Annals of Functional Analysis

Commutator ideals in C-crossed products by hereditary subsemigroups

Mamoon Ahmed

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Abstract

Let (G,G+) be a lattice-ordered abelian group with positive cone G+, and let H+ be a hereditary subsemigroup of G+. In previous work, the author and Pryde introduced a closed ideal IH+ of the C-subalgebra BG+ of (G+) spanned by the functions {1x:xG+}. Then we showed that the crossed product C-algebra B(G/H)+×βG+ is realized as an induced C-algebra IndHGˆ(B(G/H)+×τ(G/H)+). In this paper, we prove the existence of the following short exact sequence of C-algebras: 0IH+×αG+BG+×αG+IndHGˆ(B(G/H)+×τ(G/H)+)0. This relates BG+×αG+ to the structure of IH+×αG+ and B(G/H)+×βG+. We then show that there is an isomorphism ι of BH+×αH+ into BG+×αG+. This leads to nontrivial results on commutator ideals in C-crossed products by hereditary subsemigroups involving an extension of previous results by Adji, Raeburn, and Rosjanuardi.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 3 (2019), 370-380.

Dates
Received: 12 September 2018
Accepted: 9 December 2018
First available in Project Euclid: 6 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1565078422

Digital Object Identifier
doi:10.1215/20088752-2018-0036

Mathematical Reviews number (MathSciNet)
MR3989182

Zentralblatt MATH identifier
07089124

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]

Keywords
$C^{*}$-algebra lattice-ordered group crossed product commutator ideal

Citation

Ahmed, Mamoon. Commutator ideals in $C^{*}$ -crossed products by hereditary subsemigroups. Ann. Funct. Anal. 10 (2019), no. 3, 370--380. doi:10.1215/20088752-2018-0036. https://projecteuclid.org/euclid.afa/1565078422


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References

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