pinecone/core-2020-05-10-deduplication huggingface dataset:
Homology for Operator Algebras

AbstractA 4-cycle algebra is a finite-dimensional digraph algebra (CSL algebra) whose reduced digraph is a 4-cycle. A rigid embedding between such algebras is a direct sum of certain nondegenerate multiplicity one star-extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systemsAof 4-cycle algebras with rigid embeddings. The critical invariant is a binary relation inK0A⊕H1A, generalising the scale of theK0group, called the joint scale. The joint scale encapsulates other invariants and compatibility conditions of regular isomorphism. These include the scale ofH1A, the scale ofH0A⊕H1A, sign compatibility, congruence compatibility andH0H1coupling classes. These invariants are also important for liftingK0⊕H1isomorphisms to algebra isomorphisms; we resolve this lifting problem for various classes of 4-cycle algebra direct system
Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

AbstractLetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫Gf(g)ϕλ(g)dg,whereϕλdenote the elementary spherical functions onG/Kandλ⩾0. We consider the maximal operatorsS*f(t)=SupR>1∫R1f(λ)λ(a(t))|c(λ)|−2dλand prove thatS* maps boundedlyKLKs(G)→Ls(G)+L2(G) for 2n/(n+1)<s⩽2 wheren=dim(G/K). The result is sharp and it implies a.e. convergence properties of the inverse spherical transforms
Compensated Compactness, Paracommutators, and Hardy Spaces

AbstractLetB1: Rn×RN1→Rm1,B2: Rn×RN2→Rm2andQ: Rm2→Rm1be bilinear forms which are related as follows: ifμandνsatisfyB1(ξ,μ)=0 andB2(ξ,ν)=0 for someξ≠0, thenμτQν=0. Supposep−1+q−1=1. Coifman, Lions, Meyer and Semmes proved that, ifu∈Lp(Rn) andv∈Lq(Rn), and the first order systemsB1(D,u)=0,B2(D,v)=0 hold, thenuτQvbelongs to the Hardy spaceH1(Rn), provided that both (i)p=q=2, and (ii) the ranks of the linear mapsBj(ξ,·) : RNj→Rm1are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition whenp=q=2 involves the use of a deep result of Lojasiewicz from singularity theory
Singular Values of Compact Pseudodifferential Operators

AbstractThis paper investigates the asymptotic decay of the singular values of compact operators arising from the Weyl correspondence. The motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay. The problem is approached by using a Gabor frame expansion of the symbol to construct an approximating finite rank operator. This establishes a variety of sufficient conditions for the associated operator to be in a particular Schatten class. In particular, an improvement of a sufficient condition of Daubechies for an operator to be trace-class is obtained. In addition, a new development and improvement of the Calderón–Vaillancourt theorem in the context of the Weyl correspondence is given. Additional results of this type are then obtained by interpolation
Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

AbstractLetH⊂Gbe real reductive Lie groups. A discrete series representation for a homogeneous spaceG/His an irreducible representation ofGrealized as a closedG-invariant subspace ofL2(G/H). The condition for the existence of discrete series representations forG/Hwas not known in general except for reductive symmetric spaces. This paper offers a sufficient condition for the existence of discrete series representations forG/Hin the setting thatG/His a homogeneous submanifold of a symmetric space G/HwhereG⊂G⊃H. We prove that discrete series representations are non-empty for a number of non-symmetric homogeneous spaces such asSp(2n,R)/Sp(n0,C)×GL(n1,C)×…×GL(nk,C) (∑nj=n) andO(4m,n)/U(2m,j) (0⩽2j⩽n)
Manifold Derivative in the Laplace–Beltrami Equation

AbstractThis paper is concerned with the derivative of the solution with respect to the manifold, more precisely with theshape tangentialsensitivity analysis of the solution to the Laplace–Beltrami boundary value problem with homogeneous Dirichlet boundary conditions. The domain is an open subsetωof a smooth compact manifoldΓof RN. The flow of a vector fieldV(t,·) changesωinωt(andΓinΓt). The relative boundaryγtofωtinΓtis smooth enough andy(ωt) is the solution inωtof the Laplace–Dirichlet problem with zero boundary value onγt. The shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem; that elementy′Γ(ω;V) can be simply defined by the restriction toωofy−∇Γy·Vwhereyis the material derivative ofyand ∇Γyis the tangential gradient ofy. The study splits in two parts whether the relative boundaryγofωis empty or not. In both cases the shape derivative depends on the deviatoric part of the second fundamental form of the surface, on the fieldV(0) through its normal component onωand on the tangential fieldV(0)Γthrough its normal component on the relative boundaryγ. We extend the structure results for theshape tangentialderivative making use ofintrinsic geometryapproach and intensive use of extension operators
Backward Shift Invariant Operator Ranges

