In analogy to the
operator space or the
operator algebra
situation, there is an abstract characterization of operator modules (cf. [Pop00, Déf. 4.1]):
Consider, as above, two unital -algebras
with
,
and an (algebraic) -module .
We call an
abstract
-operator module, if it carries an operator space structure satisfying the following
axioms (of Ruan type):
(R1) | |||
(R2) |
For abstract operator modules holds a
representation theorem of Ruan type (cf. [Pop00, Thm. 4.7]):
Let be an abstract -operator module. Then there exist a Hilbert space
,
a complete isometry
and -representations
, of resp.
in
such that: