Picard groups of certain stably projectionless C*-algebras

We compute Picard groups of several nuclear and non-nuclear simple stably projectionless C*-algebras. In particular, the Picard group of Razak-Jacelon algebra W_2 is isomorphic to a semidirect product of Out(W_2) with R_+^\times. Moreover, for any separable simple nuclear stably projectionless C*-algebra with a finite dimensional lattice of densely defined lower semicontinuous traces, we show that Z-stability and strict comparison are equivalent. (This is essentially based on the result of Matui and Sato, and Kirchberg's central sequence algebras.) This shows if A is a separable simple nuclear stably projectionless C*-algebra with a unique tracial state (and no unbounded trace) and has strict comparison, the following sequence is exact: [{CD} {1} @>>> \mathrm{Out}(A) @>>> \mathrm{Pic}(A) @>>> \mathcal{F}(A) @>>> {1} {CD}] where $\mathcal{F}(A)$ is the fundamental group of A.Comment: 20 pages, to appear in J. London Math. So

Z-stability, finite dimensional tracial boundaries and continuous rank functions

We observe that a recent theorem of Sato, Toms–White–Winter and Kirchberg–Rørdam also holds for certain nonunital C*-algebras. Namely, we show that an algebraically simple, separable, nuclear, nonelementary C*-algebra with strict comparison, whose cone of densely finite traces has as a base a Choquet simplex with compact, finite dimensional extreme boundary, and which admits a continuous rank function, tensorially absorbs the Jiang–Su algebra Z

Z is universal

We use order zero maps to express the Jiang-Su algebra Z as a universal C*-algebra on countably many generators and relations, and we show that a natural deformation of these relations yields the stably projectionless algebra W studied by Kishimoto, Kumjian and others. Our presentation is entirely explicit and involves only *-polynomial and order relations.Comment: 12 page

Equivariant property (SI) revisited

We revisit Matui-Sato's notion of property (SI) for C*-algebras and C*-dynamics. More specifically, we generalize the known framework to the case of C*-algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly non-unital C*-algebras, in particular to establish equivariant Jiang-Su stability. Our main result is an extension of an observation by Sato: For any countable amenable group $\Gamma$ and any non-elementary separable simple nuclear C*-algebra $A$ with strict comparison, every $\Gamma$-action on $A$ has equivariant property (SI). A more general statement involving relative property (SI) for inclusions into ultraproducts is proved as well. As a consequence we show that if $A$ also has finitely many rays of extremal traces, then every $\Gamma$-action on $A$ is equivariantly Jiang-Su stable. We moreover provide applications of the main result to the context of strongly outer actions, such as a generalization of Nawata's classification of strongly outer automorphisms on the (stabilized) Razak-Jacelon algebra.Comment: v4 36 pages; this version has been accepted at Analysis & PD