Abstract matrix spaces

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

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Concrete Operator Space

• Definition. A closed subspace of
  for some Hilbert spaces and .
• We speak of an operator space in

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Abstract Operator Space

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

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Ruans consistent conditions

• Let , ,
  and
• . Then
• and

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Completely Boundedness

• Lemma. Smith. For

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Completely Boundedness(cont.)
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O.S. structure on mapping spaces

• Linear isomorphisms
• give norms on matrices over
  and
• respectively. These satisfy (R1) and (R2).

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Useful Identifications

• Remark. When the target is

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The right left h-k-matrix spaces

• Definitions. Let be an o.s. in
• Notation

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The right left h-k-matrix spaces

• Theorem. Let V be an operator space in
  
• and let h and k be Hilbert spaces. Then
• is an o.s. in
  
• 2. The natural isomorphism
• restrict to

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Properties of h-k-matrix spaces (cont.)

• 3.
• is u.w.closed
  is u.w.closed
• 5.

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h-k-matrix space lifting

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

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h-k-matrix space lifting (cont.)

• 2.
• 3.
• 4. if is CI then
  is CI too.
• In particular, if is CII then so is

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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

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Topologies on
(cont.)

• Theorem. The weak h-k-matrix topology and the
• ultraweak h-k-matrix topology coincide on
  bounded
• subsets of

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Topologies on
(cont.)

• Theorem. For
• is continuous
  in both weak and
• ultraweak h-k-matrix topologies.

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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.

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Seeking abstract description of h-k-matrix space
(cont.)

• Theorem. For a concrete o.s. , the map
• defined by
• is completely isometric isomorphism.

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The proof step 1 of 4

• Lemma. LindsayWills The map
• where
• is completely isometric isomorphism.

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The proof step 1 of 4 (cont.)

• Special case when we have a map
• where
• which is completely isometric isomorphism.

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The proof step 2 of 4

• Lemma. The map
• where
• is completely isometric isomorphism.

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The proof step 3 of 4

• Lemma. The map
• where
• is a completely isometric isomorphism.

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The proof step 4 of 4

• Theorem. The map
• where
• is a completely isometric isomorphism.

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The proof step 4 of 4 (cont.)

• The commutative diagram

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Matrix space lifting left multiplication
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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

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Topologies on
(cont.)

• Theorem. For the
  left multiplication is continuous in
  both pointwise-norm topology and restricted
  pointedwise-norm topologies.

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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

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Matrix space tensor product

• Lemma. The map
• where
• is completely isometric isomorphism.

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Matrix space tensor product
(cont.)

• Theorem. The map
• where
• is completely isometric isomorphism.

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Normal Fubini

• Theorem. Let and be ultraweakly
  closed o.ss in and
  respeectively.
• Then

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Normal Fubini

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

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Matrix space tensor products lifting

• Observation. For , an
  inclusion
• induces a CB map

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Matrix space tensor products lifting

• Theorem. Let and be
  an u.w. closed concrete o.s. Then
• such that

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Matrix space tensor products lifting

• Definition. For and
• we define a map
  as

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Matrix space tensor products lifting

• Theorem. The map corresponds to
  the composition of maps
• and
• where and
• (under the natural isomorphism
  ).

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Matrix space tensor products of maps
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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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