## Taiwanese Journal of Mathematics

### THE BANACH ALGEBRA ${\cal F}(S,T)$ AND ITS AMENABILITY OF COMMUTATIVE FOUNDATION $*$-SEMIGROUPS $S$ AND $T$

#### Abstract

In the present paper we shall first introduce the notion of the algebra ${\cal F}(S,T)$ of two topological $*$-semigroups $S$ and $T$ in terms of bounded and weakly continuous $*$-representations of $S$ and $T$ on Hilbert spaces. In the case where both $S$ and $T$ are commutative foundation $*$-semigroups with identities it is shown that ${\cal F}(S,T)$ is identical to the algebra of the Fourier transforms of bimeasures in $BM(S^*,T^*)$, where $S^*$ ($T^*$, respectively) denotes the locally compact Hausdorff space of all bounded and continuous $*$-semicharacters on $S$ ($T$, respectively) endowed with the compact open topology. This result has enabled us to make the bimeasure Banach space $BM(S^*,T^*)$ into a Banach algebra. It is also shown that the Banach algebra ${\cal F}(S,T)$ is amenable and $K\big(\sigma (\overline{{\cal F}(S,T)})\big)$ is a compact topological group, where $\sigma (\overline {{\cal F}(S,T)})$ denotes the spectrum of the commutative Banach algebra $\overline{{\cal F}(S,T)}$ as a closed subalgebra of wap $(S \times T)$, the Banach algebra of weakly almost periodic continuous functions on $S \times T$.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 787-802.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406616

Digital Object Identifier
doi:10.11650/twjm/1500406616

Mathematical Reviews number (MathSciNet)
MR2892913

Zentralblatt MATH identifier
1244.43003

#### Citation

Bami, M. Lashkarizadeh. THE BANACH ALGEBRA ${\cal F}(S,T)$ AND ITS AMENABILITY OF COMMUTATIVE FOUNDATION $*$-SEMIGROUPS $S$ AND $T$. Taiwanese J. Math. 16 (2012), no. 2, 787--802. doi:10.11650/twjm/1500406616. https://projecteuclid.org/euclid.twjm/1500406616

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