The Experts below are selected from a list of 237 Experts worldwide ranked by ideXlab platform
Eberhard Kaniuth - One of the best experts on this subject based on the ideXlab platform.
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power boundedness in fourier and fourier stieltjes Algebras and other Commutative Banach Algebras
Journal of Functional Analysis, 2011Co-Authors: Eberhard Kaniuth, Anthony Toming Lau, A UlgerAbstract:Abstract We study power boundedness in the Fourier and Fourier–Stieltjes Algebras, A ( G ) and B ( G ) , of a locally compact group G as well as in some other Commutative Banach Algebras. The main results concern the question of when all elements with spectral radius at most one in any of these Algebras are power bounded, the characterization of power bounded elements in A ( G ) and B ( G ) and also the structure of the Gelfand transform of a single power bounded element.
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the bochner schoenberg eberlein property for Commutative Banach Algebras especially fourier and fourier stieltjes Algebras
Transactions of the American Mathematical Society, 2010Co-Authors: Eberhard Kaniuth, A UlgerAbstract:The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M(G) of G. This has led to the study of the algebra of BSE-functions on the spectrum of an arbitrary Commutative Banach algebra and of the concept of a BSE-algebra as introduced by Takahasi and Hatori. Since then BSE-Algebras have been studied by several authors. In this paper we investigate BSE-Algebras in the general context on the one hand and, on the other hand, we specialize to Fourier and Fourier-Stieltjes Algebras of locally compact groups.
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multipliers of Commutative Banach Algebras power boundedness and fourier stieltjes Algebras
Journal of The London Mathematical Society-second Series, 2010Co-Authors: Eberhard Kaniuth, Anthony Toming Lau, A UlgerAbstract:Let A be a semisimple and regular Commutative Banach algebra with bounded approximate identity. We study multipliers of A, in particular power bounded ones, and the associated ideals of A and A-invariant projections of the dual space of A. Samples of the results are general versions of the classical theorems of Choquet�Deny and of Foguel about measures on locally compact abelian groups. The results are linked to sets of synthesis in the Gelfand spectrum of A, and the main applications are concerned with Fourier and Fourier�Stieltjes Algebras of locally compact groups.
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a course in Commutative Banach Algebras
2008Co-Authors: Eberhard KaniuthAbstract:General Theory of Banach Algebras.- Gelfand Theory.- Functional Calculus, Shilov Boundary, and Applications.- Regularity and Related Properties.- Spectral Synthesis and Ideal Theory.
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weak spectral synthesis in Commutative Banach Algebras iii
Journal of Functional Analysis, 2008Co-Authors: Eberhard KaniuthAbstract:Abstract Let A be a regular and semisimple Commutative Banach algebra with structure space Δ ( A ) . Continuing the investigations of [7] and [9] , we establish various results on weak spectral sets in Δ ( A ) . To each closed subset of Δ ( A ) we associate a descending sequence of subsets of Δ ( A ) which proves to be a powerful tool in the study of weak spectral synthesis. Applications concern injection type properties, unions of weak spectral sets and projective tensor products. A number of interesting examples is discussed: Algebras of m-times continuously differentiable functions and of Lipschitz functions, and L 1 ( R N ) .
Takeshi Miura - One of the best experts on this subject based on the ideXlab platform.
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classification of semisimple Commutative Banach Algebras of type i
Nihonkai mathematical journal, 2019Co-Authors: Jyunji Inoue, Takeshi Miura, Hiroyuki Takagi, Sinei TakahasiAbstract:In the first and fourth authors' paper in 2017, it was shown that there exists a BSE-algebra of type I isomorphic to no C*-Algebras, which solved negatively a question posed by the fourth author and O. Hatori. However, this result suggests a further investigation of Commutative Banach algebra of type I. In the first part of the paper, we classify type I Algebras into six families by means of BSE, BED, and Tauberian. It is shown that a Banach algebra of type I is isomorphic to a Segal algebra in some Commutative C*-algebra if and only if it is Tauberian. In the second part, we give concrete examples of type I Algebras to show that all of six families mentioned above are nonempty.
