Banach envelopes in symmetric spaces of measurable operators
We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class ( HC ) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes E ^ are also symmetric spaces. The c...
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| Published in: | Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 21; no. 1; pp. 473 - 492 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.03.2017
Springer Nature B.V |
| Subjects: | |
| ISSN: | 1385-1292, 1572-9281 |
| Online Access: | Get full text |
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| Summary: | We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (
HC
) of quasi-normed symmetric sequence or function spaces
E
for which their Banach envelopes
E
^
are also symmetric spaces. The class of symmetric spaces satisfying (
HC
) contains but is not limited to order continuous spaces. Let
M
be a non-atomic, semifinite von Neumann algebra with a faithful, normal,
σ
-finite trace
τ
and
E
be as symmetric function space on
[
0
,
τ
(
1
)
)
or symmetric sequence space. We compute Banach envelope norms on
E
(
M
,
τ
)
and
C
E
for any quasi-normed symmetric space
E
. Then we show under assumption that
E
∈
(
H
C
)
that the Banach envelope
E
(
M
,
τ
)
^
of
E
(
M
,
τ
)
is equal to
E
^
M
,
τ
isometrically. We also prove the analogous result for unitary matrix spaces
C
E
. |
|---|---|
| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 1385-1292 1572-9281 |
| DOI: | 10.1007/s11117-016-0430-4 |