Partial *-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras

Abstract

AbstractFor monotone complete C*-algebras A ⊂ B with A contained in B as a monotone closed C*-subalgebra, the relation X = AsA gives a bijection between the set of all monotone closed linear subspaces X of B such that AX + XA ⊂ X and XX* + X*X ⊂ A and a set of certain partial isometries s in the “normalizer” of A in B, and similarly for the map s ⟼ Ad s between the latter set and a set of certain “partial *-automorphisms” of A. We introduce natural inverse semigroup structures in the set of such X's and the set of partial *-automorphisms of A, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough B the homomorphism becomes surjective and all the partial *-automorphisms of A are realized via partial isometries in B. In particular, the inverse semigroup associated with a type II1 von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the C*-algebra version of these results.