}, d,d' \in \mathcal {F}$ , we have:

1. (1) $([\rho '(d),\nu _{\rho '}(d')])_I \approx _{2\varepsilon } 0,$

2. (2) $([\sigma '(d),\nu _{\sigma '}(d')])_I \approx _{2\varepsilon } 0,$

3. (3) $\rho '(d)_I\nu _{\rho '}(d')_I \approx _\varepsilon \mu _{\rho '}(d \otimes d')_I,$

4. (4) $\sigma '(d)_I\nu _{\sigma '}(d')_I \approx _\varepsilon \mu _{\sigma '}(d \otimes d')_I,$

5. (5) $\nu _{\rho '}(d)_I \approx _{2\varepsilon } \nu _{\sigma '}(d)_I$ .

If $\rho ,\sigma $ are $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon )$ -good for $[r,s],[s,t]$ , respectively, for some finite $\mathcal {F}' \supseteq \mathcal {F}$ set of contractions and for some $0 < \varepsilon ' < \varepsilon $ , then we can arrange so that $\rho ',\sigma ',\nu _{\rho '},\nu _{\sigma '}$ are $(\mathcal {F}',3\varepsilon ',\mathcal {G})$ -good for the interval I, and that the above five conditions hold with $\varepsilon '$ in place of $\varepsilon $ and $\mathcal {F}'$ in place of $\mathcal {F}$ .

Moreover, if $B \subseteq A$ is a unital inclusion of $C([0,1])$ -algebras such that $\rho (\mathcal {D}) \subseteq B, \sigma (\mathcal {D}) \subseteq B$ and $B_s \subseteq A_s$ is $\mathcal {D}$ -stable, then the images of all $\rho ',\sigma ',\mu _{\rho '},\mu _{\sigma '}$ are contained in B (as are the images of $\nu _{\rho '}$ and $\nu _{\sigma '}$ ).

Proof This is [Reference Hirshberg, Rørdam and Winter27, Lemma 4.4], except we have replaced c.c.p. maps with u.c.p. maps. One can easily check that the resulting maps are u.c.p. maps.

As for the “moreover” part, which is the only addition besides the unitality, we outline the definitions of these maps to show that the images of $\rho ',\sigma ',\mu _{\rho '},\mu _{\sigma '}$ are contained in B. As $B_s \subseteq A_s$ is $\mathcal {D}$ -stable, we can find $\kappa : A_s \to A_s$ and $\mu : \mathcal {D} \to B_s$ as in Lemma 4.21, where $\kappa (a_s) \approx a_s$ for an appropriate error whenever $a \in \mathcal {G}$ . We use Choi–Effros to find u.c.p. lifts $\tilde {\rho },\tilde {\sigma }: \mathcal {D} \to B$ for the maps $\kappa \circ \pi _s \circ \rho $ and $\kappa \circ \pi _s \circ \sigma $ , respectively (note that $\kappa \circ \pi _s \circ \rho $ and $\kappa \circ \pi _s \circ \sigma $ lie in $B_s$ , which is a *-homomorphic image of B). One then defines piece-wise linear functions $f,g: [0,1] \to [0,1]$ which attain both values 0 and 1 at the end points (their definition is not important to show the “moreover” part). Then $\rho ',\sigma '$ are defined as

(4.37) $$ \begin{align} \rho'(d) := (1-f)\cdot \rho(d) + f\cdot\tilde{\rho}(d) \text{ and } \sigma'(d) := (1-g)\cdot\sigma(d) + g\cdot \tilde{\sigma}(d). \end{align} $$

Clearly, $\rho ',\sigma '$ take values in B as $\rho ,\tilde {\rho },\sigma ,\tilde {\sigma }$ all do and $(1-f),f,(1-g),g$ are in B. Now we define u.c.p. maps $\tilde {\mu }_{\rho '},\tilde {\mu }_{\sigma '}: \mathcal {D} \otimes \mathcal {D} \to B_s$ by

(4.38) $$ \begin{align} \tilde{\mu}_{\rho'}(d \otimes d') := \rho'(d)_s\mu(d') \text{ and }\tilde{\mu}_{\sigma'}(d \otimes d') := \sigma'(d)_s\mu(d'). \end{align} $$

Now by Choi–Effros, we can take u.c.p. lifts $\mu _{\rho '}$ and $\mu _{\sigma '}$ of $\tilde {\mu }_{\rho '}$ and $\mu _{\sigma '}$ , respectively. As the images of $\tilde {\mu }_{\rho '}$ and $\mu _{\sigma '}$ lie in $B_s$ , the images of $\mu _{\rho '}$ and $\mu _{\sigma '}$ will lie in B.

