pinecone/core-2020-05-10-deduplication huggingface dataset:
Representations of theq-Deformed Lie Algebra of the Group of Motions of the Euclidean Plane

AbstractBounded an unbounded Hilbert space ∗-Representations of theq-deformed Lie algebra of the group of plan motions are studied for different choices of involutions. Integrable (“well-behaved”) representations of the corresponding ∗-algebras are defined and described up to unitary equivalence. In the case of the ∗-algebras with quadratic involutions, analytically defined representations are introduced, and irreducible analytically defined representations are described up to unitary equivalence using dynamical systems. In the case of the ∗-algebras with involutions of the first order all analytically defined representations are shown to be one- dimensional. For these ∗-algebras the problem of unitary classification of all representations defined on some dense invariant domain is shown to be equivalent to the unitary classification of arbitrary families of bounded self-adjoint operators. Integrable (“well-behaved”) representations of these ∗-algebras are defined and described up to unitary equivalence
Howe Correspondence for Real Unitary Groups

AbstractRoger Howe proved that for any reductive dual pair (G,G′) in the symplectic groupSp(2n,R), there is a one-to-one correspondence of irreducible admissible representations of some two-fold covers ofGandG′. We determine this correspondence explicitly for dual pairs of the form (U(p,q),U(r,s)) withr+s=p+q, and describe it in terms of Langlands parameters. In this case, the correspondence may be understood in a natural way as a correspondence of representations of the linear groups, rather than the appropriate covers. We show that every irreducible admissible representation ofU(p,q) occurs in the correspondence with precisely one unitary group of equal rank. This result verifies a conjecture of Harris, Kudla, and Sweet, who investigated the correspondence for unitary groups of equal size overp-adic fields. The correspondence of discrete series representations was determined by J.-S. Li. For induced representations, the correspondence is obtained in a natural way from the corresponding discrete series on the respective Levi factors of the parabolic subgroups ofU(p,q) andU(r,s). Generalizing a result of Li, we show that under the correspondence representations with nonzero cohomology are matched in an interesting way, with unitarity not necessarily preserved. The proof uses the induction principle which is due to Kudla, and an argument involvingK-types and the space of joint harmonics (Howe)
Unbounded Derivations in AT Algebras

AbstractLetAbe a simple unital AT algebra of real rank zero such that it has a unique tracial stateτandK1(A) is neither 0 norZ. For eachϕ∈Hom(K1(A),R) with dense range inRwe construct a closed derivationδinAwhich generates a one-parameter automorphism groupαofAsuch thatτ(δ(u)u*)=2πiϕ([u]) for any unitaryu∈D(δ). Furthermore we construct such anαwith the Rohlin property, which is defined in Kishimoto (Comm. Math. Phys.179(1996), 599–622), in this case the crossed productA×αRis a simple AT algebra of real rank zero. As an application we obtain that for such aC*-algebraAthe kernel of the natural homomorphism of the groupInn(A) of approximately inner automorphisms intoExt(K1(A),K0(A))⊕Ext(K0(A),K1(A)),is the group HInn(A) of automorphisms homotopic to inner automorphisms. Combining with the result of Kishimoto and Kumjian (Trans. Amer. Math. Soc., to appear),Inn(A)/HInn(A) is isomorphic to the above direct sum. As another application of the construction of derivations, we show that ifAis aC*-algebra of the above type andα∈HInn(A) has the Rohlin property and comes fromϕ∈Hom(K1(A),R) with dense range as in Kishimoto and Kumjian (preprint), then the crossed productA×αZis again of the same type; in particularA×αZis an AT algebra. (The other properties are known from Kishimoto [J. Operator Theory40(1998)].
Geometry and Curvature of Diffeomorphism Groups withH1Metric and Mean Hydrodynamics

