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Partial ∗-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras
Published online by Cambridge University Press: 20 November 2018
Abstract
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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 
$X{{X}^{*}}+{{X}^{*}}X\subset A$ 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.
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