Lemma 4.23 Let A be a unital, separable $C([0,1])$ -algebra. Suppose $\mathcal {F} \subseteq \mathcal {D}, \mathcal {G} \subseteq A$ are finite self-adjoint subsets with $1_{\mathcal {D}} \in \mathcal {F}$ and $\varepsilon> 0$ . There exists $0 < \varepsilon ' < \varepsilon $ and a finite subset $\mathcal {F}' \supseteq \mathcal {F}$ such that if $\rho ,\sigma : \mathcal {D} \to A$ are u.p.c. maps and $0 \leq r < s < t \leq 1$ are points such that $\rho $ is $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon ')$ -good for $[r,s]$ , $\sigma $ is $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon ')$ -good for $[s,t]$ and $A_s$ is $\mathcal {D}$ -stable, then there is a u.c.p. map $\psi : \mathcal {D} \to A$ which is $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon ')$ -good for $[r,t]$ .

Moreover, if $B \subseteq A$ is a unital inclusion of $C([0,1])$ -algebras such that $\rho (\mathcal {D}) \subseteq B$ , $\sigma (\mathcal {D}) \subseteq B$ and $B_s \subseteq A_s$ is $\mathcal {D}$ -stable, then $\psi (\mathcal {D}) \subseteq B$ .

Proof The first part is [Reference Hirshberg, Rørdam and Winter27, Lemma 4.5], except we have replaced c.c.p. maps with u.c.p. maps. One has to check that the resulting $\psi $ is unital, but this follows easily if $\rho $ and $\sigma $ are.

We outline the construction of $\psi $ to show unitality, as it will also be useful to show the “moreover” part, which is the only real addition. Let $u \in C([0,1],\mathcal {D} \otimes \mathcal {D})$ be a path of unitaries such that $u_0 = 1_{\mathcal {D} \otimes \mathcal {D}}$ and

(4.39) $$ \begin{align} u_1(d \otimes 1_{\mathcal{D}})u_1^* \approx_{\frac{\varepsilon}{4}} 1_{\mathcal{D}} \otimes d. \end{align} $$

We replace $\rho ,\sigma $ with $\rho ',\sigma '$ as in the above lemma and this yields u.c.p. maps $\mu _\rho ,\mu _\sigma $ satisfying the hypotheses above for some interval $I \subseteq (r,t)$ with s in its interior. Define

(4.40) $$ \begin{align} \phi_\rho,\phi_\sigma: C([0,1]) \otimes \mathcal{D} \otimes \mathcal{D} \to A \end{align} $$

by

(4.41) $$ \begin{align} \nonumber \phi_\rho(f \otimes d \otimes d') &:= f\cdot \mu_\rho(d \otimes d'), \\ \phi_\sigma(f \otimes d \otimes d') &:= f\cdot \mu_\sigma(d \otimes d'). \end{align} $$

Note that these maps are unital. Take nonzero piece-wise linear functions

(4.42) $$ \begin{align} h_1,h_2,h_3,h_4: [0,1] \to [0,1] \end{align} $$

which sum to 1 (their specific form does not matter to show unitality of $\psi $ nor the “moreover” part) and $g_\rho ,g_\sigma : [0,1] \to [0,1]$ which sum to 1 (again, their specific form does not matter to show unitality of $\psi $ nor the “moreover” part). Define unitaries $u_\rho ,u_\sigma \in C([0,1]) \otimes \mathcal {D} \otimes \mathcal {D} \simeq C([0,1],\mathcal {D} \otimes \mathcal {D})$ by

(4.43) $$ \begin{align} u_{\rho x} := u_{g_\rho(x)} \text{ and } u_{\sigma x} := u_{g_\sigma(x)}. \end{align} $$

