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 the Razak–Jacelon algebra 𝒲2 is isomorphic to a semidirect product of Out (𝒲2) with R + × . 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 that if A is a separable, simple, nuclear, stably projectionless C*-algebra with a unique tracial state (and no unbounded trace) and has strict comparison, then the following sequence is exact: 1 → Out ( A ) → Pic ( A ) → ℱ ( A ) → 1 , where ℱ(A) is the fundamental group of A.