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Basic examples of operator modules
Let
be a unital
-algebra,
a normed space, and
an operator space.
Then
resp.
are operator spaces via the identifications
resp.
.
These become
-operator bimodules, when endowed with the natural module operations as follows ([ER88, p. 140]):
for all
,
resp.
,
resp.
.
In the category of operator modules, the morphisms are the
completely bounded module homomorphisms .
For these we have a representation and an extension theorem .
Representation theorem (cf. [Hof95, Kor. 1.4]):
Let
be a Hilbert space,
a
-algebra in
, and
,
-subalgebras of
. Then the following hold true:
- (a)
- (cf. [Pau86, Thm. 7.4]) For each completely bounded
-module homomorphism
,
there exist a Hilbert space
, a
-representation
and linear operators
sharing the following properties:
- (a1)
-
for all
, i.e.
is a representation of
- (a2)
-
- (a3)
-
- (a4)
-
for all
and
for all
.
- (b)
- (cf. also [Smi91, Thm.3.1]) If, in addition,
is a von Neumann algebra
and
is a normal
completely bounded
-module homomorphism, one can require the
-representation
of part (a) to be normal.
There exist families
and
in the commutant of
and
, respectively, with the following properties
(the sums are to be taken in the WOT topology):
- (b1)
-
for all
- (b2)
-
.
Extension theorem ([Wit84a, Thm. 3.1], cf. also [MN94, Thm. 3.4]
and [Pau86, Thm. 7.2]):
Let
be an injective
-algebra, and let
be two unital
-subalgebras.
Consider furthermore two
-operator modules
and
with
.
Then for each
, there exists an extension
with
and
.
The decomposition theorem for
completely bounded module homomorphisms can be found in the corresponding chapter.
Next: Completely bounded module homomorphisms
Up: Operator modules
Previous: Operator modules
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Prof. Gerd Wittstock
2001-01-07