Now define $\zeta _\rho ,\zeta _\sigma : \mathcal {D} \to A$ by

(4.44) $$ \begin{align} \zeta_\rho(d) &:= \phi_\rho(u_\rho(1_{C([0,1])} \otimes d \otimes 1_{\mathcal{D}})u_\rho^*), \\ \nonumber \zeta_\sigma(d) &:= \phi_\sigma(u_\sigma(1_{C([0,1])} \otimes d \otimes 1_{\mathcal{D}})u_\sigma^*), \end{align} $$

which are clearly unital. Finally, the map $\psi : \mathcal {D} \to A$ is defined by

(4.45) $$ \begin{align} \psi(d) := h_1\cdot\rho(d) + h_2\cdot \zeta_\rho(d) + h_3\cdot \zeta_\sigma(d) + h_4\cdot \sigma(d). \end{align} $$

Clearly, $\psi $ is unital.

Now for the “moreover” part. If $\rho (\mathcal {D}) \subseteq B$ and $\sigma (\mathcal {D}) \subseteq B$ , clearly the first and fourth terms in the definition of $\psi $ will lie in B. So it suffices to show that $\zeta _\rho (\mathcal {D}) \subseteq B$ and $\zeta _\sigma (\mathcal {D}) \subseteq B$ , and for this it suffices to show that $\mu _\rho (\mathcal {D} \otimes \mathcal {D}) \subseteq B$ and $\mu _\sigma (\mathcal {D} \otimes \mathcal {D}) \subseteq B$ (since $h_1,h_2,h_3,h_4$ all lie in B). But this follows from the “moreover” part of the previous lemma.

With this, we get the analog of [Reference Hirshberg and Winter28, Theorem 4.6], the proof being essentially the same as well, except we insist that the our u.c.p. maps commute with a prescribed finite subset of A.

Proposition 4.24 Let $\mathcal {D}$ be strongly self-absorbing, and X be a compact Hausdorff space with finite covering dimension. Suppose that $B \subseteq A$ is a unital inclusion of $C(X)$ -algebras. Then $B_x \subseteq A_x$ is $\mathcal {D}$ -stable for all $x \in X$ if and only if $B \subseteq A$ is $\mathcal {D}$ -stable.

Proof As previously mentioned, if $B \subseteq A$ is $\mathcal {D}$ -stable, then $B_x \subseteq A_x$ is $\mathcal {D}$ -stable for all x.

For the converse, the proof is essentially the same as [Reference Hirshberg, Rørdam and Winter27, Theorem 4.6]. Using the arguments there, one can simplify to the case where we can argue this for $C([0,1])$ -algebras (by using [Reference Hurewicz and Wallman29, Theorem V.3], which says that a compact space of dimension $\leq n$ is homeomorphic to a subset of $[0,1]^{2n+1}$ , and then working component-wise). Now for $\mathcal {F} \subseteq \mathcal {D}, \mathcal {G} \subseteq A$ and $\varepsilon> 0$ , let $\mathcal {G}_x := \{a_x \mid a \in \mathcal {G}\}$ . Without loss of generality suppose that $\mathcal {F}^*=\mathcal {F},\mathcal {G}^*=\mathcal {G}$ and that $1_{\mathcal {D}} \in \mathcal {F}$ . Let $\mathcal {F}',\varepsilon '$ be as in Lemma 4.23.

By $\mathcal {D}$ -stability of the inclusion $B_x \subseteq A_x$ there are u.c.p. $(\mathcal {F}',\varepsilon ',\mathcal {G}_x)$ -embeddings $\psi _x: \mathcal {D} \to B_x \subseteq A_x$ which lift by Choi–Effros to u.c.p. maps $\psi _x': \mathcal {D} \to B$ . The norm is upper semi-continuous (Remark 4.20), and this yields intervals $I_x \subseteq [0,1]$ such that $\psi _x'$ is $(\mathcal {F}',\varepsilon ',\mathcal {G})$ -good for $\overline {I_x}$ . Note that $\psi _x'$ being $(\mathcal {F}',\varepsilon ',\mathcal {G})$ -good for the whole of $I_x$ implies that it is $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon ')$ -good for $\overline {I_x}$ . Compactness then allows us to split the interval as

