Algebra isomorphisms between standard operator algebras

Thomas Tonev; Aaron Luttman

Displaying similar documents to “Algebra isomorphisms between standard operator algebras”

Almost multiplicative functions on commutative Banach algebras

S. H. Kulkarni, D. Sukumar (2010)

Studia Mathematica

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Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by $σ_{ϵ} (a) : = λ \in ℂ : | | λ - {a | | | | (λ - a)}^{- 1} | | \geq 1 / ϵ$ with the convention that $| | λ - {a | | | | (λ - a)}^{- 1} | | = \infty$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $ϕ (a) \in σ_{δ} (a)$ for all a in A. (2) If ϕ is linear and $ϕ (a) \in σ_{ϵ} (a)$ for...

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations $A_{i} X - X B_{i} = C_{i}$ (1 ≤ i ≤ k), where $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C ₁, . . ., C_{k})$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

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We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X ̃,_{X ̃}, m)$ . Consider the associated unitary operators on $L ² (X ̃,_{X ̃}, m)$ given by $U ̃_{S} f = \sqrt (d (m \circ S) / d m) \cdot (f \circ S)$ and $φ \cdot U ̃_{S}$ , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman, Simone Warzel (2013)

Journal of the European Mathematical Society

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We analyse the spectral phase diagram of Schrödinger operators $T + λ V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $i i d$ random variables. The main result is a criterion for the emergence of absolutely continuous $(a c)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $a c$ spectrum appears at arbitrarily weak disorder $(λ ≪ 1)$ in an energy regime which extends beyond the spectrum of $\sim T$ ....

A representation theorem for tense $n \times m$ -valued Łukasiewicz-Moisil algebras

Aldo Victorio Figallo, Gustavo Pelaitay (2015)

Mathematica Bohemica

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In 2000, Figallo and Sanza introduced $n \times m$ -valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$ -valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class $_{n \times m}$ of tense $n \times m$ -valued Łukasiewicz-Moisil algebras (or tense LM $_{n \times m}$ -algebras), namely $n \times m$ -valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense...

On generalized Beurling’s theorem and symmetry of $L_{1}$ -group algebras

Zenobia Anusiak (1971)

Colloquium Mathematicae

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The third order spectrum of the p-biharmonic operator with weight

Khalil Ben Haddouch, Najib Tsouli, Zakaria El Allali (2014)

Applicationes Mathematicae

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We show that the spectrum of $Δ ²_{p} u + {2 β \cdot \nabla (| Δ u |}^{p - 2} {Δ u) + | β | ² | Δ u |}^{p - 2} Δ u = α m {| u |}^{p - 2} u$ , where $β \in ℝ^{N}$ , under Navier boundary conditions, contains at least one sequence of eigensurfaces.

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

Uniform spectral radius and compact Gelfand transform

Alexandru Aleman, Anders Dahlner (2006)

Studia Mathematica

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We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{ν} = s u p {| | (e - x)}^{- 1} {| |}_{p} {: x \in A, | | x | |}_{p} \leq 1, m a x | x ̂ | \leq ν$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{δ} = s u p | | x^{- 1} {| |}_{p} {: x \in A, | | x | |}_{p} \leq 1, m i n | x ̂ | \geq δ$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty} (A)$ , introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty} (A) < 1$ , and this result is extended to more general nonlinear extremal...

On generalized property (v) for bounded linear operators

J. Sanabria, C. Carpintero, E. Rosas, O. García (2012)

Studia Mathematica

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An operator T acting on a Banach space X has property (gw) if $σ_{a} (T) ∖ σ_{S B F ₊ ¯} (T) = E (T)$ , where $σ_{a} (T)$ is the approximate point spectrum of T, $σ_{S B F ₊ ¯} (T)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $σ (T) = σ_{a} (T)$ .

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- $n$ commutative algebras equipped with an ordered basis is an affine scheme $𝔅_{n}$ of finite type over $ℤ$ , with geometrically connected fibers. It is smooth if and only if $n \leq 3$ . It is reducible if $n \geq 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$ ). The relative dimension of $𝔅_{n}$ is $\frac{2}{27} n^{3} + O (n^{8 / 3})$ . The subscheme parameterizing étale algebras is isomorphic to ${GL}_{n} / S_{n}$ , which is of dimension only $n^{2}$ . For $n \geq 8$ , there exist algebras that are not limits of étale algebras. The dimension...

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator $𝒫_{ϕ}^{u}$ on $L^{1} (Σ)$ as a predual of weighted Koopman operator $W = u U_{ϕ}$ on $L^{\infty} (Σ)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of $𝒫_{ϕ}^{u}$ under certain conditions.

Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

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Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $(f (x_{i}, x_{i}))$ having f evaluated at the greatest common divisor $(x_{i}, x_{i})$ of...

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g (R_{F}) \int f_{x} (R_{F}) d μ (x) = h (R_{F})$ . (1) They involve functions of the kind $X ∋ x \mapsto f_{x} (R_{F}) \in B (F)$ , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

Some examples of cocycles with simple continuous singular spectrum

K. Frączek (2001)

Studia Mathematica

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We study spectral properties of Anzai skew products $T_{φ} : ² \to ²$ defined by $T_{φ} (z, ω) = (e^{2 π i α} z, φ (z) ω)$ , where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{φ}$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently...

On cardinalities of algebras of formulas for $ω_{0}$ -categorical theories

Jan Waszkiewicz (1973)

Colloquium Mathematicae

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Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

Peer C. Kunstmann, Hendrik Vogt (2007)

Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

Trivialization of $𝒞 (X)$ -algebras with strongly self-absorbing fibres

Marius Dadarlat, Wilhelm Winter (2008)

Bulletin de la Société Mathématique de France

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Suppose $A$ is a separable unital $𝒞 (X)$ -algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$ -injective $C^{*}$ -algebra $𝒟$ . We show that $A$ and $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces

Mats Andersson, Sebastian Sandberg (2003)

Studia Mathematica

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We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_{g}$ on $H^{p} (D)$ , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_{g}$ provided that $g^{- 1} (0)$ is a compact subset of D.

Compact operators whose adjoints factor through subspaces of $l_{p}$

Deba P. Sinha, Anil K. Karn (2002)

Studia Mathematica

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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K \subset \sum_{n = 1}^{\infty} α ₙ x ₙ : α ₙ \in B a l l (l_{p^{'}})$ , where p’ = p/(p-1) and $x ₙ \in l_{p}^{s} (X)$ . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $x ₙ \in l_{p}^{w} (X)$ . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p}, κ_{p})$ of p-compact operators and $(W_{p}, ω_{p})$ of weakly p-compact operators, arising from these factorizations,...

Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.