If ,
,
are operator spaces, and if
is bilinear, we can define another bilinear map in the following way
(cf.: Amplification of bilinear mappings):
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is called
completely bounded if
, and completely contractive if
.26
Compare this definition with the approach presented in
Completely bounded bilinear Mappings .
[In the sequel, for Banach algebras with unit , we will require
.]
An operator space
with a bilinear, associative and completely contractive map
, the multiplication, is called
an
abstract operator algebra (cf. [BRS90, Def. 1.4]).
Here, the multiplication on
is just
the matrix multiplication
.
In the unital case 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 be a unital Banach algebra and an
operator space. Then
is completely isometrically isomorphic to an
operator algebra if and only if the multiplication on
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 be a Banach algebra and an operator space.
Then
is completely isomorphic to an operator algebra if and only if the multiplication on
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 , one obtains the following [Ble95, Thm. 3.4]:
with the composition as multiplication, is completely isomorphic to an operator algebra if and only if
is completely isomorphic to a column Hilbert space .
- An analogue statement holds for the isometric case.
In the following result, for operator algebras and
, the assumption that
and
be (norm-) closed, is essential
(in contrast to the whole rest) [ER90b, Prop. 3.1]:
A unital complete isometry between
and
(
,
Hilbert spaces), where
,
, is already an algebra homomorphism.