Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets

Michael Didas

Displaying similar documents to “Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets”

Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

Ilona Królak (2006)

Banach Center Publications

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We study a certain class of von Neumann algebras generated by selfadjoint elements $ω_{i} = a_{i} + a ⁺_{i}$ , where $a_{i}, a ⁺_{i}$ satisfy the general commutation relations: $a_{i} a ⁺_{j} = \sum_{r, s} t_{j}^{i} r_{s} a ⁺_{r} a_{s} + δ_{i j} I d$ . We assume that the operator T for which the constants $t_{j}^{i} r_{s}$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the...

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

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Let ℳ be a von Neumann algebra with unit $1_{ℳ}$ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_{t} {(x)}_{t \geq 0}$ the generalized s-numbers of x, defined by $μ_{t} (x)$ = inf||xe||: e is a projection in ℳ i with $τ (1_{ℳ} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ \mapsto \int_{0}^{t} l o g μ_{s} (f (λ)) d s$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

Outers for noncommutative $H^{p}$ revisited

David P. Blecher, Louis E. Labuschagne (2013)

Studia Mathematica

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We continue our study of outer elements of the noncommutative $H^{p}$ spaces associated with Arveson’s subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in $H^{p}$ actually satisfy the stronger condition that there exist aₙ ∈ A with haₙ...

Multiple solutions to a perturbed Neumann problem

Giuseppe Cordaro (2007)

Studia Mathematica

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We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in $ℝ^{N}$ with boundary of class C², $α \in L^{\infty} (Ω)$ with $e s s i n f_{Ω} α > 0$ , f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $s u p_{| s | \leq t} | g (\cdot, s) | \in L^{p} (Ω)$ and $g (\cdot, t) \in L^{\infty} (Ω)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1 / 2 ξ ² - \int_{0}^{ξ} f (t) d t$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough,...

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

Giovanni Anello, Giuseppe Cordaro (2003)

Colloquium Mathematicae

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We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $- Δ_{p} u + λ (x) {| u |}^{p - 2} u = μ f (x, u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ \in L^{\infty} (Ω)$ with $e s s i n f_{x \in Ω} λ (x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

L²-homology and reciprocity for right-angled Coxeter groups

Boris Okun, Richard Scott (2011)

Fundamenta Mathematicae

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Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra $_{μ}$ containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define $L ²_{μ}$ -Betti numbers and an $L ²_{μ}$ -Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Topological algebras with pseudoconvexly bounded elements

Mati Abel (2005)

Banach Center Publications

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It is shown that every commutative sequentially bornologically complete Hausdorff algebra A with bounded elements is representable in the form of an (algebraic) inductive limit of an inductive system of locally bounded Fréchet algebras with continuous monomorphisms if the von Neumann bornology of A is pseudoconvex. Several classes of topological algebras A for which $r_{A} (a) \leq β_{A} (a)$ or $r_{A} (a) = β_{A} (a)$ for each a ∈ A are described.

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

J. Chabrowski, Shusen Yan (2002)

Colloquium Mathematicae

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We consider the Neumann problem for the equation $- Δ u - λ u = {Q (x) | u |}^{2 * - 2} u$ , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k - 1}$ and $λ_{k}$ . Applying a min-max principle based on topological linking we prove the existence of a solution.

Automorphisms of central extensions of type I von Neumann algebras

Sergio Albeverio, Shavkat Ayupov, Karimbergen Kudaybergenov, Rauaj Djumamuratov (2011)

Studia Mathematica

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Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as $T = T_{a} \circ T_{ϕ}$ , where $T_{a} (x) = a x a^{- 1}$ is an inner automorphism implemented by an element a ∈ E(M), and $T_{ϕ}$ is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type $I_{\infty}$ then every band preserving...

On certain functions that generalize von Mangoldt’s function $Λ (n)$

A. Ivić (1975)

Matematički Vesnik

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On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces

Mikio Kato, Lech Maligranda, Yasuji Takahashi (2001)

Studia Mathematica

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Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{N J} (X)$ , and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{N J} (X)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{N J} (X)$ -constant, which implies that a Banach space with $C_{N J} (X)$ -constant less than 5/4 has the fixed point property. ...

