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

TL;DR

This paper characterizes linear maps between unital locally matrix algebras that preserve normalized rank and determinant, including infinite tensor products and Clifford algebras, extending understanding of algebraic structure preservation.

Contribution

It provides a description of linear mappings preserving normalized rank and determinant in unital locally matrix algebras, including new classes like infinite tensor products and Clifford algebras.

Findings

Characterization of rank-preserving linear maps between locally matrix algebras.

Description of determinant-preserving maps over real or complex fields.

Extension of algebraic structure preservation to infinite-dimensional cases.

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.