Completely Bounded Multilinear Mappings

Going beyond the linear case, one can introduce the concept of complete boundedness for multilinear maps. The motivation mainly lies in the study of higher dimensional Hochschild cohomology over $C^*$- and von Neumann algebras⁵⁵ [Christensen/Effros/Sinclair '87].

As in the linear case, the most important properties of the completely bounded multilinear maps make their appearance⁵⁶ in representation, extension and decomposition theorems.

Let $X_1, \dots, X_k$, $Y$ be operator spaces and $\Phi: X_1 \times \dots \times X_k \rightarrow Y$ a multilinear mapping. We define [Christensen/Effros/Sinclair '87, p. 281] a multilinear mapping

$$\Phi^{(n)}:M_{n}(X_1) \times \dots \times M_{n}(X_k) \rightarrow M_n(Y)$$

$$\left(x_1, \dots, x_k \right) \mapsto \left[ \sum_{j_1, j_2, \dots, j_{k-1} = 1}^{n} \Phi(x_1^{l,j_1}, x_2^{j_1,j_2}, \dots, x_k^{j_{k-1},m})\right],$$

where $n\in{\mathbb{N}}$, the $n$th amplification of $\Phi$.

$\Phi$ is called completely bounded if $\Vert\Phi\Vert _{{\mathrm{cb}}} := {\sup}_n \Vert\Phi^{(n)}\Vert< \infty$. It is called completely contractive if $\Vert\Phi\Vert _{{\mathrm{cb}}} \le 1$.

Also compare the chapter Completely bounded bilinear maps .

Example: For bilinear forms on commutative $C^*$-algebras we have the following result on automatic complete boundedness [Christensen/Sinclair '87, Cor. 5.6]:

Let $A$ be a commutative $C^*$-algebra. Then each continuous bilinear form $\Phi: A \times A \rightarrow {\mathbb{C}}$ is automatically completely bounded and

$$\Vert\Phi\Vert \leq \Vert\Phi\Vert _{\mathrm{cb}} \leq K_G \Vert\Phi\Vert,$$

where $K_G$ denotes the Grothendieck constant. Furthermore, $K_G$ is the least such constant.

One often studies completely bounded multilinear maps by considering the linearization on the Haagerup tensor product, where the following relation holds [Paulsen/Smith '87, Prop. 1.3; cf. also Sinclair/Smith '95, Prop. 1.5.1]: If $X_1, \dots, X_n$ are operator spaces and $\H$ is a Hilbert space, then a multilinear mapping $\Phi: X_1 \times \dots \times X_n \rightarrow B(\H)$ is completely bounded if and only if its linearization $\varphi$ is a completely bounded mapping on $X_1 \otimes_h \dots \otimes_h X_n$. In this case, $\Vert\Phi\Vert _{{\mathrm{cb}}} =\Vert\varphi\Vert _{{\mathrm{cb}}}$.

Also compare the chapter: Completely bounded bilinear mappings .

Representation theorem [Paulsen/Smith '87, Thm. 3.2, cf. also Thm. 2.9; Sinclair/Smith, Thm. 1.5.4]:
Let $A_1, \dots, A_k$ be $C^*$-algebras, $X_1 \subset A_1, \dots, X_k \subset A_k$ operator spaces and $\H$ a Hilbert space. Let further be $\Phi: X_1 \times \dots \times X_k \rightarrow B(\H)$ a completely contractive multilinear mapping. Then there exist Hilbert spaces $\mathcal{K}_i$ ( $i=1, \dots, k$), $^*$-representations $\pi_i: A_i \rightarrow B(\mathcal{K}_i)$ ( $i=1, \dots, k$), contractions $T_i: \mathcal{K}_{i+1} \rightarrow \mathcal{K}_i$ ( $i=1, \dots, k-1$) and two isometries $V_i: \H\rightarrow \mathcal{K}_i$ ($i = 1, k$) such that

$$\Phi(x_1, \dots, x_k) = V_1^* \pi_1(x_1) T_1 \pi_2(x_2) T_2 \cdots T_{k-1} \pi_k(x_k) V_k.$$

