pinecone/core-2020-05-10-deduplication huggingface dataset:
Integrability of Functionals of Dirichlet Processes, Probabilistic Representations of Semigroups, and Estimates of Heat Kernels

AbstractThis paper consists of three parts. In Part I, we obtain results on the integrability of functional (of exponential type) of Dirichlet processes. In Part II, we give a striking probabilistic representation of semigroup (probably non-Markovian) associated with a non-divergence operator. Part III is devoted to perturbation bounds on the operator norm of semigroups and a new (short) proof of the off-diagonal estimates of the heat kernel associated with a divergence operator. The theory of Dirichlet forms and forward, backward martingales decompositions play a central role in the whole paper
Analysis and Geometry on Configuration Spaces

AbstractIn this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration spaceΓXover a Riemannian manifoldX. This geometry is “non-flat” even ifX=Rd. It is obtained as a natural lifting of the Riemannian structure onX. In particular, a corresponding gradient ∇Γ, divergence divΓ, and Laplace–Beltrami operatorHΓ=−divGamma;∇Γare constructed. The associated volume elements, i.e., all measuresμonΓXw.r.t. which ∇Γand divGamma;become dual operators onL2(ΓX;μ), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume elementdxonX. In particular, all these measures obey an integration by parts formula w.r.t. vector fields onΓX. The corresponding Dirichlet forms EΓμonL2(ΓX;μ) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion onΓXand which is sub- sequently identified as the usual independent infinite particle process onX. The associated heat semigroup (TΓμ(t))t>0is calculated explicitly. It is also proved that the diffusion process, when started withμ, is time-ergodic (or equivalently EGamma;μis irreducible or equivalently (TΓμ(t))t>0is ergodic) if and only ifμis Poisson measureπzdxwith intensityz dxfor somez⩾0. Furthermore, it is shown that the Laplace–Beltrami operatorHΓ=−divGamma;∇Gamma;onL2(ΓX;πz dx) is unitary equivalent to the second quantization of the Laplacian −ΔXonXon the corresponding Fock space ⊗n⩾0L2(X;z dx)⊗n. As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields onXon Poisson space. Finally, generalizations to the case wheredxis replaced by an absolutely continuous measure and also to interacting particle systems onXare described, in particular, the case where the mixed Poisson measuresμare replaced by Gibbs measures of Ruelle-type onΓX
Differential Banach *-Algebras of Compact Operators Associated with Symmetric Operators

AbstractExtensive development of noncommutative geometry requires elaboration of the theory of differential Banach *-algebras, that is, dense *-subalgebras of C*-algebras whose properties are analogous to the properties of algebras of differentiable functions. We consider a specific class of such algebras,D-algebras, and show that various *-algebras of compact operators associated with symmetric operatorsSon Hilbert spacesHareD-subalgebras of the C*-algebra of all compact operatorsC(H). We focus on how the properties of the operatorsSare reflected in the structure of these operator algebras
On Modified Logarithmic Sobolev Inequalities for Bernoulli and Poisson Measures

AbstractWe show that for any positive functionfon the discrete cube {0,1}n,Entμnp(f)⩽pqEμnp1f|Df|2whereμnpis the product measure of the Bernoulli measure with probability of successp, as well as related inequalities, which may be shown to imply in the limit the classical Gaussian logarithmic Sobolev inequality as well as a logarithmic Sobolev inequality for Poisson measure. We further investigate modified logarithmic Sobolev inequalities to analyze integrability properties of Lipschitz functions on discrete spaces. In particular, we obtain, under modified logarithmic Sobolev inequalities, some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables
On Inverse Spectral Theory for Self-Adjoint Extensions: Mixed Types of Spectra

AbstractLetHbe a symmetric operator in a separable Hilbert space H. Suppose thatHhas some gapJ. We shall investigate the question about what spectral properties the self-adjoint extensions ofHcan have inside the gapJand provide methods on how to construct self-adjoint extensions ofHwith prescribed spectral properties insideJ. Under some weak assumptions about the operatorHwhich are satisfied, e.g., provided the deficiency indices ofHare infinite and the operator (H−λ)−1is compact for one regular pointλofH, we shall show that for every (auxiliary) self-adjoint operatorM′ in the Hilbert space H and every open subsetJ0of the gapJofHthere exists a self-adjoint extensionHofHsuch that insideJthe self-adjoint extensionHofHhas the same absolutely continuous and the same point spectrum as the given operatorM′ and the singular continuous spectrum ofHinJequals the closure ofJ0inJ. Moreover we shall present a method of how to construct such a self-adjoint extensionH. Via our methods it is possible to construct new kinds of self-adjoint realizations of the Laplacian on a bounded domainΩin Rd,d>1, with spectral properties very different from the spectral properties of the self-adjoint realizations known before. Mathematics Subject classification (1991): 47A10; 47A60; 47B25; 47E05; 47F05
A Problem on Spreading Models

