next up previous contents index
Next: Jointly complete boundedness Up: Completely bounded bilinear mappings Previous: Completely bounded bilinear mappings   Contents   Index

Amplification

In the literature, there are two different notions of an amplification of a bilinear mapping. We shall call the first kind [*] of amplification the joint amplification. This joint amplification is needed to obtain a matrix duality - which is fundamental in the duality theory of operator spaces -, starting from an ordinary duality $ \langle X, X^ * \rangle$. The notion of joint amplification leads to the jointly completely bounded bilinear maps as well as the projective operator space tensor product. We will speak of the second kind [*] of an amplification as the amplification of a bilinear mapping. This notion leads to the completely bounded bilinear maps and the Haagerup tensor product. In the sequel, we will use the notation $ \Phi: X \times Y \rightarrow Z$ for a bilinear mapping and $ \tilde\Phi : X \otimes Y \rightarrow Z$ for its linearization. Both notions of an amplification of a bilinear map $ \Phi$ are formulated in terms of the amplification of its linearization:

$\displaystyle \tilde\Phi^{(n)} : M_{n}(X \otimes Y) \rightarrow M_{n}(Z).
$

  1. The joint amplification of $ \Phi$ produces the bilinear mapping

    $\displaystyle \Phi^{(p \times q)} : (x, y) \mapsto \tilde\Phi^{(pq)}(x \otimes y)
= [\Phi(x_{ij},y_{kl})] \in M_p(M_q(Z)) = M_{pq}(Z).
$

    of the operator matrices $ x = [x_{ij}] \in M_{p}(X)$ and $ y = [y_{kl}] \in M_{q}(Y)$. Here, the tensor product of operator matrices is defined via

    $\displaystyle x \otimes y
=
[x_{ij}] \otimes [y_{kl}] :=
[ x_{ij} \otimes y_{kl} ] \in M_{pq}(X \otimes Y)
=
M_p(M_q(X \otimes Y)).
$

  2. In the case of completely bounded bilinear maps one deals with the tensor matrix multiplication [Eff87]

       $\displaystyle \mbox{$
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)
$}$$\displaystyle $

    of operator matrices $ x = [x_{ij}] \in M_{n,l}(X)$ and $ y = [y_{jk}]\in M_{l,n}(Y)$. For more formulae, see: tensor matrix multiplication . The (n,l)-th amplification of a bilinear map $ \Phi: X \times Y \rightarrow Z$ is defined by
    $\displaystyle \Phi^{(n,l)}:M_{n,l}(X) \times M_{l,n}(Y)$ $\displaystyle \rightarrow$ $\displaystyle M_n(Z)$  
    $\displaystyle (x, y)$ $\displaystyle \mapsto$ $\displaystyle \tilde\Phi^{(n)}(x \odot y)
=
\left[\mbox{$\sum_{j=1}^l \Phi(x_{ij}, y_{jk})$} \right]
\in
M_{n}(Z)$  

    for $ l,n \in {\mathbb{N}}$, $ x = [x_{ij}] \in M_{n,l}(X)$, $ y = [y_{jk}]\in M_{l,n}(Y)$. In case $ n=l$, we shortly write48
    $\displaystyle \Phi^{(n)} := \Phi^{(n,n)}:
M_n(x) \otimes M_n(Y)$ $\displaystyle \rightarrow$ $\displaystyle M_n(Z)$  
    $\displaystyle (x, y)$ $\displaystyle \mapsto$ $\displaystyle \Phi^{(n)}(x,y) := \tilde\Phi^{(n)}(x \odot y).$  



Footnotes

... write48
In the literature, the amplification of a bilinear map often is defined only for quadratic operator matrices and is called the amplification $ \Phi^{(n)}$. Nevertheless, we will also be dealing with the amplification for rectangular matrices since this permits the formulation of statements where for fixed $ n\in{\mathbb{N}}$ all amplifications $ \Phi^{(n,l)}$, $ l \in {\mathbb{N}}$, are considered.

next up previous contents index
Next: Jointly complete boundedness Up: Completely bounded bilinear mappings Previous: Completely bounded bilinear mappings   Contents   Index
Prof. Gerd Wittstock 2001-01-07