We study permanence properties of the classes of stable and so-called D-stable C*-algebras, respectively. More precisely, we show that a C_0(X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for C*-algebras absorbing the Jiang--Su algebra Z tensorially). Furthermore, we prove that if D is a K_1-injective strongly self-absorbing C*-algebra, then A absorbs D tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a C*-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of Z-absorbing C*-algebras.
It is shown that a strongly self-absorbing C*-algebra is of real rank zero and absorbs the Jiang-Su algebra if it contains a nontrivial projection. We also consider cases where the UCT is automatic for strongly self-absorbing C*-algebras, and K-theoretical ways of characterizing when Kirchberg algebras are strongly self-absorbing.
Let $P \subset A$ be a inclusion of unital C*-algebras and $E\colon A \to P$ be a conditional expectation of index finite type. We introduce a Rokhlin property for $E$ and discuss about $\mathcal{D}$-absorbing proeprty, where $\mathcal{D}$ is a separable, unital, strongly self-absorbing C*-algebra. In this paper we consider permanent properties for strongly self-absorbing property under inclusions of unital C*-algebras with a finite Watatani index.
We give a shorter proof of the fact that the Jiang-Su algebra is strongly self-absorbing. This is achieved by introducing and studying so-called unitarily suspended endomorphisms of generalized dimension drop algebras. Along the way we prove uniqueness and existence results for maps between dimension drop algebras and UHF-algebras.
We introduce and study strongly self-absorbing actions of locally compact groups on C*-algebras. This is an equivariant generalization of a strongly self-absorbing C*-algebra to the setting of C*-dynamical systems. The main result is the following equivariant McDuff-type absorption theorem: A cocycle action $(α,u): G\curvearrowright A$ on a separable C*-algebra is cocycle conjugate to its tensorial stabilization with a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, if and only if there exists an equivariant and unital $*$-homomorphism from $\mathcal{D}$ into the central sequ…
Suppose $A$ is a separable unital $C(X)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra $D$. We show that $A$ and $C(X) \otimes D$ are isomorphic as $C(X)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.
This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C*-algebras. Given a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, we establish permanence properties for the class of separable C*-dynamical systems absorbing $γ$ tensorially up to cocycle conjugacy. Generalizing results of both Toms-Winter and Dadarlat-Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on…
This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable ${\cal Z}$-absorbing C*-algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their $K$-theory is finitely generated and their Elliott invariants are the same.
In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.
Let G be a finite group acting on {1,...,n}. For any C*-algebra A, this defines an action of αof G on A^{\otimes n}. We show that if A tensorially absorbs a UHF algebra of infinite type, the Jiang-Su algebra, or is approximately divisible, then A \times_α G has the corresponding property as well.
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong c…
We prove the title. This characterizes the Jiang-Su algebra Z as the uniquely determined initial object in the category of strongly self-absorbing C*-algebras.
We show that the Fraïssé limit of a category of unital separable $C^*$-algebras which is sufficiently closed under tensor products of its objects and morphisms is strongly self-absorbing, given that it has approximate inner half-flip. We use this connection between Fraïssé limits and strongly self-absorbing $C^*$-algebras to give a self-contained and rather elementary proof for the well known fact that the Jiang-Su algebra is strongly self-absorbing.
In this paper we consider a bootstrap class $\mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$\mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Γ\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly …
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbi…
Say that a separable, unital C*-algebra D is strongly self-absorbing if there exists an isomorphism $ϕ: D \to D \otimes D$ such that $ϕ$ and $id_D \otimes 1_D$ are approximately unitarily equivalent $*$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $\mathcal{O}_2$, $\mathcal{O}_{\infty}$, the UHF algebras of infinite type, the Jiang--Su algebra Z and tensor products of $\Oh_{\infty}$ with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra D we characterise when a separable C*-algebra absorbs D tensorially (i.e., is D-stable), and prove…
In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal D$ is a strongly self-absorbing C*-algebra, and $α: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $α$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. This contains several existing results of similar nature …
We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields …
Let A be a unital separable C*-algebra, and D a K_1-injective strongly self-absorbing C*-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.
We show that the Dixmier-Douady theory of continuous field $C^*$-algebras with compact operators $\mathbb{K}$ as fibers extends significantly to a more general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a strongly self-absorbing C*-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient $K$-theoretical cond…
Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\A…
We give an operator algebraic model for the first group of the unit spectrum $gl_1(KU)$ of complex topological K-theory, i.e. $[X, BGL_1(KU)]$, by bundles of stabilized infinite Cuntz C*-algebras $O_{\infty} \otimes \K$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{I}$-monoid model for the units of $K$-theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-abso…
We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and cr…
In this paper, we introduce a class of non-unital tracial approximation ${\rm C^*}$-algebras. Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips's simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo's tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\rm C^*}$-algebras.
Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…
We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. …
We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0…
We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z. Our main result is formulated in a more general …
It is determined exactly which classifiable C*-algebras have approximately inner flip. The answer includes a number of C*-algebras with torsion in their K-theory, and a number of C*-algebras that are self-absorbing but not strongly self-absorbing.
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γis elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.
For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},δ)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},δ)$-stable if and only…
Let $Λ$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $Λ$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $Λ$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement:…
We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann alg…