AbstractResults on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called H(W;B) spaces. Necessary and sufficient conditions are given for the groups Ext1A(D)(H2C, H(W;B)) to vanish whereH2Cis thedualof the vector-valued Hardy module, H2C. One condition involves an extension problem for the Hankel operator with symbolB,ΓB, but viewed as a module map from H2Cinto H(W;B). The group Ext1A(D)(H2C, H(W;B))=(0) precisely whenΓBextends to a module map from L2Cinto H(W;B) and this in turn is equivalent to the injectivity of H(W;B) in the category of contractive HilbertA(D)-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓB and the operator corona problem
Spherical Means, Wave Equations, and Hermite–Laguerre Expansions

AbstractIn this paper we study the maximal function associated to the Weyl transformW(μr) of the normalised surface measureμron the sphere |z|=rin Cn. This operator is given by the expansionW(μr)f=∑k=0∞k!(n−1)!(k+n−1)!phiv;k(r)Pkf,whereϕkare Laguerre functions of type (n−1) andPkare Hermite projection operators. We show that whenp>2n/(2n−1), the maximal operator supr>0|W(μr)f(x)| is bounded onLp(Rn). Using this we study almost everywhere convergence to initial data of solutions of the wave equation associated to the Hermite operator. The above expansion forW(μr) motivates the study of operators of the formSαtf=∑k=0∞phiv;αk(t)Pkf,wherephiv;αkare Laguerre functions of typeα. We study various mapping properties of these operators with applications to Hermite expansions and solutions of Darboux type equations
Functional Calculus for Infinitesimal Generators of Holomorphic Semigroups

AbstractWe give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner–Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups inL1(Rn) andL1(G), for a stratified Lie groupG. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0,∞)
On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model

AbstractA generalization of the standard spin-boson model is considered. The HamiltonianH(α) of the model with a coupling parameterα∈Racts in the tensor product H⊕Fbof a Hilbert space H and the boson (symmetric) Fock space FboverL2(Rν). The existence and uniqueness of ground states ofH(α) are investigated. The degeneracy of the ground states is also discussed. The results obtained arenonperturbative. The methods used are those of constructive quantum field theory and the min-max principle. An exact asymptotic formula for the ground state energy ofH(α) as |α|→∞ is also established
A Generalization of the Riesz Representation Theorem to Infinite Dimensions

AbstractLetHbe a real separable Hilbert space and letE⊂Hbe a nuclear space with the chain of Hilbert spaces {Ep:p=1,2,3,…} such thatE=∩∞p=1Ep. LetE* andE−pdenote the dual spaces ofEandEp, respectively. Let Cpbe the collection of real-valued functionsfdefined onE−psuch thatfis uniformly continuous on bounded subsets ofE−pand such that ‖f‖∞,p≔supx∈E−p{|f(x)|exp(−12 |x|2−p)} is finite. Set C∞=∩∞p=1Cp. ThenC∞is a complete countably normed space equipped with the family {‖·‖∞,p:p=1,2,3,…} of norms. In this paper it is shown that to every bounded linear functionalFin C*∞, there corresponds a signed measureνFsuch thatF(ϕ)=∫E*ϕ(x)νF(dx) forϕ∈C∞. It is also shown that there exists somepsuch that the measurable support ofνis contained inE−pand ∫E−pexp(12|x|2−p) |νF| (dx)<∞. This result extends the Riesz representation theorem to infinite dimensions. In the course of the proof, an infinite dimensional analogue of the Weierstrass approximation theorem is also established onE*
Stability and Approximations of Symmetric Diffusion Semigroups and Kernels

