Automorphisms of Certain Simple C*-algebras

Abstract

It has been observed by M. Takesaki (private communication) that for the C*-algebras 0 _n there is a bijective correspondence between unitaries in 0 _n and endomorphisms that leave the unit fixed. We use this idea to study automorphisms of 0 _n. The group Aut 0 _n is shown to contain a maximal abelian subgroup T (that is canonically associated with the construction of 0 _n from a topological Markov.chain; cf. the paper by W. Krieger and the author: A class of C*-algebras and topological Markov chains). This maximal com- mutative subgroup of Aut 0 _n is similar to a maximal torus in a semisimple Lie group in that it is an inductive limit of tori, and its Weyl group, i.e. the quotient of its normalizer in Aut 0 _n by T itself is discrete.

Then one-parameter subgroups of Aut 0 _n that are contained in T, are studied. In particular, one class of such groups, related to Bernoulli-shifts in 0 _n is shown to have unique KMS-states. On the other hand, (unlike to the von Neumann algebra case) any two different one-parameter groups in this class are not exterior equivalent, i.e. do not just differ by a unitary cocycle. Finally, automorphisms in N(T) and their action on a certain class of pure states of 0 _n, related to T, are studied.