$C^*$-extreme points

Let $A$ be an unital $C^*$-algebra and $Y\subset A$. The $C^*$-convex hull of $Y$ is the smallest $C^*$-convex set that contains $Y$.

Suppose $K\subset A$ is $C^*$-convex. Then $x\in K$ is a $C^*$-extreme point, if whenever $x=\sum_{i=1}^{n} a_i^*x_ia_i$ is a $C^*$-convex combination of $x_i\in K$ with invertible $a_i\in A$, then there are unitaries $u_i\in A$ such that $x=u_i^*x_iu_i$ for $i=1,\ldots,n$.

Suppose now $A=M_n$ and let $K\subset M_n$ be compact and $C^*$-convex. Let $\tilde{K}$ be the matrix convex hull of $K$. Then $\tilde{K}$ is a simple compact and matrix convex set in ${\mathbb{C}}$, such that $\tilde{K}_n=K$ ([Fis96]). Thus it is possible to conceive a $C^*$-convex subset of $M_n$ as a matrix convex set in ${\mathbb{C}}$. Now the matrix convex Krein-Milman theorem can be used. Moreover, it follows from the work of Farenick and Morenz that the structural elements of $\tilde{K}_n$ are exactly the not reducible $C^*$-extreme points of $K$. So following theorem holds: Let $K\subset M_n$ be compact and $C^*$-convex, then $K$ is equal to the $C^*$-convex hull of its $C^*$-extreme points.

In order to get a somewhat more general result, the definition of the extreme points can be changed. Suppose that $R$ is a hyperfinite factor and that $K\subset R$ is $C^*$-convex. Then $x\in K$ is a $R$-extreme point, if whenever $x=\sum_{i=1}^{n} a_ix_ia_i$ is a $C^*$-convex combination of $x_i\in K$ such that all $a_i\in A$ are positive and invertible, then it follows that $x=x_i$ and $a_ix=xa_i$ forr $i=1,\ldots,n$.⁶⁷

With this definition following theorem hold: Let $K\subset R$ be $C^*$-convex and weak* compact. Then $K$ is equal to the weak* closure of the $C^*$-convex hull of its $R$-extreme points.

Footnotes

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If $R=M_n$ the $R$-extreme points are exactly the $C^*$-extreme points. In general every $R$-extreme point is also $C^*$-extreme, but not vice versa.