Normalized Rank- and Determinant-Preserving Mappings of Locally Matrix Algebras

Abstract

Let $A$ be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product $\otimes M_{n_i}(\mathbb{F})$ of matrix algebras over a field $\mathbb{F}$, and (2) the Clifford algebra of a nondegenerate quadratic form on an infinite-dimensional vector space over an algebraically closed field of characteristic different from $2$. We describe linear mappings $A \to B$ between unital locally matrix algebras that preserve the normalized rank. When $\mathbb{F}$ is a field of real or complex numbers, we also describe linear mappings $A \to A$ that preserve the normalized determinant.