AbstractIt is proved that if a Banach spaceXhas a basis (en) satisfying every spreading model of a normalized block basis of (en) is 1-equivalent to the unit vector basis of ℓ1 (respectively,c0) thenXcontains ℓ1(respectively,c0). Furthermore, Tsirelson's spaceTis shown to have the property that every infinite dimensional subspace contains a sequence having spreading model 1-equivalent to the unit vector basis of ℓ1. An equivalent norm is constructed onTso that ‖s1+s2‖<2 whenever (sn) is a spreading model of a normalized basic sequence inT
On the Generators of Quantum Stochastic Flows

AbstractA time-indexed family of ∗-homomorphisms between operator algebras (jt:A→B)t∈Iis called a stochastic process in quantum probability. When EC:B→C is a conditional expectation onto a subalgebra, the composed process (kt=EC∘jt)t∈Iis no longer ∗-homomorphic, but is completely positive and contractive. In some situations, the filtered processkmay be described by a stochastic differential equation. The central aim of this paper is to study completely positive processeskwhich admit a differential description through a stochastic equation of the formdkt=kt∘θαβΛβα(t), in whichΛis the matrix of basic integrators of finite dimensional quantum stochastic calculus, andθis a matrix of bounded linear maps on the algebra. The structure required of the matrixθ, for complete positivity of the process, is obtained. The stochastic generators of contractive, unital, and ∗-homomorphic processes are also studied. These results are applied to the equationdVt=lαβVtdΛβα(t) in whichlis a matrix of bounded Hilbert space operators
Spectral Theory of Singular Elliptic Operators with Measurable Coefficients

AbstractWe study higher order elliptic operators with measurable coefficients acting on Euclidean domains. The coefficients may have degeneracies or singularities on the boundary or at infinity. We prove Gaussian-type bounds on the fundamental solution of the associated semigroup. These bounds are expressed in terms of a distanced(x, y) that reflects the singularity or degeneracy of the coefficients. The estimates are then used to extend the semigroup to otherLpspaces and to prove that theLp-spectrum isp-independent
GKS Decomposition and Spherical Dilations

AbstractUsing the generalized version of the classical F. and M. Riesz Theorem as given by Gliksberg, König, and Seever, we obtain a few decomposition theorems for tuples of commuting operators on Hilbert spaces that admit normal dilations whose joint spectra are contained in the unit sphere of Cn. Our results apply in particular to sphericaln-hypercontractions, subnormaln-tuples whose joint spectra are contained in the closed unit ball of Cn, and to spherical isometries. The questions related to the uniqueness of decompositions are addressed by appealing to a specialized version of an approximation result related to the solution of the inner function problem on the unit ball of Cn. The Henkin measures on the unit sphere play a central role in the development of the relevant theory
The Spherical Transform of a Schwartz Function on the Heisenberg Group

AbstractSuppose thatK⊂U(n) is a compact Lie group acting on the (2n+1)-dimensional Heisenberg groupHn. We say that (K,Hn) is a Gelfand pair if the convolution algebraL1K(Hn) of integrableK-invariant functions onHnis commutative. In this case, the Gelfand spaceΔ:(K,Hn) is equipped with the Godement–Plancherel measure, and the spherical transform∧:L2K(Hn)→L2(Δ(K,Hn)) is an isometry. The main result in this paper provides a complete characterization of the set SK(Hn)∧={f|f∈SK(Hn)} of spherical transforms ofK-invariant Schwartz functions onHn. We show that a functionFonΔ(K,Hn) belongs to SK(Hn)∧if and only if the functions obtained fromFvia application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions forK-invariant Schwartz functions onHnmodulo its center
L1-Factorization forC00-Contractions with Isometric Functional Calculus