Following [Ylinen '90, p. 296; cf. also Christensen/Effros/Sinclair '87] it is possible to eliminate the ``bridging maps'' $T_i$. One obtains the following simpler form for the representation theorem:

Let $A_1, \dots, A_k$ be $C^*$-algebras, $X_1 \subset A_1, \dots, X_k \subset A_k$ operator spaces and $\H$ a Hilbert space. Let further $\Phi: X_1 \times \dots \times X_k \rightarrow B(\H)$ be a completely contractive multilinear mapping. Then there exist a Hilbert space $\mathcal{K}$, $^*$-representations $\pi_i: A_i \rightarrow B(\mathcal{K})$ and two operators $V_1, V_k \in B(\H, \mathcal{K})$ such that

$$\Phi(x_1, \dots, x_k) = V_1^* \pi_1(x_1) \pi_2(x_2) \cdots \pi_k(x_k) V_k.$$

From this result one can deduce the following:

Extension theorem [cf. Paulsen/Smith '87, Cor. 3.3 and Sinclair/Smith '95, Thm. 1.5.5]:
Let $X_i \subset Y_i$ ( $i=1, \dots, k$) be operator spaces and $\H$ a Hilbert space. Let further $\Phi: X_1 \times \dots X_k \rightarrow B(\H)$ be a completely contractive multilinear mapping. Then there exists a multilinear mapping ${\widetilde{\Phi}}: Y_1 \times \dots \times Y_k \rightarrow B(\H)$ which extends $\Phi$ preserving the ${\mathrm{cb}}$-norm: $\Vert\Phi \Vert _{{\mathrm{cb}}} = \Vert {\widetilde{\Phi}} \Vert _{{\mathrm{cb}}}$.

Let $A$ and $B$ be $C^*$-algebras. For a $k$-linear mapping $\Phi: A^k \rightarrow B$ we define [Christensen/Sinclair '87, pp. 154-155] another $k$-linear mapping $\Phi^*: A^k \rightarrow B$ by

$$\Phi^*(a_1, \dots, a_k) := \Phi(a_k^*, \dots, a_2^*, a_1^*)^*,$$

where $a_1, \dots, a_k \in A$. A $k$-linear map $\Phi: A^k \rightarrow B$ is called symmetric if $\Phi=\Phi^*$. In this case, ${\Phi^{(n)}}^*=\Phi^{(n)}$ ( $n\in{\mathbb{N}}$).

A $k$-linear map $\Phi: A^k \rightarrow B$ is called completely positive if

$$\Phi^{(n)} (A_1, \dots A_k) \geq 0$$

for all $n\in{\mathbb{N}}$ and $(A_1, \dots, A_k)=(A_k^*, \dots, A_1^*) \in M_n(A)^k$, where $A_{\tfrac{k+1}{2}} \geq 0$ for odd $k$.

Caution is advised: In the multilinear case complete positivity does not necessarily imply complete boundedness! For an example (or more precisely a general method of constructing such), cf. Christensen/Sinclair '87, p. 155.

There is a multilinear version of the decomposition theorem for completely bounded symmetric multilinear mappings:

Decomposition theorem [Christensen/Sinclair '87, Cor. 4.3]:
Let $A$ and $B$ be $C^*$-algebras, where $B$ is injective, and let further $\Phi: A^k \rightarrow B$ be a completely bounded symmetric $k$-linear mapping. Then there exist completely bounded, completely positive $k$-linear mappings $\Phi_+, \Phi_-: A^k \rightarrow B$ such that $\Phi = \Phi_+ - \Phi_-$ and $\Vert\Phi\Vert _{{\mathrm{cb}}} = \Vert \Phi_+ + \Phi_- \Vert _{{\mathrm{cb}}}$.

Footnotes

... algebras⁵⁵

There is a close connection to the long-standing still open derivation problem for $C^*$-algebras (or, equivalently [Kirchberg '96, Cor. 1], the similarity problem). One should note that in the framework of operator spaces and completely bounded maps, considerable progress has been made in attacking these problems. For instance, Christensen [Christensen '82, Thm. 3.1] was able to show that the inner derivations from a $C^*$-algebra into $B(\H)$ are precisely the completely bounded ones.

... appearance⁵⁶

possibly footnote!