On a conjecture on algebras that are locally embeddable into finite dimensional algebras.
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.
scientific article; zbMATH DE number 2140257