AbstractLetTbe an absolutely continuous contraction acting on a Hilbert space H. Forx, y∈H, definex·Ty∈L1(T) by its Fourier coefficients:x·Ty∧(n)=(T*nx, y) ifn<0. The main technical result of the paper is that the vanishing condition limn→∞(‖xn·Tw‖L1/H10+‖w·Txn‖L1/H10)=0,w∈H implies that limn→∞‖xn·Tw‖L1=0,w∈H. Using known factorization techniques, we exhibit a Borel setσTsuch that for anyf∈L1(σT), there existx, y∈H such thatf=(x·Ty)|σT. In the case whereT∈A∩C00, this leads to a simple proof of the fact that for everyf∈L1(T) there existsx, y∈H such thatf=x·Ty. In this case we also show, using dilation theory in the unit disk, that every strictly positive lower semicontinuous functionϕ∈L1(T) can be written in the formϕ=x·Tx. Examples show that this is the best possible result for the class A∩C00
A Topology for Operator Modules overW*-Algebras

AbstractGiven a von Neumann algebraRon a Hilbert space H, the so-calledR-topology is introduced into B(H), which is weaker than the norm and stronger than the ultrastrong operator topology. A rightR-submoduleXof B(H) is closed in theR-topology if and only if for eachb∈B(H) the right ideal, consisting of alla∈Rsuch thatba∈X, is weak* closed inR. Equivalently,Xis closed in theR-topology if and only if for eachb∈B(H) and each orthogonal family of projectionseiinRwith the sum 1 the conditionbei∈Xfor alliimplies thatb∈X
Relative Yoneda Cohomology for Operator Spaces

AbstractWe provide two alternate presentations of the completely bounded Hochschild cohomology. One as a relative Yoneda cohomology, i.e., as equivalence classes ofn-resolutions which are relatively split, and the second as a derived functor. The first presentation makes clear the importance of certain relative notions of injectivity, projectivity and amenability which we introduce and study. We prove that every von Neumann algebra is relatively injective as a bimodule over itself and consequently,Hncb(M,M)=0 for any von Neumann algebraM. A result obtained earlier by Christensen and Sinclair. We prove that the relatively amenableC*-algebras are precisely the nuclearC*-algebras, and hence exactly those which are amenable as Banach algebras. In a similar vein we prove that the only relatively projectiveC*-algebras are finite dimensional. This result implies that the onlyC*-algebras that are projective as Banach algebras are finite dimensional, a result first obtained by Selivanov and Helemskii. In a slightly different direction we prove thatB(H) viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module. This result is in some sense a generalization of the Sz.-Nagy-Foias commutant lifting theorem or of the hypoprojectivity introduced by Douglas and the author
Real Spinor Fields and the Electroweak Interaction

AbstractThe space of real spinor fields of a given massm>0 in Minkowski space is the direct sum of two irreducibly invariant subspaces under the connected Poincaré groupP. These subspaces admit uniqueP-invariant positive-energy complex unitarizable structures, in terms of which they are unitarily and canonically equivalent to the conventional “left electron” and “right positron” subspaces defined by the Dirac equation. Space reversalPand time reversalTactP-covariantly as antiunitary operators on the real fields. They also extend to the complex fields, in terms of whichPexchanges the two subspaces, whileTacts separately on each. Whenm=0, the identical formalism produces a (rigorous) version of the conventional neutrino formalism. The restriction to the real context of the conventional spinor-vector interaction is equivalent to the conventionalV−Ainteraction together with a computationally equivalent form of quantum electrodynamics in which the right electron and left electron are redundant
Almost Commuting Unitaries and Classification of Purely Infinite SimpleC*-Algebras

AbstractWe show that, for any irrational rotational algebraAθ,Aθ⊗O2≅O2. This is proved by combining recently established results forC*-algebras of real rank zero with the following result: For anyε>0, there isδ>0, such that for any pair of unitariesu,vin any purely infinite simpleC*-algebraAwithK1(A)=0 satisfies‖uv−vu‖<δandκ(u, v)=0.There is a pair of commuting unitariesU,V∈Awith finite spectrum such that‖u−U‖<εand‖v−V‖<ε.More general results concerning almost commuting unitaries are also obtained
Analysis and Geometry on Configuration Spaces: The Gibbsian Case

AbstractUsing a natural “Riemannian geometry-like” structure on the configuration spaceΓover Rd, we prove that for a large class of potentialsφthe corresponding canonical Gibbs measures on 1 can be completely characterized by an integration by parts formula. That is, if ∇Γ is the gradient of the Riemannian structure onΓone can define a corresponding divergence divΓφsuch that the canonical Gibbs measures are exactly those measuresμfor which ∇Γ and divΓφare dual operators onL2(Γ, μ). One consequence is that for suchμthe corresponding Dirichlet forms EΓμare defined. In addition, each of them is shown to be associated with a conservative diffusion process onΓwith invariant measureμ. The corresponding generators are extensions of the operatorΔΓφ:=divΓφ∇Γ. The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brownian motion onΓperturbed by a singular drift. Another main result of this paper is the following: Ifμis a canonical Gibbs measure, then it is extreme (or a “pure phase”) if and only if the corresponding weak Sobolev spaceW1, 2(Γ, μ) onΓis irreducible. As a consequence we prove that for extreme canonical Gibbs measures the above mentioned diffusions are time-ergodic. In particular, this holds for tempered grand canonical Gibbs measures (“Ruelle measures”) provided that the activity constant is small enough. We also include a complete discussion of the free case (i.e.,φ≡0) where the underlying space Rdis even replaced by a Riemannian manifoldX
Some Uniform Ergodic Inequalities in the Nonmeasurable Case

