pinecone/core-2020-05-10-deduplication huggingface dataset:
Contractions in Von Neumann Algebras

AbstractWe associate to any contractionT(and, more generally, to any operatorTof classCρ) in a von Neumann algebraMan operator kernelKα(T) (|α|<1) which allows us to define various kinds of functional calculis forT. WhenMis finite, we use this kernel to give a short proof of the Fuglede-Kadison theorem on the location of the trace and to prove that a contractionTinMis unitary if and only if its spectrum is contained in the unit circle. By using a perturbation of the kernelKα(T) we give, for any operatorTof classCρacting on a separable Hilbert spaceH, a short proof of the power inequality for the numerical range and an accurate conjugacy (to a contraction) result forT. We also get a generalized von Neumann inequality which gives a good control of ‖f(rT*)x+g(rT)x‖ (0⩽r<1) forx∈Handf,gin the disc algebra. Finally, we associate to anyC1contraction in a Hilbert space an asymptotic kernel which allows us to describe new kinds of invariant subspaces forT, from the positive solutionsXof the operator equationT*XT=X. In particular, we recover some results of Beauzamy based on the notion of invariant subspace of “functional type.
Quantum Tori and the Structure of Elliptic Quasi-simple Lie Algebras

AbstractWe study and classify those tame irreducible elliptic quasi-simple Lie algebras which are simply laced and of rankl⩾3. The first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates. When the type isDorEthe coordinates are Laurent polynomials in ν variables, while for typeAthe coordinates can be any quantum torus in ν variables. The next step is to study the universal central extension as well as the derivation algebra of the core. These are related to the first Connes cyclic homology group of the coordinates. The final step is to use this information to give constructions of Lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras
Conjugation in Representation Categories of Multiplicative Unitaries and Their Actions on C*-Algebras

AbstractA conjugation functorFon a full subcategory ofR(V), the representation category of a multiplicative unitaryV, is defined. IfVhas a conjugate, it is also regular and the domain ofFis allR(V). Examples of selfconjugate multiplicative unitaries are discussed. A coaction of the HopfC*-algebra associated withVon the Cuntz algebraOdis canonically defined by a unitary objectWofR(V) acting on ad-dimensional Hilbert space. As in the group action case ifd=∞ andWbelongs to the domain ofF, ergodic coactions are often characterized by the absence of finite dimensional subrepresentations ofW. Furthermore model actions of compact quantum group duals onC*-algebras are defined
On the Structure of Classical and Quantum Flows

AbstractIn the framework of quantum probability, stochastic flows on manifolds and the interaction representation of quantum physics become unified under the notion ofMarkov cocycle. We prove a structure theorem forσ-weakly continuous Markov cocycles which shows that they are solutions of quantum stochastic differential equations on the largest *-subalgebra, contained in the domain of the generator of the Markov semigroup, canonically associated to the cocycle. The result is applied to prove that any Markov cocycle on the Clifford bundle of a compact Riemannian manifold, whose structure maps preserve the smooth sections and satisfy some natural compatibility conditions, uniquely determines a family of smooth vector fields and a connection, with the property that the cocycle itself is induced by the stochastic flow along the paths of the classical diffusion on the manifold, defined by these vector fields and by the Ito stochastic parallel transport associated to the connection. We use the language and techniques of quantum probability but, even when restricted to the classical case, our results seem to be new
Coherent State Transforms for Spaces of Connections

AbstractThe Segal–Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie groupGwith its normalized Haar measureμH, the Hall transform is an isometric isomorphism fromL2(G, μH) toH(GC)∩L2(GC, ν), whereGCthe complexification ofG,H(GC) the space of holomorphic functions onGC, andνan appropriate heat-kernel measure onGC. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie groupGby (a certain extension of ) the spaceA/Gof connections modulo gauge transformations. The resulting “coherent state transform” provides a holomorphic representation of the holonomyC* algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4 dimensions
Common Zero Structure of Rational Matrix Functions

