Involutive triangular matrix algebras

Abstract

In this paper we provide new examples of Banach $ \ast $-subalgebras of the matrix algebra $M_n(\mathscr{A}) $ over a commutative unital $C^*$-algebra $\mathscr{A}$. For any involutive algebra, we define two involutions on the triangular matrix extensions. We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal.*******************************************************************************