On a conjecture on algebras that are locally embeddable into finite dimensional algebras

Abstract

The notion of an algebra that is locally embeddable into finite dimensional algebras (LEF) and the notion of an LEF group was introduced by Gordon and Vershik in [GoVe]. M. Ziman proved in [Zi] that the group algebra of a group $G$ is an LEF algebra if and only if $G$ is an LEF group. He conjectured that an algebra generated as a vector space by a multiplicative subgroup $G$ of its invertible elements is an LEF algebra if and only if $G$ is an LEF group. In this paper we give a characterization of the invertible elements of an LEF algebra and use it to construct a counterexample to this conjecture.