On norm preserving conditions for local automorphisms of commutative banach algebras

Abstract

Many studies on preserver problems have been focusing on linear preserver problems in matrix theory. Kadison and Sourour showed that the local derivations of Von Neumann algebras are continous linear maps which coincide with some derivation at each point in the algebra over the field of complex numbers. Most of the studies have been focusing on the spectral norm preserver and rank preserver problems of linear maps on matrix algebras but not on norm preserver problems for local automorphism of commutative Banach algebras. In this study, we have investigated properties of local automorphism of commutative Banach algebras and determined their norms. The objectives of the study have been to: Investigate properties of local automorphisms of commutative Banach algebras; establish norm preserving conditions for local automorphisms of commutative Banach algebras and determine the norms of local automorphisms of commutative Banach algebras. The methodology involved the concept of local automorphisms and derivations introduced by Kadison and Sourour. We used Hahn-Banach extension theorem, Gelfand transform and the results developed by Molnar to investigate the preserver problems of local automorphisms. The results obtained show that local automorphisms are linear, inner, bounded, continous and their groups are algebraically reflexive. Moreover, the results on norms indicate that kφ(x) + φ(y)k = kxk + kyk and kφx(y)k = 2kyk. The results obtained in this study are useful in the applications of operator algebras and quantum mechanics.