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

Oksana Bezushchak

TL;DR

The paper addresses the problem of classifying linear preservers of the normalized rank and, over real or complex fields, the normalized determinant on unital locally matrix algebras. It extends classical finite-dimensional preservers to the locally matrix setting using corner decompositions and Steinitz-number invariants. The main contributions are a complete description of rank-preserving maps as combinations of a homomorphism and an antihomomorphism into corner algebras (with invertible multipliers), the identification of when such maps reduce to simple (anti)homomorphisms in locally finite Steinitz cases, and a determinant-preserving result showing surjective preservers arise from isomorphisms or anti-isomorphisms up to invertibles with unit determinant. These results generalize the theory of preserver problems to a broad, infinite-dimensional algebraic framework with connections to operator algebras and Clifford 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.

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

TL;DR

The paper addresses the problem of classifying linear preservers of the normalized rank and, over real or complex fields, the normalized determinant on unital locally matrix algebras. It extends classical finite-dimensional preservers to the locally matrix setting using corner decompositions and Steinitz-number invariants. The main contributions are a complete description of rank-preserving maps as combinations of a homomorphism and an antihomomorphism into corner algebras (with invertible multipliers), the identification of when such maps reduce to simple (anti)homomorphisms in locally finite Steinitz cases, and a determinant-preserving result showing surjective preservers arise from isomorphisms or anti-isomorphisms up to invertibles with unit determinant. These results generalize the theory of preserver problems to a broad, infinite-dimensional algebraic framework with connections to operator algebras and Clifford algebras.

Abstract

Let be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product of matrix algebras over a field , 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 . We describe linear mappings between unital locally matrix algebras that preserve the normalized rank. When is a field of real or complex numbers, we also describe linear mappings that preserve the normalized determinant.
Paper Structure (3 sections, 8 theorems, 36 equations)

This paper contains 3 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Rank-preserving mappings
  3. Determinant-preserving mappings: proof of Theorem \ref{['Th__2']}

Key Result

Theorem 1

Let $A$ and $B$ be unital locally matrix algebras over a field $\mathbb{F}$ whose characteristic is not $2$. Let $\varphi : A \to B$ be a linear mapping that preserves the normalized rank. Then there exist orthogonal idempotents $e_1, e_2 \in B$, $e_1 + e_2 = 1_B;$ a homomorphism $\varphi_1 : A \to

Theorems & Definitions (18)

  • Example 1
  • Example 2
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['Th__1']}
  • ...and 8 more