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a characterization of multipliers of a lau algebra constructed by semisimple Commutative Banach Algebras
Taiwanese Journal of Mathematics, 2016Co-Authors: Sinei Takahasi, Hiroyuki Takagi, Takeshi MiuraAbstract:A necessary and sufficient condition for a Lau type binary operation defined by two mappings to be an algebra-operation is given in terms of multipliers. Also a characterization of multipliers of a Lau algebra constructed by semisimple Commutative Banach Algebras is given in terms of multipliers of original Banach Algebras.
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additively spectral radius preserving surjections between unital semisimple Commutative Banach Algebras
Open Mathematics, 2010Co-Authors: Osamu Hatori, Go Hirasawa, Takeshi MiuraAbstract:Let A and B be unital, semisimple Commutative Banach Algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.
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a generalization of the Banach stone theorem for Commutative Banach Algebras
Nihonkai mathematical journal, 2010Co-Authors: Go Hirasawa, Takeshi Miura, Rumi ShindoAbstract:Let $I$ be an index set, not necessarily a subset of any Banach algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital semisimple Commutative Banach Algebras with maximal ideal spaces $M_{\mathcal{A}}$ and $M_{\mathcal{B}}$, respectively. If surjective mappings $S_1, S_2 \colon I \to \mathcal{A}$ and $T_1, T_2 \colon I \to \mathcal{B}$ satisfy $\mathrm{r}(T_1(\lambda) - T_2(\mu)) = \mathrm{r}(S_1(\lambda) - S_2(\mu))$ for all $\lambda, \mu \in I$, where $\mathrm{r}(a)$ is the spectral radius of $a$, then there exist $p, w \in \mathcal{B}$, a homeomorphism $\varphi \colon M_\mathcal{B} \to M_\mathcal{A}$ and a closed and open subset $K$ of $M_\mathcal{B}$ such that $|\hat{w}| = 1$ on $M_\mathcal{B}$ and that $$ \widehat{T_k(\lambda)}(y) - \hat{p}(y) = \begin{cases} \hat{w}(y)\widehat{S_k(\lambda)}(\varphi(y)) & y \in K \\[2pt] \hat{w}(y)\overline{\widehat{S_k(\lambda)}(\varphi(y))} & y \in M_\mathcal{B} \setminus K \end{cases} $$ for all $\lambda \in I$ $(k = 1, 2)$. In particular, if $\mathcal{A}$ and $\mathcal{B}$ are uniform Algebras, and if $S_1, S_2 \colon I \to \mathcal{A}$ and $T_1, T_2 \colon I \to \mathcal{B}$ satisfy $$ \sigma_\pi ({T_1(\lambda) - T_2(\mu)}) \cap \sigma_pi ({S_1(\lambda) - S_2(\mu)} ) \neq \emptyset \qquad (\forall \lambda, \mu \in I), $$ where $\sigma_pi (f)$ is the peripheral spectrum of $f$, then $\widehat{T_k(\lambda)}(y) = \hat{p}(y) + \widehat{S_k(\lambda)}(\varphi(y))$ for all $\lambda \in I$ and $y \in M_\mathcal{B}$ $(k = 1, 2)$.
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polynomially spectrum preserving maps between Commutative Banach Algebras
arXiv: Functional Analysis, 2009Co-Authors: Osamu Hatori, Takeshi Miura, Hiroyuki TakagiAbstract:Let $A$ and $B$ be unital semi-simple Commutative Banach Algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ \sigma (p(Tf,Tg))=\sigma (p(f,g)), \quad f,g \in A \] holds is an algebra isomorphism.
A Ulger - One of the best experts on this subject based on the ideXlab platform.
