December 26, 2005
Leiba Rodman
Title: Linear preservers of generalized invertibility
A basic result in the theory of Banach algebras, known as the Gleason-Kahane-\v{Z}elazko theorem and proved in the late 1960's, asserts that a unital linear functional on a complex unital Banach algebra that does not vanish on invertible elements, is necessarily multiplicative. This is a prototype of many results obtained in recent two decades or so on the following general theme: Linear maps on various Banach algebras that preserve certain properties are necessarily multiplicative in some sense. In the case of the Gleason-Kahane-\v{Z}elazko theorem the property that is preserved is invertibility. In the talk some recent results in this direction will be reviewed. In particular, bijective continuous unital linear maps on $B(H)$ (the Banach algebra of linear bounded operators on an infinite dimensional separable Hilbert space) that preserve generalized invertibility in both directions will be characterized in terms of multiplicative properties, a result obtained jointly with Mbekhta and \v{S}emrl.