pinecone/core-2020-05-10-deduplication huggingface dataset:
Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

AbstractThe introduction of a Feller-type condition allows the study of Markovian, cocycles adapted to a Fock filtration to be extended from von Neumann algebras to C*-algebras. It is shown that every such cocycle which, along with its adjoint cocycle, is pointwise strongly continuous and whose associated semigroups are norm continuous, weakly satisfies a quantum stochastic differential equation (QSDE). The matrix of coefficients of this equation may thereby be considered as the generator of the cocycle. The QSDE is satisfied strongly in any of the following cases: when the cocycle is completely positive and contractive, or the driving quantum noise is finite dimensional, or the C*-algebra is finite dimensional and the cocycle generator is bounded. Applying the algebra results to Fock-adapted Markovian cocycles on a Hilbert space we obtain similar characterisations. In particular a contraction cocycle whose Markov semigroup is norm continuous strongly satisfies a QSDE. A representation of cocycles in terms of a family of associated semigroups is central to the present analysis, providing the connection with QSDEs through a parallel work (2000, J. M. Lindsay and S. J. Wills, Probab. Theory Related Fields116, 505–543)
Equivalence of Hardy Submodules Generated by Polynomials

AbstractIn this paper, we obtain a complete classification under unitary equivalence for Hardy submodules on the polydisk which are generated by ideals of polynomials. Let I be an ideal of polynomials in n variables. Since I is generated by finitely many polynomials, I has a greatest common divisor p. So, I can be uniquely written as I=p L which is called the Beurling form of I. Let I1=p1L1, I2=p2L2. We prove that [I1] and [I2] are unitarily equivalent if and only if there are polynomials q1 and q2 with Z(q1)∩Dn=Z(q2)∩Dn=∅ such that |p1q1|=|p2q2| on Tn, and [p1L1]=[p1L2]. Consequently, two principal submodules [p1] and [p2] are unitarily equivalent if and only if there are polynomials q1 and q2 with Z(q1)∩Dn=Z(q2)∩Dn=∅ such that |p1q1|=|p2q2| on Tn. Furthermore, we give a complete similarity classification for submodules generated by homogeneous ideals. Finally, we point out that in the case of the Hardy module on the unit ball, [I1] and [I2] are unitarily equivalent if and only if they are equal. If I1 and I2 are homogeneous ideals, then [I1] and [I2] are quasi-similar if and only if I1=I2
The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on Z

AbstractWe prove that if X is a reflexive translation invariant Banach space of complex sequences on Z that contains all finitely supported sequences, in which the coordinate functionals are continuous, and for every sequence {c(n)} in the space the sequences {c(n)} and {c(−n)} are also in the space, then X has a nontrivial translation invariant subspace. This provides in particular a positive solution to the translation invariant subspace problem for weighted ℓp spaces on Z with even weights, for 1<p<∞. The proof is based on an intermediate result that asserts that if A is an operator on a reflexive real Banach space of dimension greater than one, and there exist non-zero vectors, u in the space and v in the dual space, such that {〈Anu, v〉}∞n=0 is a moment sequence of a finite positive Borel measure on a bounded interval on the real line, then A has a nontrivial invariant subspace
Invariant Subspaces and Nevanlinna–Pick Kernels

AbstractA theorem of Beurling–Lax–Halmos represents a subspace M of H2C(D)—the Hardy space of analytic functions with values in the Hilbert space E and square summable power series—invariant for multiplication by z as ΦH2F, where F is a subspace of E and Φ is an inner function with values in L(F, E). When the Hardy space is replaced by the Hilbert space H(k) determined by a Nevanlinna–Pick kernel k, such as the Dirichlet kernel or the row contraction kernel on the ball in Cd, the BLH Theorem survives with F an auxiliary Hilbert space and Φ a L(F, E) valued function which is inner in the sense that the operator MΦ of multiplication by Φ is a partial isometry. Under mild additional hypotheses, when E=C, Mz, the operator of multiplication by z, is cellularly indecomposable and has the codimension one property; however, if M is invariant for Mz, M⊖MzM need not be a cyclic subspace for Mz restricted to M
Weakly Elliptic Systems with Obstacle Constraints III. Complex Eigenvalues and Singular Systems

AbstractUsing traditional methods, especially that of eigenfunction expansions combined with the penalty method, sharp conditions are given for the existence of solutions to general weakly elliptic linear N×N second order systems subjected to obstacle type constraints. The result applies to systems with defining coefficient matrices A and B having complex eigenvalues as well as to the case of singular systems—a situation not well treated in the literature. These conditions are placed on the matrix pencil Aλ−B. The effectiveness of the result is then tested on some examples, first in control theory and then for the systems that arise in Part II of this series (D. R. Adams, 2000, J. Geom. Anal.10, 375–412)
Geometric Theory of Spaces of Integral Polynomials and Symmetric Tensor Products

