Noncommutative Harmonic Analysis Research Articles

Reflection positivity via Krein space analysis

We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M+) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U,J,M+) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J, and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U,J,M+), with U assumed unitary and J-self-adjoint, and M+ reflecting the choices of the associated J-positive and U-invariant subspaces. We further consider the wider setting when U is only assumed J-self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M+ will be realized as spaces G(X), graph of a second linear operator X, where the possibilities for such X, as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis.

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and Parcet.

Real Harmonic Analysis on the Special Orthogonal Group

This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. However, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. Here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operational properties, such as derivatives, sampling, and Clebsch-Gordon coefficients. Furthermore, we implement both of complex and real harmonics analysis on the special orthogonal group into an open source software package that utilizes parallel processing through the OpenMP library. The efficacy of the presented results are illustrated by benchmark studies and an application to spherical shape matching.

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of ‘Markov metric’ governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

Random walks on Ramanujan complexes and digraphs

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group G over a local field F . We show that if T is any k -regular G -equivariant operator on the Bruhat–Tits building with a simple combinatorial property (collision-free), the associated random walk on the n -vertex Ramanujan complex has cutoff at time log _k n . The high-dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of G . Via these, we show that operators T as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ( r -normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group G, previously known for groups of type \widetilde A_n and \widetilde C_2 .

Multi-Toeplitz operators and free pluriharmonic functions

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains Dqm(H)⊂B(H)n, m,n≥1, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain Dqm(H). We prove that the set of all weighted multi-Toeplitz operators coincides withA(Dqm)⁎+A(Dqm)‾WOT, where the domain algebra A(Dqm) is the norm-closed unital non-selfadjoint algebra generated by the universal model (W1,…,Wn) of the noncommutative domain Dqm(H). These results are used to study the class of free pluriharmonic functions on Dqm(H)∘. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions. Our study of weighted multi-Toeplitz operators on Fock spaces is a blend of multi-variable operator theory, noncommutative function theory, operator spaces, and harmonic analysis.

Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information

Operator space theory was the joining element of our workshop. This theory is a quantisation of the theory of Banach spaces. The talks at our meeting investigated the interactions of operator space theory with operator algebras and other areas of functional analysis, with classical and quantum probability, with noncommutative harmonic analysis, and, in particular, with quantum information theory.

Dynamic metaplectic spinor quantization: the projective correspondence for spectral dual pairs

From the historical perspective, the technique of X-ray computer tomography featured the predecessor of magnetic spin resonance tomography, which in fact is a non-invasive, high resolution, biomedical diagnostic scanning modality. Based on non-commutative harmonic analysis on the classical (2 + 1)-dimensional real Heisenberg unipotent Lie group $$\mathcal{N}$$ and the gradient controlled inversive and co-inversive chord-contact dynamics, framed by the coadjoint $$\mathcal{N}$$ -orbit model inside the real dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ , the paper provides mathematical insight into the intrinsic electromagnetic quantum field and relativistic symmetries associated to the highly resolving clinical modality of magnetic spin resonance tomography by referring to the methodology of the basic control mechanisms of the projective duality correspondence for spectral dual pairs of real Lie groups. In terms of the projective manifold $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) \cong {\mathbb {P}}_3({\mathbb {C}})$$ , which is associated with the (2 + 1)-dimensional dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ of the real Heisenberg nilpotent Lie algebra $$\mathfrak {Lie}(\mathcal{N})$$ , the smooth line bundle technique of dynamic metaplectic spinor quantization leads to the twisted action of the metaplectic Lie group $$\mathrm{Mp}(2,{\mathbb {R}}) = \widetilde{\mathrm{Sp}}(2,{\mathbb {R}})$$ . The transitive proper hyperbolic-parabolically ruled gradient action of the projective Lorentz–Mobius Lie group $$\mathrm{PSO}(1,3,{\mathbb {R}})$$ on $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ torque-records dihedrally its spectral incidence projective curve, the plane cubic $$E({\mathbb {C}}) \hookrightarrow {\mathbb {P}}_2({\mathbb {C}}) \cong \mathrm{Sym}^2\left( {\mathbb {P}}_1({\mathbb {C}})\right) $$ , on the two-dimensional pages of the open-book foliation inside the very round sphere $${\mathbb {S}}_3 \cong \mathrm{Spin}(3,{\mathbb {R}}) \cong \mathrm{SU}(2,{\mathbb {C}}) \hookrightarrow {\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ with angular momentum axis of the driving central universal Casimir metaplectic spinor. The projective correspondence for the spectral dual pair $$\left( \mathrm{Mp}(2,{\mathbb {R}}),\mathrm{PSO}(1,3,{\mathbb {R}})\right) $$ provides an efficient mathematical approach to the high resolution imaging modalities of magnetic spin resonance tomography, optical or ocular coherence tomography of neuro-ophthalmology, spin-polarized scanning tunneling microscopy, and the quantum field phenomena of the Hanbury Brown–Twiss experiment for photons and electrons. Due to the Eisenstein meromorphic calculus, the omni-directional interferometric detection of gravitational wavepackets of two polarizations is tomographically performed by the Abel–Jacobi inversion of the relativistic parabolic porism deviation of Kepplerian bifocal periodicity with its metaplectic spinor driven pair $$(\sigma ,{\bar{\sigma }})$$ of projective tangent involutions in the space $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ . The astrophysical emission of gravitational radiation is closely related to the concept of simply connected horned sphere which is homeomorphic to the compact base manifold of the Hopf principal circle bundle $${\mathbb {S}}_1 \hookrightarrow {\mathbb {S}}_3 {\mathop {\longrightarrow }\limits ^{\eta }} {\mathbb {S}}_2$$ . Riemann surface theory provides tomographic insight into the relativistic phenomenon of post-Kepplerian metaplectically driven spinor warping.

