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

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http://dx.doi.org/10.4153/CJM-2006-042-0
Canad. J. Math. 58(2006), 1144-1202
Published:2006-12-01
Printed: Dec 2006
• Masamichi Hamana
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## Abstract

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 - General theory of $C^*$-algebras 46L08 - $C^*$-modules 46L40 - Automorphisms 20M18 - Inverse semigroups
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