AbstractHolm, Marsden, and Ratiu (Adv. in Math.137(1998), 1–81) derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equationV(t)+∇U(t)V(t)−α2[∇U(t)]t·ΔU(t)=−gradp(t) where divU=0, andV=(1−α2Δ)U. In this model, the momentumVis transported by the velocityU, with the effect that nonlinear interaction between modes corresponding to length scales smaller thanαis negligible. We generalize this equation to the setting of ann-dimensional compact Riemannian manifold. The resulting equation is the Euler–Poincaré equation associated with the geodesic flow of theH1right invariant metric on Dsμ, the group of volume preserving Hilbert diffeomorphisms of classHs. We prove that the geodesic spray is continuously differentiable fromTDsμ(M) intoTTDsμ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold (Ann. Inst. Grenoble16(1966), 319–361). To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariantH1metric on Dsμis a bounded trilinear map in theHstopology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics
Primitive Triangular UHF Algebras

AbstractWe prove that a large class of triangular UHF algebras are primitive. We use two avenues to obtain our results: a direct approach in which we explicitly construct a faithful, algebraically irreducible representation of the algebra on a separable Hilbert space, as well as an indirect, algebraic approach which utilizes the prime ideal structure of the algebra. Using these results, we completely characterize the primitive ideal spaces of all lexicographic algebras: An ideal there is primitive if and only if it is closed prime. Specializing on the algebrasA(Q,ν), we obtain a complete classification of their algebraic isomorphisms and epimorphisms through the use of a new invariant involving the primitive ideal space. Finally, we characterize the primitive ideal spaces of Z-analytic and order-preserving algebras, and obtain information about their epimorphisms
The Failure of Fatou's Theorem on Poisson Integrals of Pettis Integrable Functions

AbstractIn this paper we prove that for every infinite-dimensional Banach spaceXand every 1⩽p<∞ there exists a strongly measurableX-valuedp-Pettis integrable function on the unit circle T such that theX-valued harmonic function defined as its Poisson integral does not converge radially at any point of T, not even in the weak topology. We also show that this function does not admit a conjugate function. An application to spaces of vector valued harmonic functions is given. In the case thatXdoes not have finite cotype we can construct the function with the additional property of being analytic, in the sense that its Fourier coefficients of negative frequency are null. In the general case we are able to give a countably additive vector measure, analytic in the same sense
The Apostol–Fialkow Formula for Elementary Operators on Banach Spaces

AbstractLetA=(A1,…,An) and (B1,…,Bn) ben-tuples of bounded linear operators on a Banach spaceE. The corresponding elementary operator EA,Bis the mapS↦∑ni=1AiSBionL(E), and Ea,bdenotes the induced operators↦∑ni=1aisbion the Calkin algebra C(E)=L(E)/K(E). Heret=T+K(E) forT∈L(E). We establish that ifEhas a 1-unconditional basis, thendist(Ea,b,W(C(E)))=‖Ea,b‖⩽dist (EA,B,W(L(E))),for all elementary operators EA,BonL(E), whereW(·) stands for the weakly compact operators. There is equality throughout ifE=ℓp, 1<p<∞. Our results extend and improve a corresponding structural result of Apostol and Fialkow (Canad. J. Math.38(1986), 1485–1524), which they proved forE=ℓ2using the non-commutative Weyl–von Neumann theorem due to Voiculescu. By contrast, our arguments are based on subsequence techniques from Banach space theory. As a byproduct we obtain a positive answer to the generalized Fong–Sourour conjecture for a large class of Banach spaces. We also explicitly compute the norm of the generalized derivations↦as−sbon C(ℓ2) (this improves a result due to Fong) and show that the resulting formula fails to hold on C(ℓp)
Poisson Transforms on SomeC*-Algebras Generated by Isometries

AbstractA noncommutative Poisson transform associated to a certain class of sequences of operators on Hilbert spaces, with property (P), is defined on some universalC*-algebras (resp. nonselfadjoint algebras) generated by isometries. Its properties are described and used to study these universal algebras and their representations. As consequences, we obtain a functional calculus, isometric (resp. unitary) dilations, and commutant lifting theorem for the class of sequences of operators with property (P). Our “geometrical” approach leads also to new and elementary proofs as well as extensions of some classical results
An Infinite Dimensional Version of the Schur–Horn Convexity Theorem

