An inequality in noncommutative L-spaces

Ricard, Éric

doi:10.1016/j.jmaa.2016.01.079

Cited by 3 publications

(9 citation statements)

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“…Thus the non‐commutative version of (i.e. the condition in

L_{+}^{p} false(scriptX, τ false)

) amounts to

τ [(() x^{p - 1} - y^{p - 1}) (x - y)] ⩾ τ false({|x - y|}^{p} false) for all x, y \in L_{+}^{p} false(scriptX, τ false)

and has been proved by E. Ricard for

p \geq 2

. It follows from Theorem that

{(‖‖ x - {scriptE}^{X_{0}} x)}_{L^{p} false(scriptX, τ false)} \leq {(‖‖, x)}_{L^{p} false(scriptX, τ false)} for all x \in L_{+}^{p} false(scriptX, τ false) false(p \geq 2 false);

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

Mokhtar-Kharroubi

2016

Mathematische Nachrichten

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This paper provides various “contractivity” results for linear operators of the form I−C where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces Lpfalse(μfalse), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. Afalse(λ−Afalse)−1x≤x for all x∈L+pfalse(μfalse)false(λ>0false), provided that p⩾2. We show also that this result is not true for 1 ⩽ p<2. We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X+. We deduce from this result that, in such spaces, I−C is a contraction on X+ for any positive projection C with norm 1. We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base‐norm spaces X (e.g. in real L1false(μfalse) spaces or in preduals of hermitian part of von Neumann algebras), we show that Nfalse(u−Cufalse)≤u for all u∈X where N is the canonical half‐norm in X. For any positive contraction C on order‐unit spaces X (e.g. on the hermitian part of a C* algebra), we show that I−C is a contraction on X+. Applications to relative operator bounds, ergodic projections and conditional expectations are given.

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“…Thus the non‐commutative version of (i.e. the condition in