AbstractWe investigate the nature of extreme, (weak*-) exposed, and (weak*-) strongly exposed points of the unit ball of spaces of n-homogeneous integral polynomials on, and n-fold symmetric products of, a Banach space E. For the space of integral polynomials we show the set of extreme points is contained in the set {±φn:φ∈E′, ‖φ‖=1}. We give Šmul'yan type theorems for spaces of n-homogeneous polynomials and n-fold symmetric tensors that characterise weak*-exposed (resp. weak*-strongly exposed) points in terms of Gâteaux (resp. Fréchet) differentiability of the norm on various spaces of tensor products and polynomials. Our study of the geometry of these spaces has many applications: When E has the Radon–Nikodým property we show that the spaces of n-homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n. When the dimensions of E and n are both at least 2 then the space of n-homogeneous polynomials on E is neither smooth nor rotund. For a certain class of reflexive Banach space the space of n-homogeneous approximable polynomials on E is either reflexive or is not isometric to a dual Banach space. We conclude with a Choquet Theorem for a space of homogeneous polynomials
Invariance Properties of Thematic Factorizations of Matrix Functions

AbstractWe study the problem of invariance of indices of thematic factorizations. Such factorizations were introduced by V. V. Peller and N. J. Young (1994, J. Funct. Anal.120, 300–343) for studying superoptimal approximation by bounded analytic matrix functions. As shown by Peller and Young, the indices may depend on the choice of a thematic factorization. We introduce the notion of a monotone thematic factorization. The main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function itself. We obtain similar results for so-called partial thematic factorizations
Unitary Interpolants and Factorization Indices of Matrix Functions

AbstractFor an n×n bounded matrix function Φ we study unitary interpolants U, i.e., unitary-valued functions U such that U(j)=Φ(j), j<0. We are looking for unitary interpolants U for which the Toeplitz operator TU is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener–Hopf factorization indices of U in terms of superoptimal singular values of Φ and thematic indices of Φ−F, where F is a superoptimal approximation of Φ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions Φ of class H∞+C we study unitary interpolants U of class QC
Weyl–Heisenberg Frames, Translation Invariant Systems and the Walnut Representation

AbstractWe present a comprehensive analysis of the convergence properties of the frame operators of Weyl–Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator. We show that symmetric, norm and unconditional convergence of the Walnut series are all different, but that weak and norm convergence are the same, while there are WH-systems for which the Walnut representation has none of these convergence properties. We make a detailed study of the CC-condition (a sufficient condition for WH-systems to have finite upper frame bounds) and show that (for ab rational) a uniform version of this passes to the Wexler–Raz dual. We also show that a condition of Tolimieri and Orr implies the uniform CC-condition. We obtain stronger results in the case when (g, a, b) is a WH-system and ab is rational. For example, if ab is rational, then the CC-condition becomes equivalent to the unconditional convergence of the Walnut representation—even in a more general setting. Many of the results are generalized to shift-invariant systems. We give classifications for numerous important classes of WH-systems including: (1) The WH-systems for which the frame operator extends to a bounded operator on Lp(R), for all 1⩽p⩽∞; (2) The WH-systems for which the frame operator extends to a bounded operator on the Wiener amalgam space; (3) The families of frames which have the same frame operator
Extension Properties for the Space of Compact Operators

AbstractLet Z be a fixed separable operator space, X⊂Y general separable operator spaces, and T:X→Z a completely bounded map. Z is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to Y and the Mixed Separable Extension Property (MSEP) if every such T admits a bounded extension to Y. Finally, Z is said to have the Complete Separable Complementation Property (CSCP) if Z is locally reflexive and T admits a completely bounded extension to Y provided Y is locally reflexive and T is a complete surjective isomorphism. Let K denote the space of compact operators on separable Hilbert space and K0 the c0 sum of Mn's (the space of “small compact operators”). It is proved that K has the CSCP, using the second author's previous result that K0 has this property. A new proof is given for the result (due to E. Kirchberg) that K0 (and hence K) fails the CSEP. It remains an open question if K has the MSEP; it is proved this is equivalent to whether K0 has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed
Logarithmic Sobolev Inequality on Free Loop Groups for Heat Kernel Measures Associated with the General Sobolev Spaces

AbstractIn this paper we will prove the logarithmic Sobolev inequality on free loop groups for various heat kernel measures which P. Malliavin (1989, 1991, in “Diffusion Process and Related Problems in Analysis (M. A. Pinsley, Ed.), Vol. I, Birkhäuser, Basel) constructed. Those measures are associated with the Sobolev spaces of order s (s>1/2) of the free loops in the Lie algebra. We will equipp the free loop groups with those metrics and will show that a formula of Weitzenböck type holds, which enables us to apply the method of Driver and Lohrenz (1996, J. Funct. Anal.146, 381–448)
Spectral Theory of Pauli–Fierz Operators