AbstractThe paper is concerned with description of common zero data of two square size rational matrix functions. The notion of a common null pair, which encodes the data, is introduced and studied from various points of view. The main result of the paper is a characterization of the common null pair in terms of realization of the functions in question. Connections with common minimal divisors of rational matrix functions are elucidated. The recently developed theory of null-pole triples and Sylvester data sets is extensively used in the proofs
On Some Operator Colligations and Associated Reproducing Kernel Pontryagin Spaces

AbstractLetsbe a Schur function, that is a function analytic and contractive in the unit disk D. Then the function 1−s(z)s(ω)*/1−zω* is positive in D. L. de Branges and J. Rovnyak proved that the associated reproducing kernel Hilbert space provides the state space for a coisometric realization ofs. In a previous work we extended this result to the case of operator valued functions with the denominator 1−zω* replaced bya(z)a(ω)*−b(z)b(ω)*, whereaandbare analytic functions subject to some conditions. In the present work we remove the positivity condition and allow the kernel to have a number of negative squares. Moreover, we consider functions whose values are bounded operators between Pontryagin spaces with the same index. We show that there exist reproducing kernel Pontryagin spaces which provide unitary, isometric, and coisometric realizations of the function. We also study the projective version of the above kernel
A Theory of Strongly Continuous Semigroups in Terms of Lie Generators

AbstractLetXdenote a complete separable metric space, and let C(X) denote the linear space of all bounded continuous real-valued functions onX. A semigroupTof transformations fromXintoXis said to be jointly continuous if the mapping (t, x)→T(t)xis jointly continuous from [0, ∞)×XintoX. The Lie generator of such a semigroupTis the linear operator in C(X) consisting of all ordered pairs (f, g) such thatf, g∈C(X), and for eachx∈X,g(x) is the derivative at 0 off(T(·)x). We completely characterize such Lie generators and establish the canonical exponential formula for the original semigroup in terms of powers of resolvents of its Lie generator. The only topological notions needed in the characterization are two notions of sequential convergence, pointwise and strict. A sequence in C(X) converges strictly if the sequence is uniformly bounded in the supremum norm and converges uniformly on compact subsets ofX
Homogeneous Tuples of Multiplication Operators on Twisted Bergman Spaces

AbstractLetBthe Bergman kernel on the domainΩn,mofn×mcontractive complex matrices (m⩾n⩾1). Let W=Wn,mbe the associated Wallach set consisting of theλ⩾0 for whichBλ/(m+n)is (non-negative definite and hence) the reproducing kernel of a functional Hilbert space H(λ). Forλ∈W, we examine themn-tupleM(λ)of operators on H(λ)whose components are multiplications by themnco-ordinate functions. This tuple is homogeneous with respect to the group action ofPSU(n, m) on the matrix ball. Utilising this group action we are able to determine the set of allλ∈W for which (i)M(λ)is bounded, and for which (ii)M(λ)is (bounded and) jointly subnormal. Further, the joint Taylor spectrum ofM(λ)is determined for allλas in (i). The subnormality ofM(λ)turns out to be closely tied with the representation theory ofPSU(n, m). Namely,M(λ)is subnormal precisely when the natural (projective) representation ofPSU(n, m) on the twisted Bergman space H(λ)is a subrepresentation of an induced representation of multiplicity 1. Finally, we examine the values ofλfor whichM(λ)admits its Taylor spectrum as ak-spectral set, and obtain incomplete results on this question. This question remains open and interesting onn−1 gaps, that is, forλbelonging to the union ofn−1 pairwise disjoint open intervals. Most of the techniques developed in this paper are applicable to all bounded Cartan domains, though we stick to the matrix domainsΩn, mfor concreteness
Obtaining Eigenvalues of the Schrodinger Equation by Stochastic Control Methods

AbstractThis paper shows that each eigenvalue of the stationary Schrodinger equation can be characterized as the minimum value of a performance functional associated with a stochastic control problem. The stochastic control problem is defined for regions bounded by nodes of the solution to the Schrodinger equation. A set of admissible controls is defined and it is shown that the process defined by each control does not reach the boundary of the region being considered. It is also shown that there is a unique stationary distribution associated with each admissible control. The optimal control for this problem is defined in terms of the solution to the equation
Boundedness and Compactness of Generalized Hankel Operators on Bounded Symmetric Domains

