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

Hamana, Masamichi

Canadian journal of mathematics, Tome 58 (2006) no. 6, pp. 1144-1202

Résumé

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 $X{{X}^{*}}+{{X}^{*}}X\subset A$ and a set of certain partial isometries $s$ in the “normalizer” of $A$ in $B$ , and similarly for the map $s\mapsto \text{Ad }s$ 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 $\text{I}{{\text{I}}_{1}}$ 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.

Détail

DOI : 10.4153/CJM-2006-042-0

Mots-clés : 46L05, 46L08, 46L40, 20M18

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