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
some Hilbert spaces and .
for
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
spaces
and
for concrete operator
. 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
Special case: when
: step 1 of 4 (cont.)
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
Theorem. For
(cont.)
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
o.s’s in
and
and
be ultraweakly closed
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
we define a map
and
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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