AbstractUniform and nonmeasurable versions of some classical ergodic inequalities (all of them going back to the Hopf–Yosida–Kakutani maximal ergodic theorem) are estab- lished. Usually, uniformity involves nonmeasurable suprema and all the technical difficulties arising from this. In the present paper, a simplification is achieved by extending the given operator (a positive L1-contraction) to the class of all (i.e., not necessarily measurable) functions on the underlying measure space. This not only leads to technical improvements and clarifications of the proofs, but also to remarkable generalizations of known results. In particular, it turns out that the “operator” under consideration need not even be an extension of an L1-contraction, but has only to fulfill some mild conditions such as positivity, super-additivity, and a certain contractivity property involving upper integrals
Riesz Transforms Associated with the Number Operator on the Walsh System and the Fermions

AbstractWe prove that, for 2<p<∞ andf∈lp({−1, 1}n), ‖Δ1/2(f)‖pis equivalent to ‖(∑nj=1|f−f∗δej|2)1/2‖pwhereΔis the discrete laplacian and the constants do not depend onfnorn. Such an equivalence is false for 1<p<2 and we give in this case a suitable expression equivalent to ‖Δ1/2(f)‖p. Similar results are obtained for fermionsTviewed as some 2n×2nmatrices equipped with their Schatten normCp, comparing ‖N1/2(T)‖pwith ‖(∑nj=1|Dj(T)|2s)1/2‖p, where theDj's are the annihi- lation operators on the fermions andN=∑nj=1D*jDjis the number operator
A Functional Calculus on the Heisenberg Group and the Boundary Layer Potential □−1+for the ∂-Neumann Problem

AbstractThere are two natural commuting self-adjoint operators in the enveloping algebra of the Heisenberg group: the Heisenberg sublaplacianΔHand the central elementT=−i∂/∂t. The joint spectral theory of these operators is investigated by means of the Laguerre calculus. Explicit convolution kernels are obtained for a large class of functionsΦ(−ΔH, T). In particular we find the kernels of the operators□+, α=−ΔH−αT+T2−Tthat occur in the Kohn solution of the ∂-Neumann problem for the associated Siegel domain
On Stability ofC*-Algebras

AbstractLetAbe aσ-unitalC*-algebra, i.e.,Aadmits a countable approximate unit. It is proved thatAis stable, i.e.,Ais isomorphic toA⊗K where K is the algebra of compact operators on a separable Hilbert space, if and only if for each positive elementa∈Aand eachε>0 there exists a positive elementb∈Asuch that ‖ab‖<εandx*x=a,xx*=bfor somexinA. Using this characterization it is proved among other things that the inductive limit of any sequence ofσ-unital stableC*-algebras is stable, and that the crossed product of aσ-unital stableC*-algebra by a discrete group is again stable
Extremals for Eigenvalues of Laplacians under Conformal Mapping

AbstractG. Pólya and G. Szegő showed in 1951 that for simply connected plane domains, the first eigenvalue of the Laplacian (with Dirichlet boundary conditions) is maximal for a disk, under a conformal mapping normalization. That is, iff(z) is a conformal map of a diskDonto a bounded, simply connected plane domainΩ, normalized by |f′(0)|=1, thenλ1(Ω)⩽λ1(D). Later, Pólya and M. Schiffer showed that actually∑j=1n1λj(Ω)⩾∑j=1n1λj(D),for eachn=1, 2, 3, ….This paper shows that for every convex increasing functionΦ,∑j=1nΦ1λj(Ω)⩾∑j=1nΦ1λj(D),for eachn=1, 2, 3, ….In particular, takingΦ(a)=asfor a fixeds>1 gives that the zeta function of the eigenvalues of the Laplacian is minimal for the disk, under Dirichlet boundary conditions. The bulk of the paper addresses similar questions for simply and doubly connected domains on cones and cylinders and on surfaces of variable curvature, extending the work of C. Bandle, T. Gasser, and J. Hersch. Also, letMgbe anN-dim. Riemannian manifold with boundary and for each smooth functionwon the closure ofM, writeλj(w) for thejth eigenvalue of the operatorw−1Δg, under Dirichlet boundary conditions. Note that in two dimensions,w−1Δgis the Laplace–Beltrami operator of the metricwg. It is proved here that the “zeta-type” functional ∑nj=1Φ(λj(w)−1) is convex with respect to the mass densityw
Kasparov Products and Dual Algebras

