Transcript Slide 1
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 .
on A vector space , with complete norms , satisfying (R1) (R2) Denote , for resulting Banach spaces. 4
Let Ruan’s consistent conditions , . Then , and 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.
5. is u.w.closed is u.w.closed
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.
3.
is ultraweakly closed in 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 closed concrete o.s. Then and be an u.w. 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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