A note on extreme points in the closed unit ball of upper triangular $2\times 2$ matrices over a $C^*$-algebra

Abstract

Given a unital $C^*$-algebra $A$, let $M_{m\times n}(A)$ be the $m\times n$ matrix algebra over $A$ and
$\big(M_n(A)\big)_1$ be the closed unit ball of $M_{n\times n}(A)$.
Let $x=\begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix}\in \big(M_{m+n}(A)\big)_1$ be determined by
$a\in M_{m\times m}(A), b\in M_{m\times n}(A)$ and $c\in M_{n\times n}(A)$. Some characterizations are given such that the above upper triangular matrix $x$ is an extreme point of $\big(M_{m+n}(A)\big)_1$ and $X_{m,n}(A)$ respectively, where $X_{m,n}(A)$ is the subset of $\big(M_{m+n}(A)\big)_1$ consisting of all upper triangular matrices.