AbstractThe mapb→Hb:=(I−P)MbPfrom analytic functions on the unit diskDto the associated Hankel operators on the Hardy space or the weighted Bergman spaces is known to be an important tool in studying the “size” of the functionbin terms of the “size” of Hb. Moreover, this map is equivariant, namely it intertwines the natural actions of the Möbius groupAut(D) on functions and operators. This theory extends to some extent to the context of the open unit ballBninCn, but it fails in Cartan domains of rankr>1, because in this case the mapb→Hbtrivializes as Hbis compact only if Hb=0 (andbis constant). We study generalizations Abof Hankel operators in the context of weighted Bergman spaces over a Cartan domain of tube type with rankr>1. The mapb→Abis equivariant and non-trivial. We study also in this context generalized Bloch and little Bloch spaces (B and B0respectively), and generalized BMOA and VMOA spaces with respect to weighted volume measure. The main results are that Abis bounded if and only ifb∈B if and only ifb∈BMOA, and Abis compact if and only ifb∈B0if and only ifb∈VMOA
Estimates for Sums of Eigenvalues for Domains in Homogeneous Spaces

AbstractLet&0Hgr;⊆Mbe a bounded open subset of a homogeneous Riemannian manifold, and letσk=λ1+…+λkbe the sum of the firstkeigenvalues of the Dirichlet Laplacian on&0Hgr;, and similarlyσk=λ1+…+λkfor the Neumann Laplacian. We give bounds forσkandσkgeneralizing results of Li–Yau and Kröger in the caseM=Rn. We prove a “generic theorem” which in the case of compactMsaysσk⩾p(&0Hgr;)Σ(k/p(&0Hgr;))⩾σkwherep(&0Hgr;)=|&0Hgr;|/|M| is the relative volume of&0Hgr;andΣ(x) is the eigenvalue sum function forM(interpolated linearly for non integer values). For noncompactMthe statement isσk⩾|&0Hgr;|Σ(k/|&0Hgr;|) whereΣis a renormalized eigenvalue sum function forM(defined using the spectral resolution ofΔonM). There are also estimates in the other direction of the same form with error terms. The same generic theorems hold for Laplacian onp-forms, and for subelliptic Laplacians on subRiemannian manifolds. To give life to such generic theorems it is necessary to compute theΣfunction for a variety of examples. For Euclideann-space,Σ(x)=(n/(n+2))Cnx1+2/nwhereCnis the Weyl constant, so our generic result includes the known results. We discuss the computation ofΣfor spheres, hyperbolic spaces, noncompact symmetric spaces, and the Heisenberg groups
A Probabilistic Approach to Semi-classical Approximations

AbstractWe study in this paper the semi-classical expansion of the Schrödinger equation, using a probabilistic approach based on the Wiener measure. Using almost-analytic extensions, we exhibit a probabilistic ansatz for the wave function. We show that this ansatz approximates very well the wave function in the semi-classical regime, and gives the semi-classical expansion under mild hypothesis on the potential at infinity, and no analyticity conditions. In this paper, the study takes place before the caustics
OnC*-Algebras Generated by Idempotents

AbstractThe topic of the present paper is concrete Banach andC*-algebras which are generated by a finite number of idempotents. Our first result is that, for each finitely generated Banach algebra A, there is a numbern0so that the algebra An×nof alln×nmatrices with entries in A is generated by three idempotents whenevern⩾n0, and that An×nis generated by two idempotents if and only ifn=2 and if A is singly generated. As an application we find that the algebraCn×n(K) of all continuous Cn×n-matrix-valued functions on a compactK⊂C with connected complement but without interior points, is generated by 2 or 3 idempotents in casen=2 orn>2, respectively. This result is used to construct examples ofC*-algebras which are generated by 2 idempotents but not 2 projections. For these algebras, the standard 2×2 matrix symbol fails to be symmetric. We finally show that eachC*-algebra satisfying a polynomial identity (in particular, eachC*-algebra generated by two idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one
An Integral Operator into Dolbeault Cohomology

