States and invariant maps of operator algebras

Abstract

Recently a great deal of progress has been made in the study of groups of automorphisms of operator algebras and states invariant under the automorphisms. We are then given a C*-algebra Q!, a group G of automorphisms of GY satisfying certain requirements and want to draw conclusions about the invariant states and the fixed point algebra in LY under G. The present paper is concerned both with this problem and the converse one. In the latter case we are given a state p of a C*-algebra O? and want to find a group G of auto- morphisms of G! with respect to which p is invariant, such that the above mentioned theory is applicable. In some cases we find a natural abelian subalgebra 9 of OZ, which is the fixed points of a group G and a canonical G-invariant projection @ of U onto 9 such that p = (p ) 9) 0 @. Then we can prove theorems on integral decompo- sition of states. The main portion of this paper rests largely on the results of KovPcs and Sziics [13]. Given a von Neumann algebra 9 and a group G of automorphisms (i.e. *-automorphisms) of 9, we say .J% is G- finite if there is a family 5 of normal G-invariant states of 9 such that if A is a nonzero positive operator in 9 then for some p in F, p(A) # 0. We shall for simplicity say that