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

Open Access article
 Printed: Dec 2006
  • Masamichi Hamana
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For monotone complete $C^*$-algebras $A\subset B$ 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 $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the ``normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ 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 ${\rm II}_1$ 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.
MSC Classifications: 46L05, 46L08, 46L40, 20M18 show english descriptions General theory of $C^*$-algebras
Inverse semigroups
46L05 - General theory of $C^*$-algebras
46L08 - $C^*$-modules
46L40 - Automorphisms
20M18 - Inverse semigroups

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