On the spectrum of a class of non-commutative Banach algebras

Abstract

Let $UT_m(X)$ be the Banach algebra of all $m \times m$ upper triangular matrices with entries in a unital commutative complex Banach algebra X. It is well known that $UT_m(X)$ can be identified with a closed subalgebra of the classical Banach algebra $B(X^m)$ of all continuous linear operators on $X^m.$ In the present paper, we describe the spectrum ${\mathcal {M}}(UT_m(X))$ of $UT_m(X)$, that is, the set of all nonzero, linear, and multiplicative mappings $\phi : UT_m(X)\rightarrow {\mathbb {C}}.$ Moreover, we prove the existence of a bijection between ${\mathcal {M}}(UT_m(X))$ and the set of all closed maximal ideals of $UT_m(X)$ via the map that to each $\phi \in {\mathcal {M}}(UT_m(X))$ associates the kernel of $\phi .$ As an application, we show that every maximal ideal of X is countably generated and, consequently, $UT_m(X)$ is finite dimensional whenever every maximal ideal of $UT_m(X)$ is countably generated.