The BSE property for vector-valued Frechet Lipschitz algebras

TL;DR

This paper investigates the relationship between the BSE property of a commutative semisimple Frechet algebra and its vector-valued Lipschitz algebra, establishing conditions under which the BSE property is preserved.

Contribution

It establishes the correlation between the BSE property of a Frechet algebra and its Lipschitz algebra, including conditions for equivalence when unital.

Findings

If Lip_d(X, A) is a BSE-Frechet algebra, then A is also BSE.

The converse holds if (A, p_l) is unital.

The study clarifies the transfer of BSE properties between these algebras.

Abstract

Let $( X,d )$ be a metric space with at least two elements and $( A , p_l )$ be a commutative semisimple Frechet algebra over the scalar field of complex numbers. The correlation between the BSE-property of the Frechet algebra $( A , p_l )$ and $\Lip_{d}( X , A )$ is assessed. It is found and approved that if $\Lip_{d}( X , A )$ is a BSE-Frechet algebra, then so is $A$. The opposite correlation will hold if $( A , p_l )$ is unital.