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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