Joint amplification of a duality

The matrix dualitywhich is fundamental in the duality theory of operator spaces, is a special case of the joint amplification of a bilinear mapping. The joint amplification of a duality $\langle X, X^ * \rangle$ of vector spaces is defined by

$$\langle x, \varphi \rangle^{p \times q} = \langle [x_{ij}], [\v... ...= [ \langle x_{ij}, \varphi_{\kappa\lambda} \rangle ] \in M_p(M_q) = M_{pq}$$

for $x = [x_{ij}] \in M_p(X)$, $\varphi = [\varphi_{\kappa\lambda}] \in M_q(X^*)$. Interpreting $\varphi$ as a mapping $\varphi : X \rightarrow M_q$ we have

$$\varphi^{(p)}(x) = \langle x, \varphi \rangle^{p \times q}.$$

Associated to the duality of tensor products⁷⁴ $\langle X \otimes Y, X^* \otimes Y^*\rangle$ is the joint amplification

$$\langle M_p(X \otimes Y), M_q(X^* \otimes Y^*) \rangle.$$

Especially, the equation

$${\langle x \otimes y, \varphi \otimes \psi \rangle = \langle [x_{ij}] \otimes [y_{kl}], [\varphi_{\kappa\lambda}] \otimes [\psi_{\mu\nu}] \rangle}$$

$$:= [\langle x_{ij}, \varphi_{\kappa\lambda} \rangle \langle y_{kl}, ... ...a\mu),(jl\lambda\nu)} \in M_{p_1}(M_{p_2}(M_{q_1}(M_{q_2}))) = M_{p_1p_2q_1q_2}$$

obtains, where $x = [x_{ij}] \in M_{p_1}(X)$, $y = [y_{kl}] \in M_{p_2}(Y)$, $\varphi = [\varphi_{\kappa\lambda}] \in M_{q_1}(X^*)$, $\psi = [\psi_{\mu\nu}] \in M_{q_2}(Y^*)$.

Footnotes

... products⁷⁴

The duality $\langle X \otimes Y, X^* \otimes Y^*\rangle$ is defined by $\langle x \otimes y, \varphi \otimes \psi \rangle := \langle x,\varphi \rangle \langle y,\psi \rangle$ for $x \in X$, $y \in Y$, $\varphi \in X^*$, $\psi \in Y^*$.