Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups

Authors: Sriwulan Adji, Marcelo Laca, May Nilsen and Iain Raeburn
Journal: Proc. Amer. Math. Soc. 122 (1994), 1133-1141
MSC: Primary 46L55; Secondary 46L05, 47B35, 47D03
MathSciNet review: 1215024
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Abstract: Let $ {\Gamma ^ + }$ be the positive cone in a totally ordered abelian group $ \Gamma $. We construct crossed products by actions of $ {\Gamma ^ + }$ as endomorphisms of $ {C^ \ast }$-algebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the $ {C^\ast}$-algebras generated by two semigroups V, $ W:{\Gamma ^ + } \to B(H)$ of nonunitary isometries are canonically isomorphic, thus giving a new, self-contained proof of a theorem of Murphy, which includes earlier results of Coburn and Douglas.

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Keywords: $ {C^ \ast }$-algebra, endomorphism, ordered group, covariant representation, crossed product, semigroup of isometries, Toeplitz algebra
Article copyright: © Copyright 1994 American Mathematical Society