Abstract matrix spaces and
their generalisation
Orawan Tripak
Joint work with Martin Lindsay
Outline of the talk
• Background & Definitions- Operator spaces - h-k-matrix spaces
- Two topologies on h-k-matrix spaces• Main results
- Abstract description of h-k-matrix spaces• Generalisation - Matrix space tensor products - Ampliation
2
Concrete Operator Space
Definition. A closed subspace of for some Hilbert spaces and .
We speak of an operator space in
3
Abstract Operator Space
Definition. A vector space , with complete norms on , satisfying
(R1)
(R2)
Denote , for resulting Banach spaces.
4
Ruan’s consistent conditions
Let , , and
. Then
and
5
Completely Boundedness
Lemma. [Smith]. For
6
Completely Boundedness(cont.)
7
O.S. structure on mapping spaces
Linear isomorphisms
give norms on matrices over and
respectively. These satisfy (R1) and (R2).
8
Useful Identifications
Remark. When the target is
9
The right &left h-k-matrix spaces
Definitions. Let be an o.s. in
Notation:
10
The right & left h-k-matrix spaces
Theorem. Let V be an operator space in
and let h and k be Hilbert spaces. Then
1. is an o.s. in
2. The natural isomorphism
restrict to
11
Properties of h-k-matrix spaces (cont.)
3.
4. is u.w.closed is u.w.closed
5.
12
h-k-matrix space lifting
Theorem. Let for concrete operator spaces and . Then
1.
such that
“Called h-k-matrix space lifting”
13
h-k-matrix space lifting (cont.)
2.
3.
4. if is CI then is CI too.
In particular, if is CII then so is
14
Topologies on
Weak h-k-matrix topology is the locally convex topology generated by seminorms
Ultraweak h-k-matrix topology is the locally convex topology generated by seminorms
15
Topologies on (cont.)
Theorem. The weak h-k-matrix topology and the
ultraweak h-k-matrix topology coincide on bounded
subsets of
16
Topologies on (cont.)
Theorem. For
is continuous in both weak and
ultraweak h-k-matrix topologies.
17
Seeking abstract description of h-k-matrix space
Properties required of an abstract description.
1. When is concrete it must be completely isometric to
2. It must be defined for abstract operator space.
18
Seeking abstract description of h-k-matrix space (cont.)
Theorem. For a concrete o.s. , the map
defined by
is completely isometric isomorphism.
19
The proof : step 1 of 4
Lemma. [Lindsay&Wills] The map
where
is completely isometric isomorphism.
20
The proof : step 1 of 4 (cont.)
Special case: when we have a map
where
which is completely isometric isomorphism.
21
The proof : step 2 of 4
Lemma. The map
where
is completely isometric isomorphism.
22
The proof : step 3 of 4
Lemma. The map
where
is a completely isometric isomorphism.
23
The proof : step 4 of 4
Theorem. The map
where
is a completely isometric isomorphism.
24
The proof : step 4 of 4 (cont.)
The commutative diagram:
25
Matrix space lifting = left multiplication
26
Topologies on
Pointwise-norm topology is the locally convex
topology generated by seminorms
Restricted pointwise-norm topology is the locally
convex topology generated by seminorms
27
Topologies on (cont.)
Theorem. For the left
multiplication is continuous in both
pointwise-norm topology and restricted
pointedwise-norm topologies.
28
Matrix space tensor product
Definitions. Let be an o.s. in and be an ultraweakly closed concrete o.s.
The right matrix space tensor product is defined by
The left matrix space tensor product is defined by
29
Matrix space tensor product
Lemma. The map
where
is completely isometric isomorphism.
30
Matrix space tensor product (cont.)
Theorem. The map
where
is completely isometric isomorphism.
31
Normal Fubini
Theorem. Let and be ultraweakly closed
o.s’s in and respeectively.
Then
32
Normal Fubini
Corollary. 1.
2. is ultraweakly closed in
3.
4. For von Neumann algebras and
33
Matrix space tensor products lifting
Observation. For , an inclusion
induces a CB map
34
Matrix space tensor products lifting
Theorem. Let and be an u.w. closed concrete o.s. Then
such that
35
Matrix space tensor products lifting
Definition. For and
we define a map as
36
Matrix space tensor products lifting
Theorem. The map corresponds to the composition of maps
and
where and
(under the natural isomorphism ).
37
Matrix space tensor products of maps
38
Acknowledgements
I would like to thank Prince of Songkla University, THAILAND for financial support during my research and for this trip.
Special thanks to Professor Martin Lindsay for his kindness, support and helpful suggestions.
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