Separation theorems

Suppose that $A,B\subset B(\H)$ are unital $C^*$-algebras and $Y\subset B(\H)$ is a $(A,B)$-bimodul. Then $K\subset Y$ is $(A,B)$-absolutely convex, if

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

for all $x_i\in K$ and $a_i\in A$, $b_i \in B$ such that $\sum_{i=1}^{n} a_i^*a_i$, $\sum_{i=1}^n b_i^*b_i \leq \mathrm{1\!\!\!\:l}$. Let $Y$ be a $A$-bimodul. Then $K$ is $A$-convex, 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 $\sum_{i=1}^{n} a_i^*a_i = \mathrm{1\!\!\!\:l}$. In the case $Y=A$ this definition is equivalent to the definition of $C^*$-convex sets.

There are following separation theorems: Let $A,B\subset B(\H)$ be unital $C^*$-algebras and $Y\subset B(\H)$ a $A,B$-bimodul. Let $K\subset Y$ be norm closed and $y_0\in Y\setminus K$.

1) If $A=B$, $0\in K$ and $K$ is $A$-convex, then there is a Hilbert space $H_\pi$, a cyclic representation $\pi:A\rightarrow B(H_\pi)$ and a completely bounded $A$-bimodul-homomorphism, such that for all $y\in K$

$$\mathrm{Re}\,\phi(y)\leq\mathrm{1\!\!\!\:l}$$, but $$\mathrm{Re}\,\phi(y_0)\not\leq\mathrm{1\!\!\!\:l}.$$

2)If $K$ is $(A,B)$-absolutely convex, then there is a Hilbert space $H_\pi$, representations $\pi:A\rightarrow B(H_\pi)$ and $\sigma:B\rightarrow B(\H_\pi)$ and a completely bounded $(A,B)$-bimodul-homomorphism $\phi:Y\rightarrow B(\H_\pi)$, such that for all $y\in K$

$$\Vert\phi(y)\Vert\leq 1$$, but $$\Vert\phi(y_0)\Vert>1.$$