On finite approximations of groups and algebras (Q1856423)

A group \(G\) is said to be an LEF group, i.e. \(G\) is locally embeddable into the class of finite groups, if for every finite subset \(M\subseteq G\) there exists a finite group \(H\) and an injective map \(\psi\colon M\cup M^2\to H\) such that \(\psi(x)\psi(y)=\psi(xy)\) for all \(x,y\in M\). In a similar way is defined the class of LEF algebras. Let \(\mathbb{C} G\) be the group algebra of a group \(G\) over the field \(\mathbb{C}\) of the complex numbers. The main result of this paper asserts that \(\mathbb{C} G\) is an LEF algebra if and only if \(G\) is an LEF group.