(4.46) $$ \begin{align} 0 = t_0 < t_1 < \dots < t_n = 1 \end{align} $$

and to take $\psi _i: \mathcal {D} \to B$ u.c.p. which are $(\mathcal {F},\varepsilon ,\mathcal {G};\mathcal {F}',\varepsilon ')$ -good for $[t_{i-1},t_i]$ for $i=1,\dots ,n$ ( $\psi _i = \psi _x'$ for some $x \in [0,1]$ ). Now by repeatedly using the gluing lemma (Lemma 4.23) to glue these maps together, we can find a u.c.p. map $\psi : \mathcal {D} \to B$ which is an $(\mathcal {F},\varepsilon ,\mathcal {G})$ -embedding.

5.1 (Tracial) Rokhlin properties

Here, we will restrict ourselves to finite groups for simplicity, although many results hold more generally (see [Reference Gardella and Hirshberg23, Reference Hirshberg and Orovitz26, Reference Hirshberg and Winter28]).

It is easy to see that a Rokhlin action is outer, since the central projections must commute with any unitary. The fact that weak tracial Rokhlin actions are outer is [Reference Hirshberg and Orovitz26, Proposition 5.3].

Proposition 5.5 Let A be a unital, simple, separable, nuclear, $\mathcal {Z}$ -stable C*-algebra. If $G \curvearrowright ^\alpha A$ is an action of a finite group with the weak tracial Rokhlin property, then $A^\alpha \subseteq A \rtimes _\alpha G$ is $\mathcal {Z}$ -stable.

Proof Let $k \in \mathbb {N}$ . By [Reference Hirshberg and Orovitz26, Theorem 5.6] $A \rtimes _\alpha G$ is tracially $\mathcal {Z}$ -absorbing, meaning there are tracially large (in the sense of [Reference Toms, White and Winter65]) c.p.c. order zero maps $\phi : M_k \to (A \rtimes _\alpha G)_\omega \cap (A \rtimes _\alpha G)'$ , which can be chosen to be c.p.c. order zero maps $\phi : M_k \to A_\omega \cap (A \rtimes _\alpha G)'$ by the proof of [Reference Hirshberg and Orovitz26, Lemma 5.5]. These tracially large c.p.c. order zero maps yield sequences of positive contractions $c_1 = (c_{1n}),\dots ,c_k = (c_{kn}) \in A_\omega \cap (A \rtimes _\alpha G)'$ such that if $(e_n)_{n \in \mathbb {N}} = e := 1 - \sum _i c_i^*c_i$ , we have

(5.9) $$ \begin{align} \lim_{n \to \omega} \max_{\tau \in T(A)} \tau(e_n) = 0, \inf_m \lim _{n \to \omega} \min_{\tau \in T(A)} \tau(c_{1n}^m)> 0 \end{align} $$

and $c_ic_j^* = \delta _{ij}c_1^2$ . By [Reference Gardella and Hirshberg23, Proposition 4.11] (which is much more general, applicable to all countable amenable groups), $A \subseteq A \rtimes _\alpha G$ has equivariant property (SI) since A has property (SI).Footnote ⁵ Consequently, there exists $s \in A_\omega \cap (A \rtimes _\alpha G)'$ such that $s^*s = 1 - \sum _i c_i^*c_i$ and $c_1s = s$ . Altogether,

• • $c_1 \geq 0$ ;

• • $c_ic_j^* = \delta _{ij}c_1^2$ ;

• • $s^*s + \sum _i c_i^*c_i = 1$ ;

• • $c_1s = s$ .

As mentioned in the proof of $(iv) \Rightarrow (i)$ of [Reference Matui and Sato39], $\mathcal {Z}_{n,n+1}$ is the universal C*-algebra generated by $n+1$ elements satisfying the above four relations (see [Reference Rørdam and Winter52, Proposition 5.1] and [Reference Sato54, Proposition 2.1]), and consequently we have a unital *-homomorphism $\mathcal {Z}_{n,n+1} \to A_\omega \cap (A \rtimes _\alpha G)'$ . Therefore $\mathcal {Z} \hookrightarrow A_\omega \cap (A \rtimes _\alpha G)'$ , giving that the desired inclusion is $\mathcal {Z}$ -stable by Lemma 5.2.

