Rademacher averages on noncommutative symmetric spaces
Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E ( M ) be the associated noncommutative function space. Let ( ε k ) k ⩾ 1 be a Rademacher sequence, on some probability space Ω. For finite sequences ( x k ) k ⩾ 1 of E ( M...
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| Published in: | Journal of functional analysis Vol. 255; no. 12; pp. 3329 - 3355 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
15.12.2008
Elsevier |
| Subjects: | |
| ISSN: | 0022-1236, 1096-0783 |
| Online Access: | Get full text |
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| Summary: | Let
E be a separable (or the dual of a separable) symmetric function space, let
M be a semifinite von Neumann algebra and let
E
(
M
)
be the associated noncommutative function space. Let
(
ε
k
)
k
⩾
1
be a Rademacher sequence, on some probability space
Ω. For finite sequences
(
x
k
)
k
⩾
1
of
E
(
M
)
, we consider the Rademacher averages
∑
k
ε
k
⊗
x
k
as elements of the noncommutative function space
E
(
L
∞
(
Ω
)
⊗
¯
M
)
and study estimates for their norms
‖
∑
k
ε
k
⊗
x
k
‖
E
calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if
E is 2-concave,
‖
∑
k
ε
k
⊗
x
k
‖
E
is equivalent to the infimum of
‖
(
∑
y
k
∗
y
k
)
1
2
‖
+
‖
(
∑
z
k
z
k
∗
)
1
2
‖
over all
y
k
,
z
k
in
E
(
M
)
such that
x
k
=
y
k
+
z
k
for any
k
⩾
1
. Dual estimates are given when
E is 2-convex and has a nontrivial upper Boyd index. In this case,
‖
∑
k
ε
k
⊗
x
k
‖
E
is equivalent to
‖
(
∑
x
k
∗
x
k
)
1
2
‖
+
‖
(
∑
x
k
x
k
∗
)
1
2
‖
. We also study Rademacher averages
∑
i
,
j
ε
i
⊗
ε
j
⊗
x
i
j
for doubly indexed families
(
x
i
j
)
i
,
j
of
E
(
M
)
. |
|---|---|
| ISSN: | 0022-1236 1096-0783 |
| DOI: | 10.1016/j.jfa.2008.05.002 |