AbstractA construction is given for an integral transform from sections of a vector bundle over one manifold into Dolbeault cohomology of a (related) holomorphic vector bundle over a second manifold. It is demonstrated that the transform arises naturally in appropriate homogeneous situations where it is shown to agree with a certain representation theoretic intertwining operator due to Barchini, Knapp and Zierau. Our construction is geometric with the underlying correspondence being a double fibration of the type arising in connection withX-ray transforms. From this point of view many properties of the transform are immediate
Irreducible Inclusions of Factors, Multiplicative Unitaries, and Kac Algebras

AbstractFrom an irreducible depth 2 inclusion of factors, verifying a regularity condition, we construct a multiplicative unitary, and an action, at every level of the canonical tower constructed from the inclusion; when this inclusion admits a faithful semi-finite normal operator-valued weight, stronger conditions are given, and the tower appears then as a crossed-product construction. In particular we rederive Herman and Ocneanu's results when the inclusion admits a faithful normal conditional expectation, and the tower is then the crossed-product construction, alternatively by a compact quantum group and by its dual, and, more precisely, according to Yamagami's result, by a compact type Kac algebra and by its dual
The Lattice of Ideals of a Triangular AF Algebra

AbstractWe study triangular AF (TAF) algebras in terms of their lattices of closed two-sided ideals. Not (isometrically) isomorphic TAF algebras can have isomorphic lattices of ideals; indeed, there is an uncountable family of pairwise non-isomorphic algebras, all with isomorphic lattices of ideals. In the positive direction, if A and B are strongly maximal TAF algebras with isomorphic lattices of ideals, then there is a bijective isometry between the subalgebras of A and B generated by their order preserving normalizers. This bijective isometry is the sum of an algebra isomorphism and an anti-isomorphism. Using this, we show that if the TAF algebras are generated by their order preserving normalizers and are triangular subalgebras of primitive C*-algebras, then the lattices of ideals are isomorphic if and only if the algebras are either (isometrically) isomorphic or anti-isomorphic. Finally, we use complete distributivity to show that there are TAF algebras whose 0?lattices of ideals can not arise from TAF algebras generated by their order preserving 8?normalizers. Our techniques rely on constructing a topological binary relation based on the lattice of ideals; this relation is closely connected to the spectrum or fundamental relation (also a topological binary relation) of the TAF algebra
Projections of Measures on Nilpotent Orbits and Asymptotic Multiplicities ofK-Types in Rings of Regular Functions II

AbstractLetGbe the adjoint group of a real semi-simple Lie algebragand letKbe a maximal compact subgroup ofG.KC, the complexification ofK, acts onp*C, the complexified cotangent space ofG/KateK. If O is a nilpotentKCorbit inp*C, we study the asymptotic behavior of the multiplicities ofK-types in the moduleR[O], the regular functions on the Zariski closure of O. Sekiguchi has shown that each such orbit O corresponds naturally to a nilpotentGorbitΩing*. We show that if the split rank ofgequals one, then the asymptotic behavior ofK-types is determined precisely byβΩ, the canonical Liouville measure onΩ. David Vogan has conjectured that this relationship is true in general. We show that whengis complex, this conjecture can be reduced to the case in which O is not induced from a nilpotent orbit of a proper Levi-subalgebra ofg. We also relate this conjecture to a recent result of Schmid and Vilonen that links thecharacteristic cycleof a Harish Chandra module to itsasymptotic support
Critical Point Theory for Indefinite Functionals with Symmetries

AbstractLetXbe a Hilbert space andφ∈C1(X, R) be strongly indefinite. Assume in addition that a compact Lie groupGacts orthogonally onXand thatφis invariant. In order to find critical points ofφwe develop the limit relative category of Fournieret al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem forG=Z/2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed point spaces. The proofs consist of a direct reduction to a relative Borsuk–Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract critical point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action
Regularity of Invariant Measures: The Case of Non-constant Diffusion Part

