Some properties of nonstandard hulls of Banach algebras

Abstract

We introduce the notion of generalized invertibility, a weak notion of invertibility that makes sense in the nonstandard hull of a Banach algebra. We show that a standard algebra is the closure of its invertible elements exactly when all the elements of its nonstandard hull are generalized invertible. For commutative algebras, we relate the problem of liftability of homomorphisms to the property of approximately multiplicative functionals being near to multiplicative functionals (AMNM property) and we prove a nonstandard characterization of the latter property. Finally we show that a standard superreflexive Banach algebra has the the AMNM property exactly when its nonstandard hull does.