Jiang-Su Algebra Research Articles

On dualities of actions

We introduce the weak tracial approximate representability of a discrete group action on a unital C⁎-algebra which possibly has no projections like the Jiang-Su algebra Z. Then we show a duality between the weak tracial Rokhlin property and the weak tracial approximate representability. More precisely, when G is a finite abelian group and α:G→Aut(A) is a group action on a unital simple infinite dimensional C⁎-algebra, we prove that•α has the weak tracial Rokhlin property if and only if αˆ has the weak tracial approximate representability.•α has the weak tracial approximate representability if and only if αˆ has the weak tracial Rokhlin property.

The Calkin algebra, Kazhdan's property (T), and strongly self‐absorbing C∗$\mathrm{C}^*$‐algebras

AbstractIt is well known that the relative commutant of every separable nuclear ‐subalgebra of the Calkin algebra has a unital copy of Cuntz algebra . We prove that the Calkin algebra has a separable ‐subalgebra whose relative commutant has no simple, unital, and noncommutative ‐subalgebra. On the other hand, the corona of every stable, separable ‐algebra that tensorially absorbs the Jiang–Su algebra has the property that the relative commutant of every separable ‐subalgebra contains a unital copy of . Analogous result holds for other strongly self‐absorbing ‐algebras. As an application, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra , any other Kirchberg algebra, or even the corona of the stabilization of any unital, ‐stable ‐algebra.

Between reduced powers and ultrapowers

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by the Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists a nonprincipal ultrafilter \mathcal{U} on \mathbb{N} such that for every countable (or separable metric) structure B in a countable language the quotient map from the reduced power associated with the Fréchet ideal onto an ultrapower has a right inverse. While the transfer principle is proved without appealing to additional set-theoretic axioms, the conclusion of the latter theorem relies on the Continuum Hypothesis and it is independent of the standard axioms of set theory. We also prove that in the category of C^* -algebras, tensoring with the C^* -algebra of all continuous functions on the Cantor space preserves elementary equivalence. As a side note, neither the Jiang–Su algebra \mathcal{Z} nor any UHF algebra share this property.

Generators in $\mathcal{Z}$-stable $C^*$-algebras of real rank zero

We show that every separable C^* -algebra of real rank zero that tensorially absorbs the Jiang–Su algebra contains a dense set of generators. It follows that, in every classifiable, simple, nuclear C^* -algebra, a generic element is a generator.

On the Bundle of KMS State Spaces for Flows on a $${\mathcal {Z}}$$-Absorbing C*-Algebra

We obtain three results: 1) Every compact simplex bundle with exactly one point in the fiber over 0 is the KMS bundle of a periodic flow on the Jiang-Su algebra. 2) Let A be a separable unital C*-algebra with a unique trace state. Suppose that A tensorially absorbs the Jiang-Su algebra. The (weak) cocycle-conjugacy classes of flows that are not approximately inner are uncountable. 3) Let B be a separable, simple, unital, purely infinite and nuclear C*-algebra in the UCT class. Assume that the K1 group of B is torsion free. Every proper simplex bundle with empty fiber over 0 is the KMS bundle of a periodic flow on B.

The Jiang-Su algebra is strongly self-absorbing revisited

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.

Rokhlin-type properties, approximate innerness and Z-stability

We investigate connections between actions on separable C∗-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. We show that if A admits an approximately inner group action with finite Rokhlin dimension with commuting towers then A is Z-stable. We obtain analogous results for tracial version of the Rokhlin property and approximate innerness. Going beyond approximate innerness, for actions of a single automorphism which have the Rokhlin property and are almost periodic in a suitable sense, the crossed product absorbs Z even when the original algebra does not.

Group amenability and actions on [formula omitted]-stable C*-algebras

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 F2, then there exist uncountably many, non-cocycle conjugate strongly outer actions of G on any tracial, unital, separable C*-algebra that absorbs tensorially the Jiang-Su algebra. Similar conclusions hold for outer actions on McDuff II1 factors. We moreover show that G is amenable if and only if the Bernoulli shift on any finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods are inspired by Jones' work [27], and consist in a careful study of weak containment for the Koopman representations of certain generalized Bernoulli actions.

Stabilising uniform property $\Gamma $

We introduce stabilised property Gamma, a C*-algebraic variant of property Gamma which is invariant under stable isomorphism. We then show that simple separable nuclear C*-algebras with stabilised property Gamma and $\mathrm{Cu}(A) \cong \mathrm{Cu}(A \otimes \mathcal{Z})$ absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially.