5.2 The diagonal inclusion associated with a group action

In the von Neumann setting, a certain diagonal inclusion associated with several automorphisms was considered in [Reference Burstein7, Reference Kawamuro34, Reference Popa43], and they play a role in subfactor theory. Here, we consider a unital C*-algebraic inclusion of the same form.

Definition 5.3 Let A be a C*-algebra, $\alpha _1,\dots ,\alpha _n \in \operatorname {\mathrm {Aut}}(A)$ . The diagonal inclusion associated with $\alpha _1,\dots ,\alpha _n$ is

(5.10) $$ \begin{align} B(\alpha_1,\dots,\alpha_n) = \left\{\bigoplus_{i=1}^n \alpha_i(a) \mid a \in A \right\} \subseteq M_n(A). \end{align} $$

If $G \curvearrowright ^\alpha A$ is an action of a finite group, we write

(5.11) $$ \begin{align} B(\alpha) = \left\{\bigoplus_{g \in G} \alpha_g(a) \mid a \in A\right\} \subseteq M_{|G|}(A). \end{align} $$

We note that a diagonal $B(\alpha ) \subseteq M_{|G|}(A)$ is unique up to unitary conjugation (by permutation unitaries). As $\mathcal {D}$ -stability of an inclusion is preserved under unitary conjugation, there is no ambiguity in speaking of $\mathcal {D}$ -stability of the inclusion $B(\alpha ) \subseteq M_{|G|}(A)$ .

Proposition 5.7 Let $G \curvearrowright ^\alpha A$ be an action of a countable discrete group on a unital, separable C*-algebra. If $G = \langle g_1,\dots ,g_n\rangle $ , then $A \subseteq A \rtimes _\alpha G$ is $\mathcal {D}$ -stable if and only if

(5.12) $$ \begin{align} B(\text{id}_A,\alpha_{g_1},\dots,\alpha_{g_n}) \subseteq M_{n+1}(A) \end{align} $$

is $\mathcal {D}$ -stable.

Proof First, suppose that $A \subseteq A \rtimes _\alpha G$ is $\mathcal {D}$ -stable. Let $\mathcal {F} \subseteq \mathcal {D}, \mathcal {G} \subseteq M_{n+1}(A)$ be finite and $\varepsilon> 0$ . Let $\mathcal {G}' \subseteq A$ be the set of matrix coefficients of elements of $\mathcal {G}$ , together with the identity of A, and let $L := \max \{1,\max _{a \in \mathcal {G}'}\|a\|\}$ . Relabel $\text {id}_A,\alpha _{g_1},\dots ,\alpha _{g_n}$ as $\alpha _1,\dots ,\alpha _{n+1}$ . Let

(5.13) $$ \begin{align} \delta := \frac{\varepsilon}{(4L+1)(n+1)^2}, \end{align} $$

and let $\psi : \mathcal {D} \to A$ be a u.c.p. $(\mathcal {F},\delta ,\mathcal {G}' \cup \{u_{g_i}\}_{i=1}^n)$ -embedding, where $(u_g)$ are the implementing unitaries for $\alpha $ . Let $\phi : D \to B(\alpha ) \subseteq M_{|G|}(A)$ be given by

(5.14) $$ \begin{align} \phi(d) := \bigoplus_{i=1}^{n+1} (\alpha_i \circ \psi)(d). \end{align} $$

Clearly, $\phi $ will be $(\mathcal {F},\delta )$ -multiplicative since each component is the composition of a *-homomorphism (which are contractive) with a map which is $(\mathcal {F},\delta )$ -multiplicative. Now for $d \in \mathcal {F}$ and $a = (a_{ij}) \in \mathcal {G}$ , we have

