Representation

Completely bounded bilinear forms were first studied on C$^*$-algebras $A$, $B$ [EK87]. For a bilinear form $\Phi : A \times B \rightarrow {\mathbb{C}}$ the following properties are equivalent:

(1)

$\Phi$ is completely bounded.

(2)

There is a constant $c$ such that

$$\mbox{$ \vert\sum_{j=1}^l \Phi(a_j, b_j)\vert \leq c\, \Ver... ...a_j a_j^*\Vert^\frac{1}{2} \Vert\sum_{j=1}^l b_j^* b_j\Vert^\frac{1}{2} $}$$

for all $l \in {\mathbb{N}}$, $a_j \in A$, $b_j \in B$.

(3)

There is a constant $c$ and states $\omega \in S(A)$, $\rho \in S(B)$ such that

$$\mbox{$ \vert \Phi(a, b)\vert \leq c\, \omega(a a^*)^\frac{1}{2} \rho(b^* b)^\frac{1}{2} $}$$

for all $a\in A$, $b \in B$.

(4)

There exist $^*$-representations $\pi_\omega : A \rightarrow B(\H_\omega)$ and $\pi_\rho : A \rightarrow B(\H_\rho)$ with cyclic vectors $\xi_\omega \in \H_\omega$, $\xi_\rho \in \H_\rho$ and an operator $T \in B(\H_\omega, \H_\rho)$ such that $\Phi(a,b) = \langle T \pi_\omega(a) \xi_\omega, \pi_\rho(b) \xi_\rho \rangle$ for all $a\in A$, $b \in B$.

One can choose $c = \Vert T\Vert = \Vert\Phi\Vert _\mathrm{cb}$ and $\Vert\xi_\omega\Vert = \Vert\xi_\rho\Vert = 1$.⁵²

Footnotes

....⁵²

For further references see [CS89, Sec. 4].