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power boundedness in fourier and fourier stieltjes Algebras and other Commutative Banach Algebras
Journal of Functional Analysis, 2011Co-Authors: Eberhard Kaniuth, Anthony Toming Lau, A UlgerAbstract:Abstract We study power boundedness in the Fourier and Fourier–Stieltjes Algebras, A ( G ) and B ( G ) , of a locally compact group G as well as in some other Commutative Banach Algebras. The main results concern the question of when all elements with spectral radius at most one in any of these Algebras are power bounded, the characterization of power bounded elements in A ( G ) and B ( G ) and also the structure of the Gelfand transform of a single power bounded element.
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the bochner schoenberg eberlein property for Commutative Banach Algebras especially fourier and fourier stieltjes Algebras
Transactions of the American Mathematical Society, 2010Co-Authors: Eberhard Kaniuth, A UlgerAbstract:The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M(G) of G. This has led to the study of the algebra of BSE-functions on the spectrum of an arbitrary Commutative Banach algebra and of the concept of a BSE-algebra as introduced by Takahasi and Hatori. Since then BSE-Algebras have been studied by several authors. In this paper we investigate BSE-Algebras in the general context on the one hand and, on the other hand, we specialize to Fourier and Fourier-Stieltjes Algebras of locally compact groups.
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multipliers of Commutative Banach Algebras power boundedness and fourier stieltjes Algebras
Journal of The London Mathematical Society-second Series, 2010Co-Authors: Eberhard Kaniuth, Anthony Toming Lau, A UlgerAbstract:Let A be a semisimple and regular Commutative Banach algebra with bounded approximate identity. We study multipliers of A, in particular power bounded ones, and the associated ideals of A and A-invariant projections of the dual space of A. Samples of the results are general versions of the classical theorems of Choquet�Deny and of Foguel about measures on locally compact abelian groups. The results are linked to sets of synthesis in the Gelfand spectrum of A, and the main applications are concerned with Fourier and Fourier�Stieltjes Algebras of locally compact groups.
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homomorphisms of Commutative Banach Algebras and extensions to multiplier Algebras with applications to fourier Algebras
Studia Mathematica, 2007Co-Authors: Eberhard Kaniuth, Anthony Toming Lau, A UlgerAbstract:Let A and B be semisimple Commutative Banach Algebras with bounded approximate identities. We investigate the problem of extending a homomorphism ' : A ! B to a homomorphism of the multiplier Algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms ofB (or B and ') are given that allow the construction of such extensions. We exhibit a number of classes of Banach Algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from A into A with closed range. Our results are applied to Fourier Algebras of locally compact groups. Introduction. Let A and B be semisimple Commutative Banach alge- bras and suppose that A has a bounded approximate identity (e�)�. We study homomorphisms ϕ from A to B from various aspects. Let I' be the largest ideal of B for which (ϕ(e�))� serves as an approximate identity, and let Z' denote the zero set of I' in the Gelfand spectrum � (B) of B. In Section 1 we find criteria for Z' to be open in � (B) and I' to be comple- mented by a certain ideal J' (Theorems 1.4 and 1.5). The results are applied to the related question of when a homomorphism ϕ : A → B extends to a homomorphism, φ : M(A) → M(B), between the multiplier Algebras M(A) and M(B). This extension problem is the main objective of the paper. When I' is complemented as above, we give in Theorem 2.1 (which is a basic result of the paper) an explicit construction of an extension φ : M(A) → M(B). Moreover, if in addition B is a BSE-algebra (named after Bochner-Schoenberg-Eberlein), then all homomorphisms from M(A)
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some results about the spectrum of Commutative Banach Algebras under the weak topology and applications
Monatshefte für Mathematik, 1996Co-Authors: A UlgerAbstract:LetA be a Commutative Banach algebra with a nonempty spectrum ΣA. By “weak” we denote the relative weak topology induced on ΣA by σ(A*,A**). In this note we study some properties of the topological space (ΣA, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform Algebras and to a characterization of the synthesizable ideals of the algebraA.
Osamu Hatori - One of the best experts on this subject based on the ideXlab platform.
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additively spectral radius preserving surjections between unital semisimple Commutative Banach Algebras
Open Mathematics, 2010Co-Authors: Osamu Hatori, Go Hirasawa, Takeshi MiuraAbstract:Let A and B be unital, semisimple Commutative Banach Algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.