AbstractThe Schur–Horn Convexity Theorem states that forain Rnp({U*diag(a)U:U∈U(n)})=conv(Sna),wherepdenotes the projection on the diagonal. In this paper we generalize this result to the setting of arbitrary separable Hilbert spaces. It turns out that the theorem still holds, if we take thel∞-closure on both sides. We will also give a description of the left-hand side for nondiagonalizable hermitian operators. In the last section we use this result to get an extension theorem for invariant closed convex subsets of the diagonal operators
Construction of Miniversal Deformations of Lie Algebras

AbstractWe consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra. It is known that there is in general no “universal” deformation of a Lie algebraLwith a commutative algebra baseAwith the property that for any other deformation ofLwith baseBthere exists a unique homomorphismf:A→Bthat induces an equivalent deformation. Thus one is led to seek aminiversaldeformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base specA, whereAis a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation
Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

AbstractLetSbe a connected and simply connected unimodular solvable Lie group andKa connected compact Lie group acting onSas automorphisms. We call the pair (KS) a Gelfand pair if the Banach ∗-algebraL1K(S) of allK-invariant integrable functions onSis a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K;S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K;S) we realize all irreducibleK-spherical representations ofK⋉Sfrom irreducible unitary representations ofS
Extinction of Superdiffusions and Semilinear Partial Differential Equations

AbstractA superdiffusion is a measure-valued branching process associated with a nonlinear operatorLu−ψ(u) whereLis a second order elliptic differential operator andψis a function from Rd×R+to R+. In the caseL1⩽0 (the so-called subcritical case), the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1. Most results on connections between the superdiffusion and differential equations involvingLu−ψ(u) were obtained for the subcritical case. In the present paper, we assume only thatL1 is a bounded function. In this more general setting, the probability of extinction can be smaller than 1, and we show that his happens if and only if there exists a strictly positive solution of the equationLu−ψ(u)=0 in Rd. We also establish a relation between the probability of extinction in a domainDand strictly positive solutions of equationLu−ψ(u)=0 inDthat are equal to 0 on the boundary ofD. We call such solutions special. For the equation (Δ+c)u=uα, whereΔis the Beltrami–Laplace operator on a complete Riemannian manifoldE, strictly positive solutions inEwere studied in connection with a geometrical problem: Which two functions represent scalar curvatures of two Riemannian metrics related by a conformal mapping (see [1] and references there)
Multipliers for Entire Functions and an Interpolation Problem of Beurling

AbstractWe characterize the interpolating sequences for the Bernstein space of entire functions of exponential type, in terms of a Beurling-type density condition and a Carleson-type separation condition. Our work extends a description previously given by Beurling in the case that the interpolating sequences are restricted to the real line. An essential role is played by a multiplier lemma, which permits us to link techniques from Hardy spaces with entire functions of exponential type. We finally present a characterization of the sampling sequences for the Bernstein space, also extending a density theorem of Beurling
Integrating Unitary Representations of Infinite-Dimensional Lie Groups

AbstractWe show that in the presence of suitable commutator estimates, a projective unitary representation of the Lie algebra of a connected and simply connected Lie group G exponentiates to G. Our proof does not assume G to be finite-dimensional or of Banach–Lie type and therefore encompasses the diffeomorphism groups of compact manifolds. We obtain as corollaries short proofs of Goodman and Wallach's results on the integration of positive energy representations of loop groups and Diff(S1) and of Nelson's criterion for the exponentiation of unitary representations of finite-dimensional Lie algebras
An Existence Result for Optimal Obstacles

AbstractWe consider the optimization problem min{F(g):g∈X(Ω)}, whereF(g) is a variational energy associated to the obstaclegand the classX(Ω) of admissible obstacles is given byX(Ω)={g:Ω→R:g⩽ψonΩ, ∫Ωgdx=c} withψ∈W1,p0(Ω) andc∈R fixed. Generally, this problem does not have a solution and it may happen that the “optimal” obstacle is of relaxed form. Under a monotonicity assumption onF, we prove the existence of a non-relaxed optimal obstacle in the familyX(Ω) through a new method based on the notions ofγandwγ-convergences
Spectral Flow and Bifurcation of Critical Points of Strongly-Indefinite Functionals Part I. General Theory

