An extension of the Gleason-Kahane-Żelazko theorem: A possible approach to Kaplansky's problem
Let \(A\) and \(B\) be two complex unital Banach algebras with \(B\) semisimple. In this nice survey, the authors are concerned with Kaplanski's problem that asks under which conditions every unital, invertibility preserving linear map \(\phi: A\to B\) is a Jordan homomorphism. Note that a linear map \(\phi:A\to B\) is a Jordan homomorphism if \(\phi(x^2)= \phi(x)^2 \) for all \(x\in A\). It is proved that if, additionally, \(\phi\) is a surjection that locally preserves commutativity -- meaning that \([\phi(x^2),\phi(x)]=0\), where \([x,y]=xy-yx\) is the Lie product --, then the answer is positive. As a corollary, they obtain a new proof of the Marcus--Purves theorem [\textit{M.\,Marcus} and \textit{R.\,Purves}, Can.\ J.\ Math.\ 11, 383--396 (1959; Zbl 0086.01704)] that any unital linear map \(\phi: M_n\to M_n\) preserving invertibility is a Jordan homomorphism. Here, as usual, \(M_n\) is the algebra of all \(n\times n\) complex matrices.
scientific article; zbMATH DE number 5307074
scientific article