Hölder estimates for the noncommutative Mazur maps

Abstract

For any von Neumann algebra M, the noncommutative Mazur map Mp,q fromIn analogy with the commutative case, we gather estimates showing that Mp,q is min{ p q , 1}-Hölder on balls.2010 Mathematics Subject Classification: 46L51; 47A30.

On spectral gaps of Markov maps

“…2 in[8], II 3 T (y + ) − y + 2 y + 2/p−1 2 , and so II C T (y + ) − y + some universal C. Of course, the same estimates apply to y − . But we know that T (y ± )−y ± 2 T (y) − y 2 y 2 by the proof of Corollary 2.4 in[2].…”

“…Taking now γ = min{1, p 2 }, the Mazur map M p,2 is γ-Hölder with constant Cp by the main theorem in [8] (for some universal C). Hence…”

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…The available repertoire of spaces that admit average metric dimension reduction is larger, since if p ∈ [2, ∞), then p and even S p satisfy the assumption of the following theorem, by [170] and [224], respectively. Theorem 25.…”

A remainder term for Hölder’s inequality for matrices and quantum entropy inequalities

“…The map A → D p (A) is closely related to the non-commutative Mazur map studied in [1] an [9]. For 1 ≤ p, q ≤ ∞, the Mazur map M p,q is defined on M n by M p,q (A) = A|A| (p−q)/q .…”

“…Sharp Hölder continuity bounds on M p,q in a very general von Neumann algebra setting are proved in [9], which can be consulted for further references. The norm gradient maps, which are the Mazur maps for q = p ′ , normalized to be homogeneous of degree one, are the focus of this note which concerns another setting in which they arise.…”