Tensor matrix multiplication

The definition of the completely bounded bilinear maps as well as the Haagerup tensor product relies on the tensor matrix multiplication [Eff87]

   $$\mbox{$ x \odot y = [x_{ij}] \odot [y_{jk}] := \left[\sum_{j=1}^l x_{ij} \otimes y_{jk}\right] \in M_{n}(X \otimes Y) $}$$$$$$

of operator matrices $x = [x_{ij}] \in M_{n,l}(X)$, $y = [y_{jk}]\in M_{l,n}(Y)$.

The amplification of the bilinear mapping $\otimes : X \times Y \rightarrow X \otimes Y$ is given by

$$\otimes^{(n,l)} = \odot : M_{n,l}(X) \times M_{l,n}(Y) \rightarrow M_n(X \otimes Y).$$

For scalar matrices $\alpha, \gamma \in M_n$, $\beta \in M_l$ we have

$$(\alpha x \beta) \odot (y \gamma) = \alpha (x \odot (\beta y)) \gamma.$$

We use the short hand notation $\alpha x \beta \odot y \gamma$.

For linear maps

$$\Phi = [\Phi_{ij}] : \, x \rightarrow M_{n,l}(V)$$

$$\Psi = [\Phi_{jk}] : \, x \rightarrow M_{l,n}(W)$$

we denote by $\Phi \odot \Psi$ the mapping

$$\Phi \odot \Psi = \left[\sum_{j=1}^l \Phi_{ij} \otimes \Psi_{jk}\right] : X \otimes Y \rightarrow M_{n}(V \otimes W),$$

$$\Phi \odot \Psi : x \otimes y \mapsto \left[\sum_{j=1}^l \Phi_{ij}(x) \otimes \Psi_{jk}(y) \right].$$

We then have

$$(\Phi \odot \Psi)^{(p)}(x \odot y) = (\Phi^{(p,q)}(x)) \odot (\Psi^{(q,p)}(y))$$

for $x \in M_{p,q}(X)$, $y \in M_{q,p}(Y)$. Let $\otimes_\alpha$ be an operator space tensor product . We define the tensor matrix multiplication $\odot_\alpha$ of operator matrices $S = [S_{i,j}] \in M_{n,l}(\mathit{CB}(X_1,X_2))$, $T = [T_{k,l}] \in M_{l,n}(\mathit{CB}(Y_1,Y_2))$ of completely bounded maps by setting

$$S \odot_\alpha T = \left[\sum_{j=1}^l S_{ij} \otimes_\alpha T_{jk}\right] \in M_n(\mathit{CB}(X_1 \otimes_\alpha Y_1, X_2 \otimes_\alpha Y_2)).$$