ACTA issues

$W_{\varrho }$ completely bounded maps

Ching-Yun Suen

Acta Sci. Math. (Szeged) 67:3-4(2001), 747-760

Abstract. Let $A$ be a unital $C^*$-algebra and let $L$ be a $w_{\rho }$ completely bounded map $(1\leq\rho \leq2)$. Then $\left\|L\right\|_{w_{\rho }cb}=\left\|L_{2}\mid_{D_{\rho }\otimes A}\right\|_{w_{2}cb}.$ Moreover, there exist completely positive linear maps $\phi_{i}$ from $A$ to $B(H)$ with $\left\|\phi_{i}\right\|=\left\|L\right\|_{w_{\rho }cb}(i=1,2)$ such that the linear maps $\pmatrix{\phi_{1} & \sqrt{\rho(2-\rho )}L \cr\sqrt {\rho(2-\rho )}L^* & \phi_{2}}$ and $\pmatrix{\phi_{2} & (1-\rho )L \cr(1-\rho )L^* & \phi_{2}}$ from $A\otimes M_{2}$ to $B(H)\otimes M_{2}$ are completely positive. The above properties extend the results [8, Theorem 7.3] and [16, Corollary 3.11]. Let $M$ be a subspace of a unital $C^*$-algebra $A$, $\widehat{L}$ be a linear map from $M$ to $B(H)$ with $\left\|\widehat{L}\right\|_{w_{\varrho }cb}< \infty $ ($1\leq\rho \leq2$), then there exists a linear map $\widetilde{L}$ from $A\otimes M_{2}$ to $B(H)\otimes M_{2}$ such that $\widetilde{L}|_{D_{\rho }\otimes M}=\widehat{L}_{2}|_{D_{\rho }\otimes M}$ and $\left\|\widetilde{L}\right\|_{w_{2}cb}=\left\|\widehat{L}_{2}|_{D_{\rho }\otimes M}\right\|_{w_{2}cb}=\left\|\widehat{L}\right\|_{w_{\varrho }cb}.$ When $\varrho =1$, we have Haagerup, Paulsen, and Wittstock's extension theorem [4,8,18].When $\rho =2,$ then there exists a linear map $\overline{L}$ from $A$ to $B(H)$ such that $\overline{L}\mid_{M}=L$ and $\left\|\overline{L}\right\|_{w_{2}cb}=\left\|L\right\|_{w_{2}cb}.$ Let $M$ be an operator subspace of $A$ and $L$ be a linear map from $M$ to $M_{n}(\bf C)$. We prove that $\left\|L\right\|_{cb}=\left\|L\otimes I_{n}\right\|$ [10] and $\left\|L\right\|_{w_{2}cb}=\left\|L\otimes I_{n}\right\|_{w_{2}}.$ In general, $\left\|L\right\|_{w_{\rho }cb}=$ $\left\|L\otimes I_{2n}\right\|_{w_{\rho }}(1<\rho < 2).$

AMS Subject Classification (1991): 46L05, 46L10

Received July 19, 2000, and in revised form March 14, 2001. (Registered under 2815/2009.)