Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet. "Unconditionality, Fourier multipliers and Schur multipliers." Colloquium Mathematicae 127.1 (2012): 17-37. <http://eudml.org/doc/284363>.

@article{CédricArhancet2012,
abstract = {Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T ⊗ Id_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^\{p\}(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions for an operator space to be completely isomorphic to an operator Hilbert space.},
author = {Cédric Arhancet},
journal = {Colloquium Mathematicae},
keywords = {locally compact abelian groups; noncommutative -spaces; Fourier multipliers; Schur multipliers; unconditionality},
language = {eng},
number = {1},
pages = {17-37},
title = {Unconditionality, Fourier multipliers and Schur multipliers},
url = {http://eudml.org/doc/284363},
volume = {127},
year = {2012},
}

TY - JOUR
AU - Cédric Arhancet
TI - Unconditionality, Fourier multipliers and Schur multipliers
JO - Colloquium Mathematicae
PY - 2012
VL - 127
IS - 1
SP - 17
EP - 37
AB - Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T ⊗ Id_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^{p}(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions for an operator space to be completely isomorphic to an operator Hilbert space.
LA - eng
KW - locally compact abelian groups; noncommutative -spaces; Fourier multipliers; Schur multipliers; unconditionality
UR - http://eudml.org/doc/284363
ER -