Operator algebras

In analogy to concrete operator spaces we define (cf. [BRS90, Def. 1.1]): An operator algebra is a closed, not necessarily self-adjoint subalgebra $X$ of $B({\mathcal{H}})$ ( ${\mathcal{H}}$ a Hilbert space).
Example: For selfadjoint $X$ we have the theory of $C^*$-algebras.
As in the operator space situation, one can also adopt an abstract point of view: here, this leads to considering Banach algebras which are operator spaces and are equipped with a multiplication compatible with the operator space structure. These provide an abstract characterization of the (concrete) operator algebras ( cf. below : analogue of Ruan's theorem for operator algebras).

If $X$, $Y$, $Z$ are operator spaces, and if $\Phi: X \times Y \rightarrow Z$ is bilinear, we can define another bilinear map in the following way (cf.: Amplification of bilinear mappings):

$$\Phi^{(n,l)}:M_{n,l}(X) \times M_{l,n}(Y) \rightarrow M_n(Z)$$

$$\left(\left[x_{ij}\right],\left[y_{jk}\right]\right) \mapsto \left[ \sum_{j=1}^{l} \Phi(x_{ij},y_{jk})\right] ~ (l, n \in {\mathbb{N}}).$$

This is called the bilinear amplification²⁵ of $\Phi$.

$\Phi$ is called completely bounded if $\Vert\Phi\Vert _{\mathrm{cb}} := {\sup}_n \Vert\Phi^{(n,n)}\Vert< \infty$, and completely contractive if $\Vert\Phi\Vert _{\mathrm{cb}} \le 1$.²⁶ Compare this definition with the approach presented in Completely bounded bilinear Mappings .

[In the sequel, for Banach algebras with unit $e$, we will require $\Vert e\Vert=1$.] An operator space $(X,\Vert\cdot\Vert _n)$ with a bilinear, associative and completely contractive map $m: ~ X \times X \rightarrow X$, the multiplication, is called an abstract operator algebra (cf. [BRS90, Def. 1.4]). Here, the multiplication on $M_n(X)$ is just the matrix multiplication $m_n$.
In the unital case $m$ is automatically associative [BRS90, Cor. 2.4].

We have an analogue of Ruan's theorem ([Ble95, Thm. 2.1], cf. also [BRS90, Thm. 3.1]): Let $A$ be a unital Banach algebra and an operator space. Then $A$ is completely isometrically isomorphic to an operator algebra if and only if the multiplication on $A$ is completely contractive.
This yields the following stability result:
1.) The quotient of an operator algebra with a closed ideal is again an operator algebra [BRS90, Cor. 3.2].
With this at hand, one deduces another important theorem on hereditary properties of operator algebras:
2.) The class of operator algebras is stable under complex interpolation [BLM95, (1.12), p. 320].

Adopting a more general point of view than in the Ruan type Representation Theorem above, one obtains the following [Ble95, Thm. 2.2]: Let $A$ be a Banach algebra and an operator space. Then $A$ is completely isomorphic to an operator algebra if and only if the multiplication on $A$ is completely bounded. (cf. the chapter Examples !)

Basic examples of operator algebras are provided by the completely bounded maps on some suitable operator spaces. More precisely, for an operator space $X$, one obtains the following [Ble95, Thm. 3.4]: $\mathit{CB}(X)$ with the composition as multiplication, is completely isomorphic to an operator algebra if and only if $X$ is completely isomorphic to a column Hilbert space . - An analogue statement holds for the isometric case.

In the following result, for operator algebras $A$ and $B$, the assumption that $A$ and $B$ be (norm-) closed, is essential (in contrast to the whole rest) [ER90b, Prop. 3.1]:
A unital complete isometry $\varphi$ between $A \subset B({\mathcal{H}})$ and $B \subset B(\mathcal{K})$ ( ${\mathcal{H}}$, $\mathcal{K}$ Hilbert spaces), where $\mathrm{1\!\!\!\:l}_{B({\mathcal{H}})} \in A$, $\mathrm{1\!\!\!\:l}_{B(\mathcal{K})} \in B$, is already an algebra homomorphism.

Footnotes

... amplification²⁵

In the literature, e.g. in [BRS90, p. 190], the bilinear amplification $\Phi^{(n,n)}$ is often referred to as the amplification and is noted by $\Phi^{(n)}$.

....²⁶

In order to define the notion of complete boundedness of bilinear mappings, it suffices to consider only the $\Phi^{(n,n)}$ instead of all the $\Phi^{(n,l)}$; this definition is usually chosen in the literature about completely bounded bi- and, analogously, multilinear maps [BRS90, p. 190], [CES87, p. 281].