2011
DOI: 10.1007/s00208-011-0657-0
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BMO spaces associated with semigroups of operators

Abstract: Abstract. We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < ∞, with optimal constants in p. Mathematics subject classification (2000): 46L51 (42B25 46L10 47D06)

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Cited by 57 publications
(118 citation statements)
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“…The above definitions are motivated by Hardy spaces of noncommutative martingales ( [22,37]) and of quantum Markov semigroups ( [15,17,27]). The main results of this section are summarized in the following statement which shows that the Hardy spaces on T d θ possess the properties of the usual Hardy spaces, as expected.…”
Section: Hardy Spacesmentioning
confidence: 99%
See 1 more Smart Citation
“…The above definitions are motivated by Hardy spaces of noncommutative martingales ( [22,37]) and of quantum Markov semigroups ( [15,17,27]). The main results of this section are summarized in the following statement which shows that the Hardy spaces on T d θ possess the properties of the usual Hardy spaces, as expected.…”
Section: Hardy Spacesmentioning
confidence: 99%
“…[15,27,28,16,17]). One can also include in this topic the very fresh promising direction of research on the Calderón-Zygmund singular integral operators in the noncommutative setting (cf.…”
mentioning
confidence: 99%
“…We take the natural definition using a faithful normal state which is not necessarily tracial anymore as a starting point. We extend interpolation results from [JuMe12,Theorem 5.2] to the arbitrary setting under a modularity assumption on the Markov semi-group. The modularity assumption is necessary to carry out our proof through Haagerup's reduction method and due to the fact that the probabilistic martingale BMO-spaces in [JuPe14] are studied (in principle only) in the tracial setting.…”
mentioning
confidence: 88%
“…In [JuMe12] Junge and Mei pursued the theory of non-commutative semi-group BMO-spaces associated with non-commutative measure spaces. They introduce several notions of BMO starting from a Markov semi-group on a tracial von Neumann algebra.…”
mentioning
confidence: 99%
“…Non-commutative finite BMO-spaces. We recall the following from [JuMe12]. Fix a finite von Neumann algebra (M, τ ).…”
Section: Preliminariesmentioning
confidence: 99%