AbstractWe prove regularity (i.e., smoothness) of measuresμon Rdsatisfying the equationL*μ=0 whereLis an operator of typeLu=tr(Au″)+B·∇u. HereAis a Lipschitz continuous, uniformly elliptic matrix-valued map andBis merelyμ-square integrable. We also treat a class of corresponding infinite dimensional cases where Rdis replaced by a locally convex topological vector spaceX. In this casesμis proved to be absolutely continuous w.r.t. a Gaussian measure onXand the square root of the Radon–Nikodym density belongs to the Malliavin test function space D2, 1
Entropy, Reversible Diffusion Processes, and Markov Uniqueness

AbstractConsider a symmetric bilinear form Eϕdefined on C∞c(Rd) by[formula]In this paper we study the stochastic process associated with the smallest closed markovian extension of (Eϕ, C∞c), and give a new proof of Markov uniqueness (i.e. the uniqueness of a closed markovian extension) based on purely probabilistic arguments. We also give another purely analytic one. As a consequence, we show that all invariant measures are reversible, provided they are of finite energy. The problem of uniqueness of such measures is also partially solved
Multipliers with Natural Local Spectra on Commutative Banach Algebras

AbstractSeveral notions from the abstract spectral theory of bounded linear operators on Banach spaces are investigated and characterized in the context of multipliers on a semi-simple commutative Banach algebra. Particular emphasis is given to the determination of the local spectra of such multipliers in connection with Dunford's property (C), Bishop's property (β), and decomposability in the sense of Foiaş. The strongest results are obtained for regular Tauberian Banach algebras with approximate units and for multipliers whose Gelfand transforms on the spectrum of the Banach algebra vanish at infinity. The general theory is then applied to convolution operators induced by measures on a locally compact abelian groupG. Our results give new insight into the spectral theory of convolution operators on the group algebraL1(G) and on the measure algebraM(G). In particular, we identify large classes of measures for which the corresponding convolution operators have excellent spectral properties. We also obtain a number of negative results such as examples of convolution operators onL1(G) without natural local spectra, but with natural spectrum in the sense of Zafran
The Feller Property for Absorption Semigroups

AbstractLetU=(U(t); t⩾0) be a substochastic strongly continuous semigroup onL1(X, m) whereXis locally compact andma Borel measure onX. We give conditions on absorption ratesVimplying that the (strong) Feller property carries over fromU* toU*V. These conditions are essentially in terms of the Kato class associated withU. Preparing these results we discuss the perturbation theory of strongly continuous semigroups and properties of one-parameter semigroups onL∞(m). In the symmetric case of Dirichlet forms we generalize the results to measure perturbations. For the case of the heat equation on Rdwe show that the results are close to optimal
Diffusions and Random Shadows in Negatively Curved Manifolds

AbstractLetMbe ad-dimensional complete simply-connected negatively-curved manifold. There is a natural notion of Hausdorff dimension for its boundary at infinity. This is shown to provide a notion of global curvature or average rate of growth in two probabilistic senses: First, on surfaces (d=2), it is twice the critical drift separating transience from recurrence for Brownian motion with constant-length radial drift. Equivalently, it is twice the criticalβfor the existence of a Green function for the operatorΔ/2−β∂r. Second, for anyd, it is the critical intensity for almost sure coverage of the boundary by random shadows cast by balls, appropriately scaled, produced from a constant-intensity Poisson point process
Invariance of Malliavin Fields on Ito's Wiener Space and on Abstract Wiener Space

AbstractLet (E, H, m) be an abstract Wiener space and (Ω, H, γ) be the corresponding Ito's Wiener space whereΩconsists of all the linear (but not necessarily continuous) functionals on the Hilbert spaceH. We show that one can always linearly embed (E, H, m) into (Ω, H, γ) in such a way that the family of allγ-regular measures onΩare exactly the family of the extensions of all probability measures of finite energy onE. A subsetAofEis a slim set if and only if it is a M-null set inΩ. The family of all MalliavinTr-fields onEare exactly the family of all the restrictions of MalliavinTr-fields onΩ. Moreover, the one to one mapping between Malliavin fields onΩand those onEis commutable with the gradient operator and keeps the Sobolev norms invariant. Hence most of the results of Malliavin calculus known for abstract Wiener space can be transferred to the Ito's Wiener space and vice versa