English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of Calgebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the C* -algebras generated by two semigroups V, W : P" —» 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. Introduction Let T be a totally ordered abelian group with positive cone T+. Murphy showed in [6] that the C*-algebras generated by two semigroups V, W: T+ -+ 5(77) of nonunitary isometries are canonically isomorphic [6, Theorem 2.9], generalising well-known theorems of Coburn [3] for the case Y = Z, and of Douglas [5] for subgroups of R. His proof involved an analysis of the ideal structure of the ordered group T, and used results of Olesen and Pedersen [9, §8.11] on the primitivity of certain crossed products. Here we obtain a characterization of faithful representations of crossed products by semigroups of endomorphisms, and use it to give a short, self-contained proof of Murphy's theorem. In § 1, we discuss our notion of crossed products by semigroups of endomorphisms of C*-algebras, which is based on ideas of Stacey [11]. Our characterization of faithful representations in Theorem 1.2 extends that for crossed products by single endomorphisms in [1]; we prove a more general theorem than we need later, because we want to fill a gap in [1] — see the remarks preceding Lemma 1.3. In §2, we show that the universal C*-algebra C*(r+) for isometric representations of T+ is such a crossed product (Proposition 2.2). To prove the main result, Theorem 2.4, we need to verify that the hypotheses of Theorem 1.2 hold when the isometric representation consists of nonunitary isometries, and we do this by adapting an argument from [4]. It follows immediately that the Toeplitz algebra of an ordered group is isomorphic to C*(T+). Received by the editors March 17, 1993. 1991 Mathematics Subject Classification. Primary 46L55; Secondary 06F15, 47B35.

Crossed Products by Semigroups of Endomorphisms and the Toeplitz Algebras of Ordered Groups