Fourier supports of K-finite Bessel integrals on classical tube domains

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

Noncommutative harmonic analysis and image registration

The image registration problem on a group G asks, given two functions f , g : G → R that are related by a translation f ( x ) = g ( s − 1 ⋅ x ) by an element s ∈ G , to find s . For abelian groups, the Fourier transform provides an elegant and fast solution to this problem. For nonabelian groups, the problem is much more involved. This paper shows how this applied problem can shed light on the constructions of noncommutative harmonic analysis, in particular the theory of Gelfand pairs. The abstract theory then suggests a novel two-step approach to solving such problems. The Gelfand pair ( SO ( 3 ) , SO ( 2 ) ) then provides us with an intuitive solution of the registration problem for images on S 2 .

Noncommutative harmonic analysis on semigroups

In this paper we obtain some noncommutative multiplier theorems and maximal inequalities on semigroups. As applications, we obtain the corresponding individual ergodic theorems. Our main results extend some classical results of Stein and Cowling on one hand, and simplify the main arguments of Junge-Le Merdy-Xu's related work [15].

Noncommutative harmonic analysis on semigroup and ultracontractivity

We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t)_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\mathcal{M}),$ and let the functions $\phi$ and $\psi$ be regularly related. We prove that the semigroup $(T_t)_{t>0}$ is $\phi$-ultracontractive, i.e. $\|T_t x\|_\infty \leq C \phi(t)^{-1} \|x\|_1$ for all $x\in L_1(\mathcal{M})$ and $ t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\|\psi(L)^{-\alpha} x\|_q \leq C'\|x\|_p$ for all $x\in L_p(\mathcal{M}),$ where $1<p<q<\infty$ and $\alpha =\frac{1}{p}-\frac{1}{q}.$ We establish some noncommutative spectral multiplier theorems and maximal function estimates for generator of $\phi$-ultracontractive semigroup. We also show the equivalence between $\phi$-ultracontractivity and logarithmic Sobolev inequality for some special $\phi$. Finally, we gives some results on local ultracontractivity.

Sofic entropy of Gaussian actions

Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc.23(2010) 217–245], Kerr and Li [Invent. Math.186(2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and$C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.

Stochastic Optimal Motion Planning for the Attitude Kinematics of a Rigid Body With Non-Gaussian Uncertainties

This paper investigates global uncertainty propagation and stochastic motion planning for the attitude kinematics of a rigid body. The Fokker–Planck equation on the special orthogonal group is numerically solved via noncommutative harmonic analysis to propagate a probability density function along flows of the attitude kinematics. Based on this, a stochastic optimal control problem is formulated to rotate a rigid body while avoiding obstacles within uncertain environments in an optimal fashion. The proposed intrinsic, geometric formulation does not require the common assumption that uncertainties are Gaussian or localized. It can be also applied to complex rotational maneuvers of a rigid body without singularities in a unified way. The desirable properties are illustrated by numerical examples.