AbstractSpectral flow is a well-known homotopy invariant of paths of selfadjoint Fredholm operators. We describe here a new construction of this invariant and prove the following theorem:Letψ;:I×U→Rbe aC2function on the product of a real intervalI=[a,b]with a neighborhood U of the origin of a real separable Hilbert space H and such that for each λ in I, 0is a critical point of the functional ψλ≡ψ(λ,·).Assume that the Hessian Lλof ψλat0is Fredholm and moreover that Laand Lbare nonsingular. If the spectral flow of the path{Lλ}λ∈Idoes not vanish, then the interval I contains at least one bifurcation point from the trivial branch for solutions of the equation∇xψ(λ,x)=0.Equivalently: every neighborhood of I×{0}contains points of the form(λ,x) wherex≠0 is a critical point ofψλ
Pure Subnormal Operators Have Cyclic Adjoints

AbstractWe shall prove that a pure subnormal operator has a cyclic adjoint. This answers a question raised by J. Deddens and W. Wogen in 1976. We first prove that a subnormal operatorSon a Hilbert space H has a cyclic adjoint if and only if there exists a compactly supported Borel measureμin the complex plane and a one-to-one linear mapA:H→L2(μ) such thatAS=NμAwhereNμ=MzonL2(μ) . Second, we show that for any pure subnormal operatorS, there exists a one-to-one map intertwiningSand a *-cyclic normal operator. This technique of intertwining is also used to give new proofs of some known results on the cyclicity of adjoints. An application of the main result shows that every pure subnormal operator has a matrix representation that is “almost” lower triangular
Derived Projective Limits of Topological Abelian Groups

AbstractIn this paper, we prove that the category TAbof topological Abelian groups is quasi-Abelian. Using results about derived projective limits in quasi-Abelian categories, we study exactness properties of the projective limit functor in TAb. IfXis a projective system of TAbindexed by a filtering ordered set, we give a necessary and sufficient condition for the derived projective limit ofXto be strict. We also characterize the countable projective systems of complete metrizable Abelian groups which are [equation] -acyclic in TAb
The Paley–Wiener Theorem for the Hua System

AbstractE. Damek, A. Hulanicki, and R. Penney (J. Funct. Anal., in press) studied a canonical system of differential equations (the Hua system) denotedHJKwhich is definable on any Kählerian manifoldM. Functions annihilated by this system are called “Hua-harmonic.” In the case whereMis a bounded homogeneous domain inCnwith its Bergman metric, it was shown that every bounded Hua-harmonic function has a boundary value on the Bergman–Shilov boundary and that the function is reproducible from the Shilov boundary by integration against the reduction of the Poisson kernel for the Laplace–Beltrami operator to the Shilov boundary. This then provided a partial generalization of the results of Johnson and Korányi to the stated context. Significantly, however, no characterization of the resulting space of boundary functions so obtained was given. The current work extends these results in several ways. We show that for a tube domain (i.e., a Siegel domain of type I), the Cauchy–Szegö Poisson kernel also reproduces the Hua-harmonic functions. Since the two kernels agree only in the symmetric case, it follows that the space of boundary functions is dense inL∞if and only if the domain is symmetric. We also show that anL2function is the boundary function for a Hua-harmonic function if and only if its Fourier transform is supported in a certain (typically non-convex) cone. This cone is characterized in terms of the Fourier transformation of the Cauchy–Szegö Poisson kernel
On Weakly Locally Uniformly Rotund Banach Spaces

