arXiv subjects
Publications and source records attributed to Samuel Evington.
We show that every stable UCT Kirchberg algebra has a principal \'etale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our methods. In particular, we obtain a principal \'etale groupoid model for the Cuntz algebra $\mathcal{O}_\infty$.
We prove that a factorial tracially complete C*-algebra with CPoU has real rank zero and stable rank one. This leads to an essentially complete description of the Cuntz semigroup of these algebras. In particular, the results of this paper hold for the uniform tracial completions of $\mathcal{Z}$-stable C*-algebras.
We compute the dynamic asymptotic dimension of the principal groupoid models for the Cuntz algebras $\mathcal{O}_k$ for $2 \leq k < \infty$ that have arisen from work of Winter and the authors. Our method generalises to a wide class of Deaconu-Renault groupoids. As an application of our results, we prove that $\mathcal{O}_2$ has infinitely many non-conjugate C$^*$-diagonals with Cantor spectrum, and we generalise this result to other Cuntz algebras by combining the main result with work of Kopsacheilis-Winter and Brown-Clark-Sierakowski-Sims.
We prove that there exists a C*-diagonal with Cantor spectrum in the Cuntz algebra $\mathcal{O}_k$ for $2 \le k < \infty$. Our method generalises to an uncountable family of UCT Kirchberg algebras with distinct K-theory. Moreover, we construct principal \'etale groupoid models for these Cuntz algebras and UCT Kirchberg algebras.
We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.
We prove that a C$^*$-algebra $A$ has uniform property $\Gamma$ if the set of extremal tracial states, $\partial_e T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in \partial_e T(A)$, the von Neumann algebra $\pi_\tau(A)''$ arising from the GNS representation has property $\Gamma$.
The uniform tracial completion of a C*-algebra A with compact non-empty trace space T(A) is obtained by completing the unit ball with respect to the uniform 2-seminorm $\|a\|_{2,T(A)}=\sup_{\tau \in T(A)} \tau(a^*a)^{1/2}$. The trace problem asks whether every trace on the uniform tracial completion is the $\|\cdot\|_{2,T(A)}$-continuous extension of a trace on A. We answer this question positively in the case of C*-algebras that tensorially absorb the Jiang-Su algebra, such as those studied in the Elliott classification programme.
We introduce a notion of equivariant $\mathcal{D}$-stability for actions of unitary tensor categories on C$^*$-algebras. We show that, when $\mathcal{D}$ is strongly self-absorbing, equivariant $\mathcal{D}$-stability of an action is equivalent to a unital embedding of $\mathcal{D}$ into a certain subalgebra of Kirchberg's central sequence algebra. We use this to show $\mathcal{Z}$-stability for a large class of AF-actions.
We introduce a new class of operator algebras -- tracially complete C*-algebras -- as a vehicle for transferring ideas and results between C*-algebras and their tracial von Neumann algebra completions. We obtain structure and classification results for amenable tracially complete C*-algebras satisfying an appropriate version of Murray and von Neumann's property gamma for II_1 factors. In a precise sense, these results fit between Connes' celebrated theorems for injective II_1 factors and the unital classification theorem for separable simple nuclear C*-algebras. The theory also underpins arguments for the known parts of the Toms-Winter conjecture.
We classify $^*$-homomorphisms from nuclear $C^*$-algebras into uniform tracial sequence algebras of nuclear $\mathcal Z$-stable $C^*$-algebras via tracial data.
We study the $H^3$ invariant of a group homomorphism $ϕ:G \rightarrow \mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $\mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $\mathrm{Hilb}(G, ω)$ for non-trivial $ω\in H^3(G, \mathbb{T})$ cannot act on $\mathcal{Z}$.
We obtain an improved upper bound for the nuclear dimension of extensions of $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebras. In particular, we prove that the nuclear dimension of a full extension of an $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebra by a stable AF algebra is one.
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.
We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions.
We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C*-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C*-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
We further examine the concept of uniform property Gamma for C*-algebras introduced in our joint work with Winter. In addition to obtaining characterisations in the spirit of Dixmier's work on central sequence in II$_1$ factors, we establish the equivalence of uniform property Gamma, a suitable uniform version of McDuff's property for C*-algebras, and the existence of complemented partitions of unity for separable nuclear C*-algebras with no finite dimensional representations and a compact (non-empty) tracial state space. As a consequence, for C*-algebras as in the Toms-Winter conjecture, the combination of strict comparison and uniform property Gamma is equivalent to Jiang-Su stability. We also show how these ideas can be combined with those of Matui-Sato to streamline Winter's classification-by-embeddings technique.
We prove that Z-stable, simple, separable, nuclear, non-unital C*-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Z-stability for simple, separable, nuclear, non-elementary C*-algebras.
We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.