(5.15) $$ \begin{align} \nonumber \|[\phi(d),(a_{ij})]\| &\leq \sum_{i,j=1}^{n+1}\|\alpha_i(\psi(d))a_{ij} - a_{ij}\alpha_j(\psi(d))\| \\ \nonumber &\leq \sum_{i,j=1}^{n+1}\| \alpha_i(\psi(d))a_{ij} - \psi(d)a_{ij}\| \\ &\ \ \ + \|\psi(d)a_{ij} - a_{ij}\psi(d)\| + \|a_{ij}\psi(d) - a_{ij}\alpha_j(\psi(d))\| \\ \nonumber &\leq \sum_{i,j=1}^{n+1}\|a_{ij}\|\left(\|\alpha_i(\psi(d)) - \psi(d)\| + \|\psi(d) - \alpha_j(\psi(d))\|\right) \\ \nonumber &\ \ \ + \|[\psi(d),a_{i,j}]\| \\ \nonumber &< (n+1)^2(2L(\delta + \delta) + \delta) \\ \nonumber &= (n+1)^2(4L+1)\delta = \varepsilon. \end{align} $$

Conversely, if the associated diagonal inclusion is $\mathcal {D}$ -stable, we note that if $(x_k) \subseteq B(\text {id}_A,\alpha _{g_1},\dots ,\alpha _{g_n})$ is central for $M_{n+1}(A)$ , writing

(5.16) $$ \begin{align} x_k = \bigoplus_{i=1}^{n+1}\alpha_i(a_k) \end{align} $$

yields that $(a_k) \subseteq A$ is central for A and is asymptotically fixed by $\alpha _{g_i}, i=1,\dots ,n$ . In particular, if $\mathcal {D} \hookrightarrow B(\text {id}_A,\alpha _{g_1},\dots ,\alpha _{g_n})_\omega \cap (M_{n+1}(A))'$ , then $\mathcal {D} \hookrightarrow A_\omega \cap (A \rtimes _\alpha G)'$ .

Corollary 5.8 Let $G \curvearrowright ^\alpha A$ be an action of a finite group on a unital, separable C*-algebra. Then $A \subseteq A \rtimes _\alpha G$ is $\mathcal {D}$ -stable if and only if

(5.17) $$ \begin{align} B(\alpha) \subseteq M_{|G|}(A) \end{align} $$

is $\mathcal {D}$ -stable.

6.1 Non-examples

We first start with some non-examples. Villadsen’s C*-algebras with perforation will be useful (see [Reference Toms and Winter68] for good exposition). Let $\mathcal {Q} = \bigotimes _n M_n$ denote the universal UHF C*-algebra.

Theorem 6.1 ([Reference Toms64, Reference Villadsen69])

There exists a unital, simple, separable, nuclear C*-algebra C satisfying the UCT such that $C \not \simeq C \otimes \mathcal {Z}$ and C contains the universal UHF algebra unitally. Moreover, C is tracial and can be chosen to be AH with

(6.1) $$ \begin{align} (K_0(C),K_0(C)^+,[1]_0,K_1(C)) = (\mathbb{Q},\mathbb{Q}_+,1,0). \end{align} $$

Corollary 6.2 There exists an embedding $\mathcal {Q} \hookrightarrow \mathcal {Q}$ which is not $\mathcal {Z}$ -stable. In particular, it is not $\mathcal {Q}$ -stable.

Proof Let C be as above. Note that $\mathcal {Q} \subseteq C$ so we must find an embedding $C \hookrightarrow \mathcal {Q}$ . As C is unital, separable, exact, satisfies the UCT and has a faithful amenable trace (it has traces, and every such trace will be faithful and amenable since C is nuclear and simple) and there is clearly a morphism between $K_0$ -groups, [Reference Schafhauser55, Theorem D] gives an embedding $C \hookrightarrow \mathcal {Q}$ . Consequently, there is an embedding

(6.2) $$ \begin{align} \mathcal{Q} \hookrightarrow C \hookrightarrow \mathcal{Q} \end{align} $$

which is not $\mathcal {Q}$ -stable since there is an intermediate C*-algebra C with $C \not \simeq C \otimes \mathcal {Z}$ .

Corollary 6.3 There is an embedding $\mathcal {Z} \hookrightarrow \mathcal {Q}$ which is not $\mathcal {Z}$ -stable.

