Annals of Functional Analysis

Spectral properties of the Lau product A×θB of Banach algebras

Prakash A. Dabhi and Savan K. Patel

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Abstract

Let A and B be commutative Banach algebras. Then a multiplicative linear functional θ on B induces a multiplication on the Cartesian product space A×B given by (a,b)(c,d)=(ac+θ(d)a+θ(b)c,bd) for all (a,b),(c,d)A×B. We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on θ.

Article information

Source
Ann. Funct. Anal. (2018), 12 pages.

Dates
Received: 7 March 2017
Accepted: 31 May 2017
First available in Project Euclid: 7 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1512637229

Digital Object Identifier
doi:10.1215/20088752-2017-0048

Subjects
Primary: 46J05: General theory of commutative topological algebras
Secondary: 46Jxx: Commutative Banach algebras and commutative topological algebras [See also 46E25]

Keywords
commutative Banach algebra Gelfand space Shilov boundary unique uniform norm property spectral extension property multiplicative Hahn–Banach property

Citation

Dabhi, Prakash A.; Patel, Savan K. Spectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebras. Ann. Funct. Anal., advance publication, 7 December 2017. doi:10.1215/20088752-2017-0048. https://projecteuclid.org/euclid.afa/1512637229


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