Picard groups of certain stably projectionless C*-algebras

TL;DR

This paper computes the Picard groups of certain stably projectionless C*-algebras, revealing their structure and establishing conditions under which Z-stability and strict comparison are equivalent, with implications for their automorphism groups.

Contribution

It provides explicit calculations of Picard groups for specific stably projectionless C*-algebras and links Z-stability with strict comparison in this context.

Findings

Picard group of Razak-Jacelon algebra W_2 is isomorphic to a semidirect product of Out(W_2) with R_+^×

Z-stability and strict comparison are equivalent for certain nuclear stably projectionless C*-algebras

Exact sequence relating Out(A), Pic(A), and the fundamental group for algebras with unique trace

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.