The $\bar{\partial}$ -Neumann operator and commutators of the Bergman projection and multiplication operators

Friedrich Haslinger (2008)

Czechoslovak Mathematical Journal

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We prove that compactness of the canonical solution operator to $\bar{\partial}$ restricted to $(0, 1)$ -forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫, \bar{M}]$ defined on the whole $L_{(0, 1)}^{2} (Ω),$ where $\bar{M}$ is the multiplication by $\bar{z}$ and $𝒫$ is the orthogonal projection of $L_{(0, 1)}^{2} (Ω)$ to the subspace of $(0, 1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar{\partial}$ -Neumann operator restricted to $(0, 1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the...

The $L^{p}$ -Helmholtz projection in finite cylinders

Tobias Nau (2015)

Czechoslovak Mathematical Journal

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In this article we prove for $1 < p < \infty$ the existence of the $L^{p}$ -Helmholtz projection in finite cylinders $Ω$ . More precisely, $Ω$ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^{1}$ -boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $Ω$ is solved, which implies existence and a representation of the $L^{p}$ -Helmholtz projection...

A minimal regular ring extension of C(X)

M. Henriksen, R. Raphael, R. G. Woods (2002)

Fundamenta Mathematicae

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Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C (X, τ_{δ})$ of continuous real-valued functions on the space $(X, τ_{δ})$ , where $τ_{δ}$ is the smallest Tikhonov topology on X for which $τ \subseteq τ_{δ}$ and $C (X, τ_{δ})$ is von Neumann regular. The compact and metric spaces for which $G (X) = C (X, τ_{δ})$ are characterized. Necessary, and different sufficient,...

Noncommutative function theory and unique extensions

David P. Blecher, Louis E. Labuschagne (2007)

Studia Mathematica

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We generalize, to the setting of Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő $L^{p}$ -distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of $H^{\infty}$ from the 1960’s. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

A nonsmooth exponential

Esteban Andruchow (2003)

Studia Mathematica

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Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set $ℳ_{s a}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{ξ}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $e x p (ξ) = e^{i L_{ξ}}$ , is continuous but not differentiable. The same holds for the Cayley transform $C (ξ) = (L_{ξ} - i) {(L_{ξ} + i)}^{- 1}$ . We also show that the unitary group $U_{ℳ} \subset L ² (ℳ, τ)$ with the strong operator topology is not an...

Classical boundary value problems for integrable temperatures in a $C^{1}$ domain

Anna Grimaldi Piro, Francesco Ragnedda (1991)

Annales Polonici Mathematici

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Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^{1}$ -base and data in $h_{c}^{1}$ , a subspace of $L$ 1. We derive our results, considering the action of an adjoint operator on $B_{T} M O C$ , a predual of $h_{c}^{1}$ , and using known properties of this last space.

On a Kleinecke-Shirokov theorem

Vasile Lauric (2021)

Czechoslovak Mathematical Journal

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We prove that for normal operators $N_{1}, N_{2} \in ℒ (ℋ),$ the generalized commutator $[N_{1}, N_{2}; X]$ approaches zero when $[N_{1}, N_{2}; [N_{1}, N_{2}; X]]$ tends to zero in the norm of the Schatten-von Neumann class $𝒞_{p}$ with $p > 1$ and $X$ varies in a bounded set of such a class.

Strongly elliptic linear operators without coercive quadratic forms I. Constant coefficient operators and forms

Gregory C. Verchota (2014)

Journal of the European Mathematical Society

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A family of linear homogeneous 4th order elliptic differential operators $L$ with real constant coefficients, and bounded nonsmooth convex domains $Ω$ are constructed in $ℝ^{6}$ so that the $L$ have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces $W^{2, 2} (Ω)$ .

The equation $\bar{\partial}u=f$ the intersection of pseudoconvex domains

Alessandro Perotti (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Viene studiata l'equazione $\bar{\partial}u=f$ per le forme regolari sulla chiusura dell'intersezione di $k$ domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme $\mathbf{C}^{k}$ .

Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and q-Gaussian Operators

Artur Buchholz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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For $(P_{k})$ being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants $C_{2 n}$ , n∈ ℕ, in the inequality $∥ \sum_{k = 1}^{N} A_{k} \otimes P_{k} ∥_{2 n} \leq {[C_{2 n}]}^{1 / 2 n} m a x {∥ (\sum_{k = 1}^{N} A *_{k} A_{k}}^{1 / 2} ∥_{L_{2 n}}, ∥ (\sum_{k = 1}^{N} A_{k} A *_{k}$ 1/2∥L2n $,$ valid for all finite sequences of operators $(A_{k})$ in the non-commutative $L_{2 n}$ space related to a semifinite von Neumann algebra with trace. In particular, $C_{2 n} = (2 n r - 1)!!$ for the Rademacher and Fermion sequences.