AbstractLet (Pt)t⩾0and (Pt)t⩾0be two diffusion semigroups on Rd(d⩾2) associated with uniformly elliptic operatorsL=∇·(A∇) and L=∇·(A∇) with measurable coefficientsA=(aij) and A=(ãij), respectively. The corresponding diffusion kernels are denoted bypt(x,y) and pt(x,y). We derive a pointwise estimate on |pt(x,y)− pt(x,y)| as well as anLp-operator norm bound, wherep∈[1,∞], forPt−Ptin terms of the localL2-distance betweenaijandãij. This implies in particular that |pt(x,y)− pt(x,y)| converges to zero uniformly in (x,y)∈Rd×Rdand that theLp-operator norm ofPt−Ptconverges to zero uniformly inp∈[1,∈] whenaij−ãijgoes to zero in the localL2-norm for each 1⩽i,j⩽n
On the Cauchy Problem for the Zakharov System

AbstractWe study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data (u(0),n(0), ∂tn(0))∈Hk⊕Hlscr;⊕Hℓ−1, in arbitrary space dimensionν. We define a natural notion of criticality according to which the critical values of (k,ℓ) are (ν/2−3/2,ν/2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of (k,ℓ). The results cover the whole subcritical range forν⩾4. Forν⩽3, they cover only part of it and the lowest admissible values are (k,ℓ)=(1/2,0) forν=2,3 and (k,ℓ)=(0,−1/2) forν=1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of (k,ℓ)
Heat Kernels of Second Order Complex Elliptic Operators and Applications

AbstractWe study the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on Rn. We obtain Gaussian bounds without further assumption ifn⩽2, and when the principal part has Hölder continuous coefficients ifn⩾3. A thorough study of the boundedness properties of the Green operator is made in dimension 1. We construct the fundamental solution of these operators in dimension 2. Boundedness results of the maximal accretive square roots onLpSobolev spaces and Hölder spaces are obtained whenn⩾2 under Hölder continuity assumption on the coefficients. BoundedH∞functional calculi onLp, 1<p<+∞, are also discussed
Regularity Results for Elliptic Equations in Lipschitz Domains

AbstractWe develop a simple variational argument based on the usual Nirenberg difference quotient technique to deal with the regularity of the solutions of Dirichlet and Neumann problems for some linear and quasilinear elliptic equation in Lipschitz domains. We obtain optimal regularity results in the natural family of Sobolev spaces associated with the variational structure of the equations. In the linear case, we obtain in a completely different way some of the results of D. Jerison and C. E. Kenig about the Laplace equatio
The Segal–Bargmann Transform for Path-Groups

AbstractLetKbe a connected Lie group of compact type and letW(K) denote the set of continuous paths inK, starting at the identity and with time-interval [0, 1]. ThenW(K) forms an infinite-dimensional group under the operation of pointwise multiplication. Letρdenote the Wiener measure onW(K). We construct an analog of the Segal–Bargmann transform forW(K). LetKCbe the complexification ofK,W(KC) the set of continuous paths inKCstarting at the identity, andμthe Wiener measure onW(KC). Our transform is a unitary map ofL2(W(K),ρ) onto the “holomorphic” subspace ofL2(W(KC),μ). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform forW(K) is nicely related by means of the Itô map to the classical Segal–Bargmann transform for the path-space in the Lie algebra ofK
Matrix Regular Operator Spaces

AbstractThe concept of the regular (or Riesz) norm on ordered real Banach spaces is generalized to matrix ordered complex operator spaces in a way that respects the matricial structure of the operator space. A norm on an ordered real Banach spaceEis regular if: (1) −x⩽y⩽ximplies that ‖y‖⩽‖x‖; and (2) ‖y‖<1 implies the existence ofx∈Esuch that ‖x‖<1 and −x⩽y⩽x. A matrix ordered operator space is called matrix regular if, at each matrix level, the restriction of the norm to the self-adjoint elements is a regular norm. In such a space, elements at each matrix level can be written as linear combinations of four positive elements. The concept of the matrix ordered operator space is made specific in such a way as to be a natural generalization of ordered real and complex Banach spaces. For the case whereVis a matrix ordered operator space, a natural cone is defined on the operator spaceX*⊗hV⊗hX, with ⊗hindicating the Haagerup tensor product, so as to make it a matrix ordered operator space. Exploiting the advantages gained by takingXto be the column Hilbert spaceHc, an equivalence is established between the matrix regularity of a space and that of its operator dual. This concept of matrix regularity also provides for more accessible proofs of the Christensen–Sinclair Multilinear Representation and Multilinear Decomposition theorems
Sewing Yang–Mills Measures and Moduli Spaces over Compact Surfaces

