A note on isomorphism rigidity for Banach algebras defined by multipliers

Document Type : Original Article

Author

Department of Mathematics, ST.C., Islamic Azad University, Tehran, Iran

10.22091/maa.2026.16884.1063

Abstract

‎Let $\A$ be a Banach algebra and $T$ a bounded two-sided multiplier on $\A$‎. ‎We study the Banach algebra $\A_T$ with product‎ ‎$a\star_T b=aT(b)$ and ask what follows from the existence of a Banach algebra‎ ‎isomorphism $\A_T\cong \A$‎. ‎For a right faithful Banach algebra‎, ‎such an isomorphism forces $T$ to be injective‎. ‎If‎, ‎moreover‎, ‎$\cl{\spn(\A^2)}=\A$‎, ‎it also forces $\cl{\A T(\A)}=\A$‎. ‎In the unital case‎, ‎these necessary conditions become a complete characterization‎: ‎$\A_T\cong \A$ if and only if $T$ is invertible in the multiplier algebra‎. ‎Equivalently‎, ‎if $T(a)=ma$ with $m=T(1)$‎, ‎then $\A_T\cong \A$ iff $m$ is invertible in $\A$‎. ‎We also derive corresponding consequences for semigroup algebras‎. ‎For a discrete semigroup $S$ and $t\in S$‎, ‎the isomorphism‎ ‎$\ell^1(S)_{L_t}\cong \ell^1(S)$ forces injectivity of the left translation‎ ‎$s\mapsto ts$ whenever $\ell^1(S)$ is right faithful and‎, ‎under the same multiplier hypotheses‎, ‎yields a natural obstruction in terms of the ideal $StS$‎.

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References

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Measure Algebras and Applications

Articles in Press, Accepted Manuscript
Available Online from 18 August 2026

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