AbstractFundamental to the analyticK-homology theory of G. Kasparov [7, 8] is the construction of the external product inK-homologyKi(A)⊗Kj(B)→Ki+j(A⊗B).This construction is modeled on the “sharp product” of elliptic operators over compact manifolds [2], and involves some deep functional-analytic considerations which at first sight may appear somewhatad hoc. A different approach to Kasparov's theory has recently been expounded by N. Higson [5], following the lead of W. Paschke [9]. He constructs a “dual algebra” D(A) for any separableC*-algebraA, in such a way thatKi(A) is canonically identified with the ordinaryK-theory of the dual algebra,K1−i(D(A)). Higson's treatment covers the exactness and excision properties ofK-homology, but stops short of the Kasparov product; it is natural to ask whether the product itself can be given a “dual” interpretation, in terms of the external product in ordinaryK-theory. It is the purpose of this article to show that this can indeed be done. A more leisurely exposition ofK-theory andK-homology from this perspective will appear in [6]
Shifts on the HyperfiniteII1Factor

AbstractR. T. Powers has constructed a family of unital endomorphisms of the hyperfiniteII1factorR. The range of each Powers shiftσis a subfactor of index 2 inR. A cocycle conjugacy invariant for a Powers shift is its commutant index, i.e., the first index k for which the range ofσkhas a nontrivial relative commutant inR. Previously we have shown that all Powers shifts of commutant index 2 are cocycle conjugate. In this paper results are obtained on the classification of the cocycle conjugacy classes of Powers shifts of higher commutant index
C*-Algebras Generated by Operator Systems

AbstractAn operator systemX, such thatX** is aC*-algebra and such that the canonical embedding ofXinX** is a unital complete isometry, is called aC*-system. IfAis a unitalC*-algebra andLis a closed left ideal ofA, thenA/(L+L*) has a natural operator-system structure relative to which it is aC*-system. It is shown that any separableC*-system is of this form for some separableA, and that an arbitrary inseparableC*-system is an inductive limit of separableC*-systems. Associated with any operator systemXis a universalC*-algebraC*u(X) which containsXas a generating subsystem and is maximal among suchC*-algebras in the sense that any other suchC*-algebra is canonically a *-homomorphic image ofC*u(X). IfXis additionally aC*-system, then theC*-subalgebraC*r(X) ofX** generated byXis minimal amongC*-algebras containingXas a generating subsystem in the sense thatC*r(X) is canonically a *-homomorphic image of any other suchC*-algebra. It follows that there is a canonical homomorphismσ:C*u(X)→C*r(X) which extends the identity map onX. WhenXis aC*-algebra of dimension greater than 1,σis never injective. One of the main results of the paper is that there exists a separable nuclearC*-systemXcontainingM2(C) for whichσis an isomorphism. This implies that for any embedding ofXas an operator subsystem of aC*-algebraA, theC*-subalgebra ofAgenerated byXis isomorphic toC*u(X). MoreoverC*u(M2(C)) is notC*-exact andC*u(M2(C)⊆C*u(X), from which it follows thatC*u(X), and henceA, are not exactC*-algebras. ThusX, though a nuclear operator system, cannot be embedded in a nuclearC*-algebra
Discrete Product Systems and Twisted Crossed Products by Semigroups

AbstractA product systemEover a semigroupPis a family of Hilbert spaces {Es:s∈P} together with multiplicationsEs×Et→Est. We viewEas a unitary-valued cocycle onP, and consider twisted crossed productsA⋊β, EPinvolvingEand an actionβofPby endomorphisms of aC*-algebraA. WhenPis quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations ofE, and consider a twisted crossed productBP⋊τ, EPwhich is universal for covariant representations ofEwhenEhas finite-dimensional fibres, and in general is slightly larger. In particular, whenP=Nand dimE1=∞, our algebraBN⋊τ, EN is a new infinite analogue of the Toeplitz-Cuntz algebras TOn. Our main theorem is a characterisation of the faithful representations ofBP⋊τ, EP