AbstractConsider a closed surfaceΣobtained by pasting two surfacesΣ1andΣ2along some of their boundary components. A formula is derived expressing quantum Yang–Mills functional integrals forΣin terms of the corresponding integrals for the surfacesΣi. Using this, a formula is derived which expresses the symplectic volume measure on the moduli space of flatSU(2) connections over a closed Riemann surfaceΣin terms of the corresponding volume measures for the surfacesΣ1andΣ2, when the latter have one boundary component each
Stein–Weiss Operators and Ellipticity

AbstractStein and Weiss (1968,Am. J. Math.90, 163–196) introduced the notion ofgeneralized gradients: equivariant first order differential operatorsGbetween irreducible vector bundles with structure group SO(n) or Spin(n). Among other things, they proved ellipticity for certain systems, analogous to the Cauchy–Riemann equations, and to the (Riemannian signature) Maxwell and Dirac equations, built from these generalized gradients. In this paper, we classify all systems of this type which are elliptic; the answer is valid for all Riemannian or (if spin structure enters) Riemannian spin manifolds. In particular, we find that ellipticity may be attained by assembling surprisingly few generalized gradients. The method employed yields a side benefit: the spectral resolution ofG*Gon the standard sphereSn, for each generalized gradientG. This spectral resolution was previously understood only for operators on “small bundles”—for example, the differential form operatorsδd, anddδ, and the square of the Dirac operator
Classification ofC*-Algebras of Real Rank Zero and UnsuspendedE-Equivalence Types

AbstractIn this article, examples are given to prove that the graded scaled orderedK-group is not the complete invariant for aC*-algebra in the class of unital separable nuclearC*-algebras of real rank zero and stable rank one, even for aC*-algebra in the subclass which consists of those real rank zero, stable rank oneC*-algebras being expressed as inductive limits of ⊕kni=1M[n, i](C(Xn, i)), whereXn, iare two-dimensional finite CW complexes and [n, i] are positive integers. (In the case of simple suchC*-algebras, it has been proved that the above invariant is the complete invariant by George Elliott and the author.) These examples prove that the classification conjecture of Elliott for the case of non simple real rank zeroC*-algebras should be revised—one needs extra invariants. The obstruction preventing two suchC*-algebras with the same graded scaled orderedK-group from being isomorphic is that they have different unsuspendedE-equivalence types (a refinement ofKK-equivalence type ofC*-algebras due to Connes and Higson). In this article, it is proved that for the above class of inductive limitC*-algebras, the obstruction of unsuspendedE-equivalence type is the only obstruction (i.e., if twoC*-algebras in the class are unsuspendedE-equivalence, then they are isomorphic). It is a surprise that in the case of simple suchC*-algebras, or even the case ofC*algebras with finitely many ideals, the obstruction will disappear (see Section 4)
Semigroups of *-Endomorphisms, Dirichlet Series, and Phase Transitions

AbstractWe study phase transitions of C*-dynamical systems (A, σ) in whichAis the crossed product of a C*-algebra by a lattice semigroup of *-endomorphisms, andσis a one-parameter subgroup of the dual action, determined by a real-valued scale on the semigroup. We show that the KMS equilibrium condition is equivalent to a Markov-type condition on the (predual) semigroup dynamical system, and give criteria for the existence of equilibrium states and of phase transitions in terms of the semigroup action. When the semigroup is an integer lattice, the partition function of the real scale is a Dirichlet series having an Euler product expansion; on their convergence region there is a phase transition whose geometry is fully determined by the ground states and is independent of the chosen scale, generalizing a model recently constructed by Bost and Connes motivated by earlier work of Julia. As applications we simplify part of the proof the Bost–Connes theorem on phase transition with spontaneous symmetry breaking and we discuss generalizations to other systems associated to subsets of primes and to number fields of class number 1
Involutive Functionals, Infinite Dimensional Tori, and Neighboring Tori