Foliation [formula omitted]-algebras on multiply fibred manifolds

Motivated by index theory for semisimple groups, we study the relationship between the foliation C⁎-algebras on manifolds admitting multiple fibrations. Let F1,…,Fr be a collection of smooth foliations of a manifold X. We impose a condition of local homogeneity on these foliations which ensures that they generate a foliation F under Lie bracket of tangential vector fields. We then show that the product of longitudinal smoothing operators Ψc−∞(F1)⋯Ψc−∞(Fr) belongs to the C⁎-closure of Ψc−∞(F). An application to noncommutative harmonic analysis on compact Lie groups is presented.

Neurogeometry of neural functional architectures

The term “neurogeometry” denotes the geometry of the functional architecture of visual areas. The paper reviews some elements of the neurogeometry of the functional architecture of the first visual area V1 and explains why contact geometry, sub-Riemannian geometry, and noncommutative harmonic analysis are brought in as natural tools. It emphasizes the fact that these geometries are radically different from Riemannian geometries.

Geometric and harmonic analysis on homogeneous spaces and applications

Lie group representation theory and harmonic analysis on Lie groups and on their homogeneous spaces have built up enormous stores of knowledge as a significant and indispensable area of mathematics since the end of the World War II when the research in these areas began. These areas are closely interrelated with various other mathematical fields such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics, and are still growing and developing. In these circumstances, Ali Baklouti and Takaaki Nomura got a plan to organize joint seminars. The first seminar took place on November 2009 in Kerkennah Islands. This seminar was focused upon analysis with the keyword harmonic analysis as a core in such vast spread of research areas within the framework of academic program under the cooperation of Ministry of Higher Education, Scientific Research and Technology in Tunisia (MHESRT) and Japan Society for the Promotion of Science (JSPS). Many researchers on the forefront in both countries, experts in the concerned research areas from different corners, brought together the newest research results and discussed further progress of their research. This was a boosting reason for Ali Baklouti and Takaaki Nomura to continue the collaboration through the organization of a second Tunisian–Japanese meeting which took place in Sousse (December 2011) and is planned to emphasis direction axes of research toward commutative harmonic analysis (Fourier analysis on Euclidean spaces), analysis on homogeneous spaces, uncertainty principles and geometric analysis and some of their applications such as stochastic analysis and spectral theory. This seminar is also combined with an exchange based on the reciprocity program between Faculty of Mathematics, Kyushu University and Faculty of Science, Sfax University. In this sense, we list up post-doctoral students at Kyushu University, young researchers who graduated from Kyushu University, and young researchers in Sfax University. We also include young mathematicians from other than the two universities in the research area of non-commutative harmonic analysis. Thus we keep our eyes also on possible future exchanges by these young mathematicians, and the seminar was not a single one-time exchange between senior researchers. It is not so often that mathematicians who just made their debut

Stochastic description of geometric phase for polarized waves in random media

We present a stochastic description of multiple scattering of polarized waves in the regime of forward scattering. In this regime, if the source is polarized, polarization survives along a few transport mean free paths, making it possible to measure an outgoing polarization distribution. We consider thin scattering media illuminated by a polarized source and compute the probability distribution function of the polarization on the exit surface. We solve the direct problem using compound Poisson processes on the rotation group SO(3) and non-commutative harmonic analysis. We obtain an exact expression for the polarization distribution which generalizes previous works and design an algorithm solving the inverse problem of estimating the scattering properties of the medium from the measured polarization distribution. This technique applies to thin disordered layers, spatially fluctuating media and multiple scattering systems and is based on the polarization but not on the signal amplitude. We suggest that it can be used as a non-invasive testing method.

Markov measures and extended zeta functions

In this paper we study a family of representations of the Cuntz algebras O p where p is a prime. These algebras are built on generators and relations. They are C ∗-algebras and their representations are a part of non-commutative harmonic analysis. Starting with specific generators and relations we pass to an ambient C ∗-algebra, for example in one of the Cuntz-algebras. Our representations are motivated by the study of frequency bands in signal processing: We construct induced measures attached to those representations which turned out to be related to a class of zeta functions. For a particular case those measures give rise to a class of Markov measures and q-Bernoulli polynomials. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter q, and an associated family of q-relations where the number q is a “quantum-deformation,” and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this q-zeta function.

Hyperbolic Wavelets and Multiresolution in $H^{2}(\mathbb{T})$

In signal processing and system identification for $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ . The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space $H^{2}(\Bbb{T})$ . The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.