## Annals of Functional Analysis

### Spectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebras

#### Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times\mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta(d)a+\theta(b)c,bd)$ for all $(a,b),(c,d)\in\mathcal{A}\times\mathcal{B}$. 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 $\theta$.

#### Article information

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

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

https://projecteuclid.org/euclid.afa/1512637229

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

#### 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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