Range-compatible homomorphisms on matrix spaces

TL;DR

This paper classifies all range-compatible group homomorphisms on certain matrix spaces over a field, showing they are essentially evaluation maps under specific codimension constraints, with implications for operator space reflexivity.

Contribution

It provides a complete classification of range-compatible homomorphisms on matrix subspaces with small codimension, extending understanding of their structure and applications.

Findings

Range-compatible homomorphisms are evaluation maps under specified conditions.

The codimension bound for classification is proven to be optimal.

New conditions for algebraic reflexivity of operator spaces are established.

Abstract

Let K be a (commutative) field, and U and V be finite-dimensional vector spaces over K. Let S be a linear subspace of the space L(U,V) of all linear operators from U to V. A map F from S to V is called range-compatible when F(s) belongs to the range of s for all s in S. Obvious examples of such maps are the evaluation maps s -> s(x), with x in U. In this article, we classify all the range-compatible group homomorphisms on S provided that the codimension of S in L(U,V) is less than or equal to 2 dim(V)-3, unless this codimension equals 2 dim(V)-3 and K has only two elements. Under those assumptions, it is shown that the linear range-compatible maps are the evaluation maps, and the above upper-bound on the codimension of S is optimal for this result to hold. As an application, we obtain new sufficient conditions for the algebraic reflexivity of an operator space and, with the above conditions on the codimension of S, we give an explicit description of the range-restricting and range-preserving homomorphisms on S.