AbstractWe study spectral properties of Pauli–Fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field. We give precise estimates of the location and multiplicity of the singular spectrum of such operators. Applications of these estimates, which will be discussed elsewhere, concern spectral and ergodic theory of non-relativistic QED. Our proof has two ingredients: the Feshbach method, which is developed in an abstract framework, and Mourre theory applied to the operator restricted to the sector orthogonal to the vacuum
A Reflected Stochastic Heat Equation as Symmetric Dynamics with Respect to the 3-d Bessel Bridge

AbstractWe prove that a stochastic heat equation with reflection at 0, on the spatial interval [0, 1] with Dirichlet boundary conditions and additive white-noise, admits an explicit symmetrizing invariant measure on C([0, 1]): the 3-d Bessel Bridge, i.e., the law of the modulus of a 3-dimensional Brownian motion conditioned to be 0 at time 1, a classical measure in probability theory, also connected with the theory of excursions of Brownian motion. This is a non-trivial example of a Gibbs-type measure being singular with respect to the reference Gaussian measure and concentrated on the convex set of positive, continuous functions on [0, 1]
Asymptotic Expansion of the Solution to the Nonlinear Schrödinger Equation with Nonlocal Interaction

AbstractThis paper deals with the equationiut+(1/2)Δu=λ(|x|−1*|u|2)u,u(0, x)=u0(x).Here, u is a complex-valued function of (t, x)∈R×Rn, n⩾2, and λ is a real number. If u0 is small in L2, s with s>(n/2)+2, then the solution u(t) behaves asymptotically asu(t, x)=(it)−n/2exp((i|x|2/2t)−iS(t, x/t))×φ(x/t)+t−1∑j=02ψ1, j(x/t)(logt)j+o(t−(n/2)−1)uniformly in Rn as t→∞. Here φ is a suitable function called the modified scattering state, and the functions S, ψ1, j, j=0, 1, 2, are represented explicitly by using φ
Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials

AbstractNo infinite dimensional Banach space X is known which has the property that for m⩾2 the Banach space of all continuous m-homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m-homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an unconditional basis (x*i), the approximable (nuclear) m-homogeneous polynomials on X have an unconditional basis if and only if the monomial basis with respect to (x*i) is unconditional. Moreover, we determine an asymptotically correct estimate for the unconditional basis constant of all m-homogeneous polynomials on ℓnp and use this result to narrow down considerably the list of natural candidates X with the above property
Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras

AbstractWe develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule XE over a C*-algebra AE such that the crossed product A⋊EN is naturally isomorphic to AE⋊XEZ. The Cuntz–Pimsner algebra OE is isomorphic to AE⋊XEZ where AE and XE are quotients of AE, resp. XE. If E is full and the left action is by generalized compact operators, then XE is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general, XE is merely an essential Hilbert C*-bimodule. Slightly extending previous results on crossed products by equivalence bimodules, we apply our dilation theory to show that for full C*-correspondences over unital C*-algebras, OE is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani
The Unitary Implementation of a Locally Compact Quantum Group Action

AbstractIn this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact quantum group (M, Δ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion Nα⊂N↪Mα⋉N is a basic construction. Here Nα denotes the fixed point algebra and Mα⋉N is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion Nα⊂N is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal.137, 466–543; 1998, J. Funct. Anal.154, 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand.84, 297–319): every integrable outer action with infinite fixed point algebra is a dual action
Entropy of Automorphisms of II1-Factors Arising from the Dynamical Systems Theory

AbstractLet a countable amenable group G act freely and ergodically on a Lebesgue space (X, μ), preserving the measure μ. If T∈Aut(X, μ) is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism αT of the II1-factor M=L∞(X, μ)⋊G. We prove that if T commutes with the action of G then H(αT)=h(T), where H(αT) is the Connes–Størmer entropy of αT, and h(T) is the Kolmogorov–Sinai entropy of T. We also prove that for given s and t, 0⩽s⩽t⩽∞, there exists a T such that h(T)=s and H(αT)=t
Extremal Properties of Central Half-Spaces for Product Measures

AbstractWe deal with the isoperimetric and the shift problem for subsets of measure 1/2 in product probability spaces. We prove that the canonical central half-spaces are extremal in particular cases: products of log-concave measures on the real line satisfying precise conditions and products of uniform measures on spheres, or balls. As a corollary, we improve the known log-Sobolev constants for Euclidean balls. We also give some new results about the related question of estimating the volume of sections of unit balls of ℓp-sums of Minkowski spaces
The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

