On Defining AW*-algebras and Rickart C*-algebras

Kazuyuki Saito; J. D. Maitland Wright

doi:10.1093/qmath/hav015

On Defining AW-algebras and Rickart C-algebras

Kazuyuki Saito
, J. D. Maitland Wright

Mathematical Science

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

6 Downloads (Pure)

Abstract

Let A be a C∗-algebra. It is shown that A is an AW∗-algebra if, and only if, each maximal abelian self-adjoint (m.a.s.a.) subalgebra of A is monotone complete. An analogous result is proved for Rickart C∗-algebras; a C∗-algebra is a Rickart C∗-algebra if, and only if, it is unital and each m.a.s.a. subalgebra of A is monotone σ-complete.

Original language	English
Pages (from-to)	979-989
Number of pages	11
Journal	Quarterly Journal of Mathematics
Volume	66
Issue number	3
Early online date	20 May 2015
DOIs	https://doi.org/10.1093/qmath/hav015
Publication status	Published - 20 May 2015

Bibliographical note

Acknowledgements
It is a pleasure to thank Dr A. J. Lindenhovius, whose perceptive questions triggered this paper.

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10.1093/qmath/hav015Licence: Unspecified

https://arxiv.org/abs/1501.02434Licence: Unspecified

Cite this

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title = "On Defining AW*-algebras and Rickart C*-algebras",

abstract = "Let A be a C∗-algebra. It is shown that A is an AW∗-algebra if, and only if, each maximal abelian self-adjoint (m.a.s.a.) subalgebra of A is monotone complete. An analogous result is proved for Rickart C∗-algebras; a C∗-algebra is a Rickart C∗-algebra if, and only if, it is unital and each m.a.s.a. subalgebra of A is monotone σ-complete.",

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N2 - Let A be a C∗-algebra. It is shown that A is an AW∗-algebra if, and only if, each maximal abelian self-adjoint (m.a.s.a.) subalgebra of A is monotone complete. An analogous result is proved for Rickart C∗-algebras; a C∗-algebra is a Rickart C∗-algebra if, and only if, it is unital and each m.a.s.a. subalgebra of A is monotone σ-complete.

AB - Let A be a C∗-algebra. It is shown that A is an AW∗-algebra if, and only if, each maximal abelian self-adjoint (m.a.s.a.) subalgebra of A is monotone complete. An analogous result is proved for Rickart C∗-algebras; a C∗-algebra is a Rickart C∗-algebra if, and only if, it is unital and each m.a.s.a. subalgebra of A is monotone σ-complete.

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