Strongly self‐absorbing C∗‐algebras and Fraïssé limits

We show that the Fra\"iss\'e 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 Fra\"iss\'e 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.

ON A GENERALIZED FRAÏSSÉ LIMIT CONSTRUCTION AND ITS APPLICATION TO THE JIANG–SU ALGEBRA

AbstractIn this paper, we present a version of Fraïssé theory for categories of metric structures. Using this version, we show that every UHF algebra can be recognized as a Fraïssé limit of a class of C*-algebras of matrix-valued continuous functions on cubes with distinguished traces. We also give an alternative proof of the fact that the Jiang–Su algebra is the unique simple monotracial C*-algebra among all the inductive limits of prime dimension drop algebras.

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

In this paper we show that for an almost finite minimal ample groupoid G, its reduced \(C^*\)-algebra \(C_r^*(G)\) has real rank zero and strict comparison even though \(C_r^*(G)\) may not be nuclear in general. Moreover, if we further assume G being also second countable and non-elementary, then its Cuntz semigroup \({\mathrm{Cu}}(C_r^*(G))\) is almost divisible and \({\mathrm{Cu}}(C_r^*(G))\) and \({\mathrm{Cu}}(C_r^*(G)\otimes {\mathcal {Z}})\) are canonically order-isomorphic, where \({\mathcal {Z}}\) denotes the Jiang-Su algebra.

Generalized quasidiagonality for extensions

We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a K-theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale C∗-algebras, including certain continuous-scale hereditary C∗-subalgebras of the stabilized Jiang–Su algebra.

Regularity of Villadsen algebras and characters on their central sequence algebras

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

The tracial Rokhlin property for actions of amenable groups on $C^*$-algebras

In this paper, we present a definition of the tracial Rokhlin property for (cocyclic) actions of countable discrete amenable groups on simple $C^*$-algebras, which generalize Matui and Sato's definition. We show that generic examples, like Bernoulli shift on the tensor product of copies of the Jiang-Su algebra, has the weak tracial Rokhlin property, while it is shown in {Hirshberg 2014} that such an action does not have finite Rokhlin dimension. We further show that forming a crossed product from actions with the tracial Rokhlin property preserves the class of $C^*$-algebras with real rank $0$, stable rank $1$ and has strict comparison for pro\-jec\-tions, generalizing the structural results in {Osaka 2006}. We use the same idea of the proof with significant simplification. In another joint paper with Chris Phillips and Joav Orovitz, we shall show that pureness and $\mathcal{Z} $-stability could be preserved by crossed product of actions with the weak tracial Rokhlin property. The combination of these results yields an application to the classification program, which is discussed in the aforementioned paper. These results indicate that we have the correct definition of tracial Rokhlin property for actions of general countable discrete amenable groups.

Characterization of 2-Local Derivations and Local Lie Derivations on Some Algebras

We prove that each 2-local derivation from the algebra Mn(A ) (n > 2) into its bimodule Mn(M) is a derivation, where A is a unital Banach algebra and M is a unital A -bimodule such that each Jordan derivation from A into M is an inner derivation, and that each 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, the Jiang–Su algebra, and UHF algebras.

Spectral triples on the Jiang-Su algebra

We construct spectral triples on a class of particular inductive limits of matrix-valued function algebras. In the special case of the Jiang-Su algebra, we employ a particular AF-embedding.

The C∗-algebra of a minimal homeomorphism of zero mean dimension

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, \sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang-Su algebra.

Strongly self-absorbing C⁎-dynamical systems, III

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 cocycle conjugacy.Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly Z-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi–Matui, we obtain the following main result. If G is a torsion-free abelian group and D is one of the known strongly self-absorbing C⁎-algebras, then strongly outer G-actions on D are unique up to (very strong) cocycle conjugacy. This is new even for Z3-actions on the Jiang–Su algebra.

THE JIANG–SU ALGEBRA AS A FRAÏSSÉ LIMIT

AbstractIn this paper, we give a self-contained and quite elementary proof that the class of all dimension drop algebras together with their distinguished faithful traces forms a Fraïssé class with the Jiang–Su algebra as its limit. We also show that the UHF algebras can be realized as Fraïssé limits of classes of C*-algebras of matrix-valued continuous functions on [0,1] with faithful traces.