a note on isomorphism rigidity for banach algebras defined by multipliers

‎let $a$ be a banach algebra and $t$ a bounded two-sided multiplier on $a$‎. ‎we study the banach algebra $a_t$ with product‎ ‎$astar_t b=at(b)$ and ask what follows from the existence of a banach algebra‎ ‎isomorphism $a_tcong a$‎. ‎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: ‎$a_tcong a$ if and only if $t$ is invertible in the multiplier algebra‎. ‎equivalently‎, ‎if $t(a)=ma$ with $m=t(1)$‎, ‎then $a_tcong a$ iff $m$ is invertible in $a$‎. ‎we also derive corresponding consequences for semigroup algebras‎. ‎for a discrete semigroup $s$ and $tin s$‎, ‎the isomorphism‎ ‎$ell^1(s)_{l_t}cong ell^1(s)$ forces injectivity of the left translation‎ ‎$smapsto ts$ whenever $ell^1(s)$ is right faithful and‎, ‎under the same multiplier hypotheses‎, ‎yields a natural obstruction in terms of the ideal $sts$‎.