Compact Sets in the Free Topology

TL;DR

This paper characterizes subsets of matrix tuples that are closed under direct sums and compact in the free topology, showing they are contained in the hull of a single point through dilation theory.

Contribution

It provides a dilation theoretic characterization of closed and compact subsets in the free topology of matrix tuples.

Findings

Subsets closed under direct sums are characterized.

Compact subsets are contained in the hull of a single point.

Dilation theory is used to analyze free topology properties.

Abstract

Subsets of the set of $g$-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, contained in the hull of a single point.