On a conjecture on algebras that are locally embeddable into finite dimensional algebras.
type: Item
instance of: scholarly article
title: On a conjecture on algebras that are locally embeddable into finite dimensional algebras.
zbMATH Open document ID: 1085.16024
author: Kira Samol
author: Achim Tresch
published in: Illinois Journal of Mathematics
publication date: 2005-03-01T00:00:00Z
review text: All algebras are associative and over a field \(K\). An algebra \(A\) is called LEF algebra if for any subspace \(M\subset A\), \(\dim M<\infty\), there exist a partial algebra monomorphism \(\varphi\) of \(M\) into a finite dimensional algebra (in particular, \(\varphi(xy)=\varphi(x)\varphi(y)\) for \(x,y,xy\in M\)). Analogously, a group \(G\) is called LEF group if for any finite subset \(M\subset G\) there exists a partial group monomorphism of \(M\) into a finite group. These definitions are due to \textit{E. I. Gordon} and \textit{A. M. Vershik} [St. Petersbg. Math. J. 9, No. 1, 49-67 (1998); translation from Algebra Anal. 9, No. 1, 71-97 (1997; Zbl 0898.20016)]. \textit{E. Ziman} [Ill. J. Math. 46, No. 3, 837-838 (2002; Zbl 1025.20001)] proved that a group algebra \(K(G)\) is an LEF algebra if and only if \(G\) is an LEF group. He proved that if an algebra \(A=K\langle U(A)\rangle\), where \(U(A)\) is the group of units of \(A\), is an LEF algebra then the group \(U(A)\) is an LEF group, and he conjectured that the converse is also true. In this paper a counterexample to this conjecture is constructed.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 2140257
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: finite groups
zbMATH Keywords: local embeddings
zbMATH Keywords: groups of units
zbMATH Keywords: Banach algebras
zbMATH Keywords: locally embeddable groups
zbMATH Keywords: surjective homomorphisms
zbMATH Keywords: LEF-groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: group algebras
zbMATH Keywords: LEF algebras
reviewed by: Leonid A. Bokut'
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2140257