The Automorphism group of a simple tracially AI algebra

Abstract

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is totally disconnected. Another consequence of our results is the following: Suppose $A$ is the transformation group \cstar-algebra of a minimal Furstenberg transformation $(\mbb{T}^{n}, h_{n})$ with a unique $h_n$-invariant probability measure on $\mbb{T}^{n}$. Then the automorphism group of $A$ is an extension of a simple topological group by the discrete group $\mrm{Aut} (\totalk(A))_{+,1}$.

35 pages

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Cited by in corpus (2)

• Automorphisms of Cuntz-Krieger algebras
• Commutative C*-subalgebras of simple stably finite C*-algebras with real rank zero