$C^*$-convexity

Let $A$ be a unital $C^*$-algebra. A subset $K\subset A$ is a $C^*$-convex set, if

$$\sum_{i=1}^{n} a_i^*x_ia_i \in K$$

for all $x_i\in K$ and $a_i\in A$ such that $\sum_{i=1}^{n} a_i^*a_i = \mathrm{1\!\!\!\:l}$. The sum $\sum_{i=1}^{n} a_i^*x_ia_i$ is called a $C^*$-convex combination.

A subset $K\subset A$ is a $C^*$-absolutely convex set, if

$$\sum_{i=1}^{n} a_i^*x_ib_i \in K$$

for all $x_i\in K$ and $a_i$, $b_i\in A$ such that $\sum_{i=1}^{n} a_i^*a_i$, $\sum_{i=1}^{n} b_i^*b_i \leq \mathrm{1\!\!\!\:l}$.

In particular $C^*$-convex sets are convex and $C^*$-absolutely convex sets are absolutely convex.

Example: Let $\H$ be a Hilbert space and $T\in B(\H)$. The n-th matrix range of $T$ is the set

$$W(T)_n :=\{\varphi(T)\vert\varphi:B(\H)\rightarrow M_n$$ completely positive and unital$$\}.$$

$W(T)_n$ is a compact and $C^*$-convex subset of $M_n$ and to every compact and $C^*$-convex subset $K\subset M_n$ there exist a separable Hilbert space $\H$ and $S\in B(\H)$ such that $K=W(S)_n$ ([LP81, Prop. 31]).

The sets

$$\mathrm{Ball}(B(\H))=\{x\in B(\H)\vert\Vert x\Vert\leq 1\}$$    and    $$P=\{x\in B(\H)\vert\leq x\leq \mathrm{1\!\!\!\:l}\}$$

are $C^*$-convex und wot-compact subsets of $B(\H)$. $\mathrm{Ball}(B(\H))$ is also $C^*$-absolutely convex.

Loebl and Paulsen introduced $C^*$-convex sets in [LP81]. At the beginning of the nineties Farenick and Morenz studied $C^*$-convex subsets of $M_n$ ([Far92],[FM93]). Eventually Morenz succeeded in proving an analogue of the Krein-Milman theorem for a compact $C^*$-convex subset of $M_n$ ([Mor94]). At the end of the nineties Magajna generalized the notion of $C^*$-convex sets to the setting of operator modules and proved some separation theorems [Mag00, Th. 1.1] and also an analogue of the Krein-Milman theorem [Mag98, Th. 1.1].