AbstractWe prove the following new characterization of Cp (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The “twisted tube” method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces
Singular Integrals with Flag Kernels and Analysis on Quadratic CR Manifolds

AbstractWe study a class of operators on nilpotent Lie groups G given by convolution with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0)⊂V1⊂…⊂Vk⊂…⫋G. We show that product kernels can be written as finite sums of flag kernels, that flag kernels can be characterized in terms of their Fourier transforms, and that flag kernels have good regularity, restriction, and composition properties. We then apply this theory to the study of the □b-complex on certain quadratic CR submanifolds of Cn. We obtain Lp regularity for certain derivatives of the relative fundamental solution of □b and for the corresponding Szegö projections onto the null space of □b by showing that the distribution kernels of these operators are finite sums of flag kernels
A Factorization Problem for Normal Completely Bounded Mappings

AbstractGiven an operator space X and a von Neumann algebra A, we consider a contractive mapping q:A⊗ehX⊗ehA→NCB(X*, A) formally defined by q(∑aj⊗xjk⊗bk)=∑xjk⊗ajbk, from the extended Haagerup tensor product A⊗ehX⊗ehA into the space of w*-continuous completely bounded maps from X* into A. We characterize elements of the range space Im(q) by a factorization property involving decomposable operators and investigate various properties of that space. In the case when X=B∗ is the predual of a von Neumann algebra B, Im(q) is included in the space DEC(B, A) of decomposable operators from B into A. Regarding q as having values in that space, we show that q is a quotient map onto its range. Then we prove that DEC(B, A) is a normal dual operator A-bimodule and that Im(q)⊂DEC(B, A) is a strong operator A-bimodule
The Standard Form of an E0-Semigroup

AbstractWe show that the conjugacy classes of continuous semigroups of *-endomorphisms of B(H) that possess a pure, normal, absorbing state (i.e., standard E0-semigroups) are indexed by the orbits of the natural action of the group of local cocycles on the set of intertwining semigoups of isometries. We use this to show that, in passing from an arbitrary E0-semigroup α to a cocycle perturbation in standard form, the boundary representation is compressed to a hereditary subalgebra of the domain D(δα) of α
The Range of Vector Measures over Topological Sets and a Unified Liapunov Theorem

AbstractLet (X, Σ, μ) be a finite, nonatomic, measure space. Let G=span{g1, g2, …, gn}⊆L1, and let the support of G be X, a.e. For f∈L∞, put M(f)=(∫Xfg1dμ, ∫Xfg2dμ, …, ∫Xfgndμ). Then Q={M(f):f∈B(L∞)} is a compact convex set. Liapunov's classsical theorem is that also Q={M(s):s∈extB(L∞)}. This paper characterizes when the functions, s, in Liapunov's theorem can further be restricted to being the signs of continuous functions. That is, suppose there is a topology on X, and that Σ is the Baire sets. Let S be the collection of supports of continuous non-negative functions. Theorem.The following are equivalent: (i)Q={M(s):s∈extB(L∞),ands=sgnf for some f∈C(X)}, (ii)for every g∈G, {x:g(x)>0}and{x:g(x)⩽0} are in S, a.e. The theorem is proved in a general setting. If the σ-field in the general theorem is arbitrary, the theorem becomes Liapunov's Theorem. The theorem above results when the σ-field is the Baire sets. A setting with the Borel sets produces Q as the range of M over extreme functions that are both lower semi-continuous a.e. and upper semi-continuous a.e
The Non-commutative Flow of Weights on a Von Neumann Algebra

AbstractThe flow of weights of Connes and Takesaki is a canonical functor from the category of separable factors to the category of ergodic flows. The non-commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi-finite von Neumann algebras equipped with trace scaling one parameter automorphism groups with conjugations as morphisms. The constructions of these two functors are very similar. The flow of weights functor is obtained by looking at all semi-finite normal weights on a factor with the Murray–von Neumann equivalence relation. The non-commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi-finite normal weights by the Connes cocycle. Not only does this construction put a period to the search for a canonical construction of the core of a factor of type III, but it also allows us to put the characteristic square of a factor obtained by Katayama, Sutherland, and Takesaki in a new perspective. The power of this new approach is seen in an ultimate solution to a long standing question of extending the extended modular automorphism of a dominant weight to an arbitrary weight, which has been left open ever since the introduction of extended modular automorphisms by Connes and Takesaki over 20 years ago. The construction of the functor ties together the theory of Lp-spaces of Haagerup, Kosaki, Hilsum, Terp, and Izumi to the structure theory of a factor of type III. In fact, the non-commutative flow of weights is obtained by the analytic continuation of Lp-spaces to a pure imaginary value of p