A Short Proof that $M_{n}(A)$ is local if $A$ is local and Fr\'echet

TL;DR

This paper provides a concise, general proof that the property of being local or closed under holomorphic functional calculus in a dense Fréchet subalgebra of a Banach algebra is preserved when tensoring with matrix algebras.

Contribution

It offers a new, simplified proof demonstrating the stability of local and holomorphic calculus properties under matrix tensoring for dense Fréchet subalgebras.

Findings

Property of being local is preserved under tensoring with matrix algebras.

Closedness under holomorphic functional calculus is maintained after tensoring.

The proof is concise and applies generally to Fréchet subalgebras.

Abstract

We give a short and very general proof of the fact that the property of a dense Fr\'echet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the $n\times n$ matrix algebra of the complex numbers.