Completely bounded module homomorphisms

Let $A, B \subset B({\mathcal{H}})$ be $C^*$-algebras with $\mathrm{1\!\!\!\:l}_{{\mathcal{H}}} \in A, B$, and let $E$ and $F$ be two $(A,B)$-operator modules, i.e. (algebraic) $A$-left-$B$-right-modules. A mapping $\phi \in L(E,F)$ is called $(A,B)$-module homomorphism (in case $A=B$ $A$-bimodule homomorphism) if

$$\phi(axb)=a \phi(x) b$$

for all $a\in A$, $b \in B$, $x \in E$.
Furthermore we will write $\mathit{CB}_{(A,B)}(E,F)$ for the set of all completely bounded $(A,B)$-module homomorphisms between $E$ and $F$. The space $\mathit{CB}_{(A,B)}(E)$ with the composition of operators as multiplication is a Banach algebra.

Let $A_1, A_2 \subset B(\H)$ be $C^*$-algebras such that ${\mathrm{1\!\!\!\:l}}_{\H} \in A_1, A_2$. Let further $A \subset A_1 \cap A_2$ be a unital $^*$-subalgebra of $A_1$ and $A_2$ with ${\mathrm{1\!\!\!\:l}}_{\H} \in A$. An $A$-bimodule homomorphism

$$\Phi: A_1 \rightarrow A_2$$

is called self-adjoint if

$$\Phi(x)^* = \Phi(x^*)$$

for all $x \in A_1$.

Dealing with completely bounded module homomorphisms, we have at our disposal a representation theorem, an extension theorem and the following decomposition theorem of Wittstock ([Wit81, Satz 4.5] and cf. [Pau86, Thm. 7.5]):

Let $A$, $E$ and $F$ be unital $C^*$-algebras. Let moreover $F$ be injective, and $A$ be a subalgebra of $E$ and $F$ with $\mathrm{1\!\!\!\:l}_E=\mathrm{1\!\!\!\:l}_F=\mathrm{1\!\!\!\:l}_A$. Then for each self-adjoint completely bounded $A$-bimodule homomorphism $\phi: E \rightarrow F$, there exist two completely positive $A$-bimodule homomorphisms $\phi_1$ and $\phi_2$ sharing the properties $\phi=\phi_1 - \phi_2$  and  $\Vert\phi\Vert _{\mathrm{cb}}=\Vert\phi_1+\phi_2\Vert _{\mathrm{cb}}$ .

Consider two von Neumann algebras $M$ and $N$, and two $C^*$-algebras $A_1,~A_2 \subset B(H)$, where $\mathrm{1\!\!\!\:l}_{{\mathcal{H}}} \in A_1$, $A_2$ and $A_1 \cup A_2 \subset M \cap N$. We then have the decomposition theorem of Tomiyama-Takesaki (cf. [Tak79, Def. 2.15]): Each operator $\phi \in \mathit{CB}_{(A_1,A_2)}(M,N)$ has a unique decomposition $\phi=\phi^{\sigma}+\phi^s$, $\phi^{\sigma},~\phi^s \in \mathit{CB}_{(A_1,A_2)}(M,N)$ normal resp. singular, where $\Vert\phi^{\sigma}\Vert _{\mathrm{cb}},~ \Vert\phi^s\Vert _{\mathrm{cb}} \le \Vert\phi\Vert _{\mathrm{cb}}$. We thus obtain the algebraically direct sum decomposition:

$$\mathit{CB}_{(A_1,A_2)}(M,N) = \mathit{CB}^{\sigma}_{(A_1,A_2)}(M,N) \oplus \mathit{CB}^s_{(A_1,A_2)}(M,N).$$ (2)

Here, the notions "normal" and "singular", repectively, are built in analogy to the framework of linear functionals on a von Neumann algebra $M$.²⁴

We list some basic facts about the spaces and mappings mentioned in ():

(a)

In case $M=N$, all the spaces in () are Banach algebras.

(b)

The following properties of $\phi$ are hereditary for the normal part $\phi^{\sigma}$ and the singular part $\phi^s$ : completely positive, homomorphism, $^*$-homomorphism.

(c)

If $\alpha \in {\rm {Aut}}(M)$ and $\beta \in {\rm {Aut}}(N)$ are $^*$-automorphisms, we have $(\beta \phi \alpha)^{\sigma} = \beta \phi^{\sigma} \alpha$ and $(\beta \phi \alpha)^s = \beta \phi^s \alpha$ .

(d)

For $\phi \in \mathit{CB}(B({\mathcal{H}}))$, ${\mathcal{H}}$ a Hilbert space, we have: $\phi \in \mathit{CB}^s(B({\mathcal{H}})) \Leftrightarrow \phi\vert _{K({\mathcal{H}})} \equiv 0$ .

Let ${\mathcal{H}}$ be a Hilbert space, and let $A_1, A_2 \subset B({\mathcal{H}})$ be two $C^*$-algebras with $\mathrm{1\!\!\!\:l}_{{\mathcal{H}}} \in A_1, A_2$. Then we obtain [Pet97, Prop. 4.2.5]:

$$\mathit{CB}^{\sigma}_{(A_1,A_2)}(B({\mathcal{H}})) \stackrel{{\rm {cb}}}{=} \mathit{CB}_{(A_1,A_2)}(K({\mathcal{H}}),B({\mathcal{H}}))$$ (3)

$$\mathit{CB}^s_{(A_1,A_2)}(B({\mathcal{H}})) \stackrel{{\rm {cb}}}{=} \mathit{CB}_{(A_1,A_2)}(Q({\mathcal{H}}),B({\mathcal{H}}))$$ (4)

completely isometrically, where $Q({\mathcal{H}})=B({\mathcal{H}}) / K({\mathcal{H}})$ denotes the Calkin algebra.

Let $X$ be an arbitrary operator space. Then the space of all completely bounded $(A_1,A_2)$-module homomorphisms between $X$ and $B({\mathcal{H}})$ can be identified with the dual of a module Haagerup tensor product in the following way ([Pet97, p. 67], cf. also [ER91, Cor. 4.6], [Ble92b, Prop. 2.3]):

$$\mathit{CB}_{(A_1, A_2)}(X, B({\mathcal{H}})) \stackrel{{\rm {cb}... ...\overline{{\mathcal{H}}}} \otimes_{hA_1} X \otimes_{hA_2} C_{{\mathcal{H}}})^*$$

completely isometrically. Hence we see that $\mathit{CB}(B({\mathcal{H}}))$ itself and (looking at (), ()), just so, $\mathit{CB}^\sigma_{(A_1,A_2)}(B({\mathcal{H}}))$ and $\mathit{CB}^s_{(A_1,A_2)}(B({\mathcal{H}}))$ are dual operator spaces [Pet97, p. 70].

Footnotes

....²⁴

Let $M_*$ denote the (unique) predual of $M$. Then we have the $\ell_1$-direct sum decomposition

$$M^* = M_* \oplus_{\ell_1} (M^*)^{s}$$

of $M^*$ into normal (i.e. $w^*$-continuous) and singular functionals. [In the literature, one usually writes $M_*^{\perp}$ instead of $M^{*s}$, corresponding to $M_*(=M^{*\sigma})$.] Analogously, an operator $\phi \in B(M,N)$, $M$, $N$ von Neumann algebras, is called normal (i.e. $w^*$-$w^*$-continuous), if $\phi^*(N_*) \subset M_*$, and it is called singular, if $\phi^*(N_*) \subset M^{*s}$.