AbstractWe show that every normed spaceEwith a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm. After obtaining aσ-discrete network of the unit sphereSEfor the weak topology we deduce that the spaceEmust have a countable cover by sets of small local diameter, which in turn implies the renorming conclusion. This solves a question posed by Deville, Godefroy, Haydon, and Zizler. For a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the Kadec property. Moreover, Banach spaces having a countable cover by sets of small local diameter coincide with the descriptive Banach spaces studied by Hansell, so we present here some new characterizations of them
Disk Envelopes of Functions, II

AbstractGiven a Suslin functionφon a domainYin a Banach spaceXone can define two envelopes ofφrelated to the pluripotential theory. The first one, denoted byPφ, is the supremum of all plurisubharmonic functions less thanφ. The value of the other one, denoted byDφ, at some pointzis equal to the infimum of integrals over the boundaries of all closed analytic disks centered atz. In this paper we study the relationship between these envelopes. We show thatDφ⩽P*φfor a large class of functions on domains in Banach spaces, whereP*φis the upper regularization ofPφ. IfX=Cn, then we prove thatD*φis always equal toP*φ
Characteristic Spaces and Rigidity for Analytic Hilbert Modules

AbstractIt is well known that a polynomial in one variable is completely determined by its zeros (counting multiplicities). We generalize this result to an ideal of polynomials in several variables by introducing the characteristic spaces of the ideal. One finds that the ideal is completely determined by its characteristic spaces on a characteristic set. In particular, a primary ideal is completely determined by its characteristic space at any zero point. Some straightforward applications of the above results yield the algebraic reduction theorem for analytic Hilbert modules in several variables. Also, we obtain some general rigidity results for analytic Hilbert modules by using the techniques of AF-envelopes of analytic Hilbert modules
Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

AbstractWe will show that the relation of the heat kernels for the Schrödinger operators with uniform magnetic fields on the hyperbolic planeH2(the Maass Laplacians) and for the Schrödinger operators with Morse potentials onRis given by means of a one-dimensional Fourier transform in the framework of stochastic analysis, where the Brownian motion onH2plays an important role. By using this relation, we will give the explicit forms of the Green functions. As a typical related problem, we will discuss the Selberg trace formula. The close relation of the trace formula with the corresponding classical mechanics will also be discussed
Random Vectors in the Isotropic Position

AbstractLetybe a random vector in Rn, satisfyingEy⊗y=id.LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by finite sums of the tensorsyi⊗yi. We prove that for some absolute constantCE1M∑i=1Myi⊗yi−id⩽C·lognM·(E‖y‖logM)1/logM,provided that the last expression is smaller than 1. We apply this estimate to improve a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position
On Lp-Contractivity of Semigroups Generated by Linear Partial Differential Operators

AbstractThis paper is devoted to the study of contraction semigroups generated by linear partial differential operators. It is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on (Lp)Nforp∈[1, ∞) unlessp=2. Ifp>1 and theLp-dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real-valued coefficients, it is proven that the criterion still fails for operators of order higher than two, except for some fourth order operators if 3/2⩽p⩽3. A class of such fourth order operators is also presented
The Segal–Bargmann Transform on a Symmetric Space of Compact Type

AbstractWe study the Segal–Bargmann transform on a symmetric space X of compact type, mapping L2(X) into holomorphic functions on the complexification XC. We invert this transform by integrating against a “dual” heat kernel measure in the fibers of a natural fibration of XC over X. We prove that the Segal–Bargmann transform is an isometry from L2(X) onto the space of holomorphic functions on XC which are square integrable with respect to a natural measure. These results extend those of B. Hall in the compact group case
Some Properties of Laplacians on Fractals

AbstractKigami has defined an analog of the Laplacian on a class of self-similar fractals, including the familiar Sierpinski gasket. We study properties of this operator. We show that there is a maximal principle for solutions of certain nonlinear equations of the formΔu(x)=F(x, u(x)). We discuss the extension of the Laplacian to non-compact fractal blow-ups, and show that it is essentially self-adjoint, and we prove an analog of Liouville's theorem in some cases. We also give an explicit algorithm for solving the Dirichlet problem for certain domains in the Sierpinski gasket and give a characterization of all harmonic functions on those domains