Abstract matrix spaces and their generalisation

1. Abstract matrix spaces and their generalisation Orawan Tripak Joint work with Martin Lindsay

2. 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

3. Concrete Operator Space Definition. A closed subspace of for some Hilbert spaces and . We speak of an operator space in 3

4. Abstract Operator Space Definition.A vector space , with complete norms on , satisfying (R1) (R2) Denote , for resulting Banach spaces. 4

5. Ruan’s consistent conditions Let,, and . Then and 5

6. Completely Boundedness Lemma.[Smith]. For 6

7. Completely Boundedness(cont.) 7

8. O.S. structure on mapping spaces Linear isomorphisms give norms on matrices over and respectively. These satisfy (R1) and (R2). 8

9. Useful Identifications Remark.When the target is 9

10. The right &left h-k-matrix spaces Definitions. Letbe an o.s. in Notation: 10

11. The right & left h-k-matrix spaces Theorem. Let V be an operator space in andlethandkbe Hilbert spaces. Then • is an o.s. in 2. The natural isomorphism restrict to 11

12. Properties of h-k-matrix spaces (cont.) 3. • is u.w.closed is u.w.closed 5. 12

13. h-k-matrix space lifting Theorem. Let for concrete operator spaces and . Then 1. such that “Called h-k-matrix space lifting” 13

14. 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

15. Topologies on Weak h-k-matrix topologyisthe locally convex topology generated by seminorms Ultraweak h-k-matrix topologyis the locally convex topology generated by seminorms 15

16. Topologies on (cont.) Theorem.The weak h-k-matrix topology and the ultraweak h-k-matrix topology coincide on bounded subsets of 16

17. Topologies on(cont.) Theorem.For is continuous in both weak and ultraweak h-k-matrix topologies. 17

18. Seeking abstract description of h-k-matrix space Properties required of an abstract description. • When is concrete it must be completely isometric to 2. It must be defined for abstract operator space. 18

19. Seeking abstract description of h-k-matrix space(cont.) Theorem. For a concrete o.s. , the map defined by is completely isometric isomorphism. 19

20. The proof : step 1 of 4 Lemma.[Lindsay&Wills]The map where is completely isometric isomorphism. 20

21. The proof : step 1 of 4(cont.) Special case: when we have a map where which is completely isometric isomorphism. 21

22. The proof : step 2 of 4 Lemma.The map where is completely isometric isomorphism. 22

23. The proof : step 3 of 4 Lemma. The map where is a completely isometric isomorphism. 23

24. The proof : step 4 of 4 Theorem.The map where is a completely isometric isomorphism. 24

25. The proof : step 4 of 4(cont.) The commutative diagram: 25

26. Matrix space lifting = left multiplication 26

27. Topologies on Pointwise-norm topologyis the locally convex topology generated by seminorms Restricted pointwise-norm topology is the locally convex topology generated by seminorms 27

28. Topologies on(cont.) Theorem.For the left multiplication is continuous in both pointwise-norm topology and restricted pointedwise-norm topologies. 28

29. 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

30. Matrix space tensor product Lemma.The map where is completely isometric isomorphism. 30

31. Matrix space tensor product (cont.) Theorem. The map where is completely isometric isomorphism. 31

32. Normal Fubini Theorem.Let andbe ultraweakly closedo.s’sin and respeectively. Then 32

33. Normal Fubini Corollary. 1. • is ultraweakly closed in 3. 4. For von Neumann algebras and 33

34. Matrix space tensor products lifting Observation.For , an inclusion induces a CB map 34

35. Matrix space tensor products lifting Theorem. Letand be an u.w. closed concrete o.s. Then such that 35

36. Matrix space tensor products lifting Definition. For and we define a map as 36

37. Matrix space tensor products lifting Theorem.The map corresponds to the composition of maps and whereand (under the natural isomorphism). 37

38. Matrix space tensor products of maps 38

39. 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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