AbstractIn the class of smooth periodic functions, we consider Hamiltonian equationsut=K(u), constants of the motionFm(u) that are in involution, and prove an infinite-dimensional version of Liouville's Theorem. We explain in what sense the generic level set of the functionalsFm(u) is an infinite-dimensional torus, why the solution of the Hamiltonian equation is almost periodic in time, and describe how neighboring generic infinite-dimensional tori are related. The proof of the Theorem is independent of the method of inverse spectral theory and the viewpoint of algebraic curves
Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems

AbstractWe consider the semilinear elliptic problem[formula]whereλis a nonnegative parameter andgis a positive, nondecreasing, convex nonlinearity. There exists a valueλ* of the parameter which is extremal in terms of existence of solution. We study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameterλ=λ*. In some cases, this linearized problem has discrete and positiveH10-spectrum. However, we prove that there always exists a positive weak eigenfunction inL1(Ω) with eigenvalue zero for this linearized problem. The zeroL1-eigenvalue is coherent with the nonexistence of solutions of the semilinear problem forλ>λ*. Finally, we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem whenΩis the unit ball andg(u)=euorg(u)=(1+u)p
Cauchy–Dirichlet Problem for First Order Nonlinear Systems

AbstractLetΩ⊂Rnbe open,u:Ω→Rmand thus the gradient matrixDu∈Rm×n. We letE⊂Rm×nbe compact and denote byRcoEandPcoEthe rank one convex and polyconvex hull ofE, respectively. We show that ifRcoE=PcoE(and two other hypotheses, named the segment property and the extreme points property) and ifϕ∈C1(Ω; Rm) is such thatDϕ(x)∈E∪intPcoE,x∈Ωthen there exists (a dense set of)u∈W1, ∞(Ω; Rm) such that[formula]We apply this existence theorem to some relevant examples studied in the literature, as well as to problems with (x, u) dependence
Stochastic Boundary Values and Boundary Singularities for Solutions of the EquationLu=uα

AbstractWe investigate positive solutions of a nonlinear equationLu=uαwhereLis a second order elliptic differential operator in a Riemannian manifoldEand 1<α⩽2. The restrictionα⩽2 is imposed because our main tool is (L, α)-superdiffusionXwhich is not defined forα>2. We establish a 1-1 correspondence between the set U of positive solutions and a class Z of functionals ofXwhich we call linear boundary functionals (they depend only on the behavior ofXnear the Martin boundaryE′). The class Z is a closed convex cone andu∈U is a subadditive function ofZ∈Z. Special roles belong to moderate solutions corresponding toZwith finite mathematical expectations and to a family of solutions determined by the range ofX. A new formula is deduced connectingu,ZandL-diffusions conditioned to hit the boundaryE′ at a given pointy. A concept of a singular boundary point foruis introduced in terms of the conditioned diffusion
A Splitting Theorem for the Space of Smooth Functions

AbstractWe will show using purely linear functional analytic methods that each exact complex0→F→(C∞(Ω))s0→p0(C∞(Ω))s1→p1(C∞(Ω))s2→p2…,wherepiare matrices of convolution operators (in particular, linear differential operators with constant coefficients), splits fromp1on (in the category of topological vector spaces). Moreover, we characterize when these complexes split completely. The obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi. The result is a consequence of studying splitting of short exact sequences0→F→jX→qG→0(∗)whereF,X,Gare graded Fréchet spaces, i.e., Fréchet spaces equipped with fixed reduced spectra representing them andj,qare consistent with the graded structures. We characterize splitting of (∗) in case: (i)ForGare spacesC∞(Ω), whereΩis an open subset in Rn, or (ii)Gis the space of all sequences
Von Neumann Algebra Invariants of Dirac Operators

AbstractIn this paper we define and study certain von Neumann algebra invariants associated to the Dirac operator acting onL2spinors on the universal covering space of a compact, Riemannian spin manifold. We first study a Novikov–Shubin type invariant, which is aconformal invariantbut which isnotindependent of the choice of metric. However, we prove results which give evidence that this invariant may always be positive. When the Novikov–Shubin type invariant is positive, we can define the von Neumann algebra determinant of the Dirac Laplacian and the corresponding element in the determinant line of the space ofL2harmonic spinors on the universal covering space, which we also compute for certain locally symmetric spaces. Finally, we study a von Neumann algebra eta invariant associated to the Dirac operator and we show that it is sometimes an obstruction to the existence of metrics of positive scalar curvature