Proof Take C as above and take the chain of embeddings (noting that $\mathcal {Q}$ is $\mathcal {Z}$ -stable)

(6.3) $$ \begin{align} \mathcal{Z} \hookrightarrow \mathcal{Q} \otimes \mathcal{Z} \simeq \mathcal{Q} \hookrightarrow C \hookrightarrow \mathcal{Q}. \end{align} $$

The only method we have used to show that an inclusion is not $\mathcal {D}$ -stable is by finding an intermediate algebra which is not $\mathcal {D}$ -stable. There are plenty of examples of stably finite C*-algebras with perforation or higher-stable rank (in particular, non- $\mathcal {Z}$ -stable C*-algebras [Reference Rørdam50]) [Reference Elliott and Villadsen19, Reference Hirshberg, Rørdam and Winter27, Reference de Lacerda Mortari37, Reference Tikuisis60, Reference Toms62–Reference Toms64, Reference Toms and Winter68–Reference Villadsen70]. This gives rise to the following two questions.

1. (1) Is there a unital inclusion $B \subseteq A$ of separable C*-algebras such that whenever C is such that $B \subseteq C \subseteq A$ , we have $C \simeq C \otimes \mathcal {D}$ but $B \subseteq A$ is not $\mathcal {D}$ -stable? Is $\mathcal {D}$ -stability equivalent to every intermediate C*-algebra being $\mathcal {D}$ -stable?

2. (2) To get non-examples, we use stably finite C*-algebras with perforation in between sufficiently regular C*-algebras. Is there a way to do this for purely infinite C*-algebras, or is finiteness the only obstruction? Thus we can ask: if $\mathcal {D}$ is a purely infinite strongly self-absorbing C*-algebra, is every embedding of $\mathcal {D}$ into itself $\mathcal {D}$ -stable? More specifically, if $B \subseteq A$ is a unital inclusion of simple, separable, purely infinite C*-algebras, is the inclusion $\mathcal {O}_{\infty }$ -stable?

Our third question asks if we can get non-examples arising from dynamical systems.

1. (3) Is there a unital, separable $\mathcal {D}$ -stable C*-algebra and a (finite) group action $G \curvearrowright ^\alpha A$ such that $A \rtimes _\alpha G$ is $\mathcal {D}$ -stable, but the inclusion is not? One would need $A \rtimes _\alpha G$ to be $\mathcal {D}$ -stable for non-dynamical reasons.

6.2 Cyclically permuting tensor powers

Here, we give a dynamical example to illustrate the discussion in Section 5. In particular, we can look at a consequence of Corollary 3.5.

Example 6.6 Let $p,q \in \mathbb {N}$ be coprime and consider the qth tensor power of the UHF algebra $A = M_{p^{\infty }}^{\otimes q}$ . Let us examine the action $\mathbb {Z}_q \curvearrowright ^\sigma A$ given by cyclically permuting the tensors:

(6.4) $$ \begin{align} \sigma(a_1 \otimes \cdots \otimes a_q) = a_2 \otimes \cdots \otimes a_q \otimes a_1. \end{align} $$

One can prove directly or use [Reference Hirshberg and Orovitz26] or [Reference Amini, Golestani, Jamali and Christopher Phillips1] in order to conclude that this action has the weak tracial Rokhlin property, or that this action is $\mathcal {Z}$ -equivariantly absorbing, and consequently that $A^{\sigma } \subseteq A \rtimes _\sigma \mathbb {Z}_q$ is $\mathcal {Z}$ -stable.

Alternatively, one can use techniques similar to [Reference Handelman and Rossmann24], [Reference Handelman and Rossmann25], or [Reference Hirshberg and Winter28] in order to compute the K-theory of the fixed point algebra $A^{\sigma }$ to be

