$C^{\ast }$-extreme points

Abstract:

Let $\mathcal {A}$ be a ${C^ \ast }$-algebra and let $\mathcal {S}$ be a subset of $\mathcal {A}$. $\mathcal {S}$ is ${C^ \ast }$-convex if whenever ${T_1},{T_2}, \ldots ,{T_n}$ are in $\mathcal {S}$ and ${A_1}, \ldots ,{A_n}$ are in $\mathcal {A}$ with $\sum \nolimits _{i = 1}^n {A_i^ \ast {A_i} = I}$, then $\sum \nolimits _{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ is in $\mathcal {S}$. An element $T$ in $\mathcal {S}$ is called ${C^ \ast }$-extreme in $\mathcal {S}$ if whenever $T = \sum \nolimits _{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ with ${T_i}$ and ${A_i}$ as above and with ${A_i}$ invertible, then ${T_i}$ is unitarily equivalent to $T$ for each $i$. We investigate the linear extreme points and ${C^ \ast }$-extreme points for three sets: first, the unit ball of operators in Hilbert space; next, the set of $2 \times 2$ matrices with numerical radius bounded by $1$; and last, the unit interval of positive operators on Hilbert space. In particular we find that for the second set, the linear and ${C^ \ast }$-extreme points are different.