Uniqueness of Norm and Faithfulness of some Product Banach Algebras

TL;DR

This paper investigates the stability of the faithful and uniqueness of norm properties in various product Banach algebras, identifying conditions under which these properties are preserved or fail, with implications for algebraic structure analysis.

Contribution

It establishes the stability of norm properties in several product Banach algebras and provides a sufficient condition related to algebra norm for the finite co-dimension of the square of the algebra.

Findings

Faithful and uniqueness of norm are stable in direct-sum, convolution, and module product algebras.

These properties are not stable in null product algebra.

A sufficient condition for finite co-dimension of $\

Abstract

We prove that the faithful and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra. Further, we exhibit that these properties are not stable in null product algebra, and also give a common sufficient condition in terms of algebra norm for the co-dimension of $\mathcal{A}^2 = \text{span} \{ ab : a,b \in \mathcal{A}\}$ to be finite in $\mathcal{A}$ and $\mathcal{A}^{2} = \mathcal{A} \ ( \text{when } \overline{\mathcal{A}^2} = \mathcal{A})$.