38 research outputs found
Harmonic analysis and BMO-spaces of free Araki-Woods factors
No full textWe consider semigroup BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in particular we L∞ → BMO multipliers. We also provide Lp-bounds for a natural generalization of the Hilbert transform.</p
Gradient forms and strong solidity of free quantum groups
No full textConsider the free orthogonal quantum groups ON+(F) and free unitary quantum groups UN+(F) with N≥ 3. In the case F= id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L∞(ON+) is strongly solid. Moreover, Isono obtains strong solidity also for L∞(UN+). In this paper we prove for general F∈ GLN(C) that the von Neumann algebras L∞(ON+(F)) and L∞(UN+(F)) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.Analysi
Riesz transforms on compact quantum groups and strong solidity
No full textOne of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call 'approximate linearity with almost commuting intertwiners'. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-S2 condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann-Ostrand property; in particular, the same strong solidity results follow again (now following [27]).</p
A Sobolev estimate for radial lp-multipliers on a class of semi-simple lie groups
No full textLet G be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup K. Let ΩK be minus the radial Casimir operator. Let 1 4 dim(G/K) < SG < 1 2 dim(G/K), s ∈ (0, SG] and p ∈ (1,∞) be such that(1 p - 1 2 )< s 2SG . Then, there exists a constant CG,s,p > 0 such that for every m ∈ L∞(G) ∩ L2(G) bi-K-invariant with m ∈ Dom(Ωs K) and Ωs K(m) ∈ L2SG/s(G) we have, (0.1) ∥Tm : Lp(G) → Lp( G)∥ ≤ CG,s,p∥Ωs K(m)∥ L2SG/s(G), where Tm is the Fourier multiplier with symbol m acting on the noncommutative Lp-space of the group von Neumann algebra of G. This gives new examples of Lp-Fourier multipliers with decay rates becoming slower when p approximates 2.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Analysi
Non-commutative differentiation and estimates on operator integrals
No full textIn 2017 Martijn Caspers, Fedor Sukochev and Dmitriy Zanin published a paper which generalises the proof of Davies' 1988 paper, and thus resolves the Nazarov-Peller conjecture. The proofs of these papers have been presented in this thesis. They have been expanded with a proof that generalises the conjecture to arbitrary Schatten classes. The optimality of the estimates in the conjecture is also studied, following the example of the 2016 paper by Coine et al. In addition, the quantum mechanical context is provided to interpret the presented results.Applied Mathematic
On the complete bounds of L<sub>p</sub> -Schur multipliers
No full textWe study the class Mp of Schur multipliers on the Schatten-von Neumann class Sp with 1 ≤ p≤ ∞ as well as the class of completely bounded Schur multipliers Mpcb. We first show that for 2 ≤ p< q≤ ∞ there exists m∈Mpcb with m∉ Mq, so in particular the following inclusions that follow from interpolation are strict: Mq⊊ Mp and Mqcb⊊Mpcb. In the remainder of the paper we collect computational evidence that for p≠ 1 , 2 , ∞ we have Mp=Mpcb, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.Analysi
The Structure of Hecke Operator Algebras
No full textThis dissertation is concerned with the study of the structure of certain deformations of operator algebras associated with Coxeter groups. These operator algebras, called Hecke C*-algebras and Hecke-von Neumann algebras, are operator algebraic completions of Iwahori-Hecke algebras. They occur as natural abstractions of certain endomorphism rings occurring in the representation theory of Lie groups and play a role in knot theory, combinatorics, the theory of buildings, quantum group theory, non-commutative geometry, and the local Langlands program. In this thesis we mainly focus on the ideal structure of Hecke C*-algebras, on approximation properties, and the rigidity of Hecke-von Neumann algebras. On our way we encounter and study several other concepts such as (Khintchine inequalities of) graph products of operator algebras, topological dynamics associated with boundaries and compactifications of graphs and (Coxeter) groups, C*-simplicity methods, the relative Haagerup property of sigma-finite unital inclusions of von Neumann algebras, approximation properties of operator algebras, and the rigidity theory of von Neumann algebras.Analysi
Gradient flow and quantum Markov semigroups with detailed balance
No full textIn this thesis we present a study of quantum Markov semigroups. In particular, we mainly consider quantum Markov semigroups with detailed balance that are defined on finite-dimensional C*-algebras. They have an invariant density matrix ρ. Carlen and Maas showed that the evolution on the set of invertible density matrices that is given by such a semigroup is gradient flow for the relative entropy with respect to ρ for some Riemannian metric. This result is a non-commutative analog of certain diffusion equations that are gradient flow in the second order Wasserstein space. We provide a self-contained and accessible account to these issues. Moreover, we give a complete introduction to Tomita-Takesaki theory which has a close relation with quantum Markov semigroups satisfying detailed balance. Finally, we present some examples of these semigroups that arise from quantum theory.Applied Mathematic
Weak (1,1) estimates for multiple operator integrals and generalized absolute value functions
No full textConsider the generalized absolute value function defined by a(t) = | t| tn−1, t∈ ℝ, n∈ ℕ≥ 1. Further, consider the n-th order divided difference function a[n]: ℝn+1 → ℂ and let 1 < p1, …, pn < ∞ be such that ∑l=1npl−1=1. Let Spl denote the Schatten-von Neumann ideals and let S1,∞ denote the weak trace class ideal. We show that for any (n + 1)-tuple A of bounded self-adjoint operators the multiple operator integral Ta[n]A maps Sp1×⋯×Spn to S1,∞ boundedly with uniform bound in A. The same is true for the class of Cn+1-functions that outside the interval [−1, 1] equal a. In [CLPST16] it was proved that for a function {atf} in this class such boundedness of Tf[n]A from Sp1×⋯×Spn to S1 may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences.Accepted author manuscriptAnalysi
Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity
No full textIn deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp class p 2 [2; 1/, then the n-fold tensor power HΓ˝n for n ≥ p2 is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in Sp for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation.Analysi
