Picard groups of certain stably projectionless C*-algebras

Abstract

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.

20 pages, to appear in J. London Math. Soc

References in corpus (14)

• The stable and the real rank of Z-absorbing C*-algebras
• K-Theory for operator algebras. Classification of C${}^{\ast }$-algebras
• Strict comparison and Z-absorption of nuclear C*-algebras
• Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras
• Z-stability of crossed products by strongly outer actions
• Regularity for stably projectionless, simple C*-algebras
• Characterizing classifiable AH algebras
• Fundamental group of simple $C^{\ast }$-algebras with unique trace
• Remarks on Z-stable projectionless C*-algebras
• Fundamental group of simple $C^{\ast }$-algebras with unique trace III
• A simple, monotracial, stably projectionless C*-algebra
• Pullbacks, $C(X)$-algebras, and their Cuntz semigroup
• Fundamental group of simple $C^{\ast }$-algebras with unique trace II
• A note on trace scaling actions and fundamental groups of $C^{\ast }$-algebras

Cited by in corpus (11)

• Nuclear dimension, Z-stability, and algebraic simplicity for stably projectionless C*-algebras
• Nuclear dimension and Z-stability of non-simple C*-algebras
• Equivariant property (SI) revisited
• Trace scaling automorphisms of the stabilized Razak-Jacelon algebra
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• A characterization of the Razak-Jacelon algebra
• Finite group actions on certain stably projectionless C*-algebras with the Rohlin property
• Fundamental group of finite von Neumann algebras with finite dimensional normal trace space
• Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra
• Rohlin actions of finite groups on the Razak-Jacelon algebra
• Fundamental group of $AF$-algebras with finite dimensional trace space