On local automorphisms of group algebras of compact groups

Abstract

We show that with few exceptions every local isometric automorphism of the group algebra LP(G) of a compact metric group G is an isometric automorphism. In the last decade considerable work has been done concerning certain local maps of operator algebras. The originators of this research are Kadison and Larson. In [Kad], Kadison studied local derivations on a von Neumann algebra R. A continuous linear map on R7 is called a local derivation if it agrees with some derivation at each point (the derivations possibly differring from point to point) in the algebra. This investigation was motivated by the study of Hochschild cohomology of operator algebras. It was proved in [Kad] that in the above setting, every local derivation is a derivation. Independently, Larson and Sourour proved in [LaSo] that the same conclusion holds true for local derivations of the full operator algebra (3(X), where X is a Banach space. For other results on local derivations of various algebras see, for example, [Bre, BrSel, Cri, Shu, ZhXi]. Besides derivations, there is at least one additional very important class of transformations on Banach algebras which certainly deserves attention. This is the group of automorphisms. In [Lar, Some concluding remarks (5), p. 298], from the view-point of reflexivity, Larson raised the problem of local automorphisms (the definition should be self-explanatory) of Banach algebras. In his joint paper with Sourour [LaSo], it was proved that if X is an infinite dimensional Banach space, then every surjective local automorphism of B(X) is an automorphism (see also [BrSel]). For a separable infinite dimensional Hilbert space H, it was shown in [BrSe2] that the above conclusion holds true without the assumption on surjectivity, i.e. every local automorphism of 3B(JC) is an automorphism. For other results on local automorphisms of various operator algebras, we refer to [BaMo, Mol2, Mol3, Mol4]. Received by the editors March 4, 1998. 1991 Mathematics Subject Classification. Primary 43A15, 43A22, 46H99.