(6.5) $$ \begin{align} K_0((M_{p^{\infty}}^{\otimes q})^{\sigma}) \simeq \underset{\to}{\lim}\left(\mathbb{Z}^q, \begin{pmatrix} p + \frac{p^q - p}{q} & \frac{p^q -p}{q} & \cdots & \frac{p^q - p}{q} \\ \frac{p^q - p}{q} & p + \frac{p^q - p}{q} & \cdots & \frac{p^q - p}{q} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{p^q - p}{q} & \frac{p^q - p}{q} & \cdots & p + \frac{p^q - p}{q} \end{pmatrix} \right), \end{align} $$

from which one can show that $K_0(A^{\sigma })$ is p-divisible. Then using the fact that $K_0(A^{\sigma })$ is p-divisible and $A^{\sigma }$ is AF, it follows that $A^{\sigma }$ is $M_{p^{\infty }}$ -stable. Using Corollary 3.5, we then see that $M_{p^{\infty }} \hookrightarrow (A^{\sigma })_\omega \cap A'$ . In particular, we have that $A^{\sigma } \subseteq A \rtimes _\sigma \mathbb {Z}_q$ is $M_{p^{\infty }}$ -stable (since clearly if this embedding is fixed by $\mathbb {Z}_q$ , it will commute with the implementing unitaries as well).

Example 6.7 Following up on the previous example, if we consider the embedding

(6.6) $$ \begin{align} B := \left\{ \begin{pmatrix} x \\ & \sigma(x) \\ & & \ddots \\ & & & \sigma^{q-1}(x) \end{pmatrix} \mid x \in M_{p^{\infty}}^{\otimes q} \right\} \subseteq M_q(M_{p^{\infty}}^{\otimes q}) := A, \end{align} $$

then $B \subseteq A$ is $M_{p^{\infty }}$ -stable by Proposition 5.7.

6.3 The canonical inclusion of the CAR algebra in $\mathcal {O}_2$

Example 6.8 Let $\mathcal {O}_2 = C^*(s_1,s_2)$ be the Cuntz algebra generated by two isometries [Reference Cuntz15], and consider the inclusion

(6.7) $$ \begin{align} M_{2^{\infty}} \simeq \overline{\text{span}}\{s_\mu s_\nu^* \mid |\mu| = |\nu|\} \subseteq \mathcal{O}_2, \end{align} $$

where for a word $\mu = \{i_1,\dots ,i_p\} \in \{1,2\}^p$ , $s_\mu = s_{i_1}\dots s_{i_p}$ . This copy of the CAR algebra is precisely the fixed point subalgebra of the gauge action (see [Reference Raeburn45]). Consider the endomorphism $\lambda : \mathcal {O}_2 \to \mathcal {O}_2$ given by

(6.8) $$ \begin{align} \lambda(x) := s_1xs_1^* + s_2xs_2^*. \end{align} $$

We note that a sequence $(x_n)_{n \in \mathbb {N}}$ is $\omega $ -asymptotically central for $\mathcal {O}_2$ if and only if it is $\omega $ -asymptotically fixed by $\lambda $ . Indeed, if $(x_n)_{n \in \mathbb {N}}$ is central, then $\|\lambda (x_n) - x_n\| \to ^{n \to \omega } 0$ since $[x_n,s_i] \to 0$ for $i=1,2$ . On the other hand, if $(x_n)_{n \in \mathbb {N}}$ is asymptotically fixed by $\lambda $ , then the inequalities

(6.9) $$ \begin{align} \nonumber \|s_ix_n - x_ns_i\| &= \|s_1x_ns_1^*s_i + s_2x_ns_2^*s_i - x_ns_i\| \leq \|\lambda(x_n) - x_n\|\|s_i\| \\ \|s_i^*x_n - x_ns_i^*\| &= \|s_i^*x_n - s_i^*s_1s_1^* - s_i^*s_2x_ns_2^*\| \leq \|s_i^*\|\|\lambda(x_n) - x_n\| \end{align} $$

imply that $(x_n)_{n \in \mathbb {N}}$ is asymptotically central.

We note that $\lambda |_{M_{2^{\infty }}}$ is the forward tensor shift if we identify $M_{2^{\infty }} = \bigotimes _{\mathbb {N}} M_2$ (see, for example, [Reference Davidson17, Section V.4]). Now [Reference Bratteli, Størmer, Kishimoto and Rørdam5] gives an embedding $\xi : M_2 \hookrightarrow (M_{2^{\infty }})_{\omega }$ such that $\lambda _\omega \circ \xi = \xi $ . In particular, $M_{2^{\infty }} \hookrightarrow (M_{2_{\infty }})_{\omega } \cap \mathcal {O}_2'$ so that this inclusion is $M_{2^{\infty }}$ -stable.