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polynomially spectrum preserving maps between Commutative Banach Algebras
arXiv: Functional Analysis, 2009Co-Authors: Osamu Hatori, Takeshi Miura, Hiroyuki TakagiAbstract:Let $A$ and $B$ be unital semi-simple Commutative Banach Algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ \sigma (p(Tf,Tg))=\sigma (p(f,g)), \quad f,g \in A \] holds is an algebra isomorphism.
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multiplicatively spectrum preserving and norm preserving maps between invertible groups of Commutative Banach Algebras
arXiv: Functional Analysis, 2009Co-Authors: Osamu Hatori, Takeshi Miura, Hiroyuki TakaggiAbstract:Let $A$ and $B$ be unital semisimple Commutative Banach Algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $\sigma (TfTg)=\sigma (fg)$; (2) $\sigma_{\pi}(TfTg-\alpha)\cap \sigma_{\pi}(fg-\alpha)\ne \emptyset$; (3) $\mathrm{r} (TfTg-\alpha )=\mathrm{r}(fg-\alpha)$ hold for some non-zero complex number $\alpha$ and for every $f, g\in A^{-1}$, where $\sigma (\cdot)$ (resp. $\sigma_{\pi}(\cdot)$) denotes the (resp. peripheral) spectrum and $\rr(\cdot)$ denotes the spectral radius. Under each of the hypotheses we show representations for $T$ and under additional assumptions we show that $T$ is extended to an algebra isomorphism. In particular, if $T$ is a surjective group homomorphism such that $T$ preserves the spectrum or $T$ is a surjective isometry with respect to the spectral radius, then $T$ is extended to an algebra isomorphism. Similar results holds for maps from $A$ onto $B$.
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an example of multiplicatively spectrum preserving maps between non isomorphic semi simple Commutative Banach Algebras
Nihonkai mathematical journal, 2007Co-Authors: Osamu Hatori, Takeshi Miura, Hirokazu OkaAbstract:ABSTRACT.In this paper we givean example ofamultiplicativelypreserving spectrum- mapbetweentwo non-unital Commutative semisimipleBanach Algebraswhich are notdgebraicallyisomorphictoeachother. 1. Moln\’ar INTRODUCTION [6] initiated the study ofmultiplicativelyspectrum-preserving maps on Banach Algebras and proved among other theorems that a map $T$ from a Banach algebra $C(\mathcal{X})$ all of complex-valued continuous functions ona first countable com- pact Hausdorff space $\mathcal{X}$ ontois itselfan almost isomorphism in the sense that $T$ is an algebra isomorphism times a weight with the valuesin $\{-1,1\}$ if $T$ is mul- tiplicatively spectrum preserving in the sense that the spectrum the product of ofany two functions $f$ and $g\in C(\mathcal{X})$ equals to the spectrum the product of of $Tf$ . $Tg$ and Rao and [7] Roy generalized the result an for arbitrary uniform algebraonto itself. Hatori, Miura and [3] Takagi studied the maps between arbitrarytwo uniform Algebras which are multiplicatively
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unital and multiplicatively spectrum preserving surjections between semi simple Commutative Banach Algebras are linear and multiplicative
Journal of Mathematical Analysis and Applications, 2007Co-Authors: Osamu Hatori, Takeshi Miura, Hiroyuki TakagiAbstract:Abstract Let T be a surjective map from a unital semi-simple Commutative Banach algebra A onto a unital Commutative Banach algebra B . Suppose that T preserves the unit element and the spectrum σ ( f g ) of the product of any two elements f and g in A coincides with the spectrum σ ( T f T g ) . Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a Commutative C*-algebra into itself such that σ ( T f T g ) = σ ( f g ) holds for every f , g are given. We also show an example of a surjective unital map from a Commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ ( T f T g ) ⊂ σ ( f g ) holds for every f , g .
Nikolai Vasilevski - One of the best experts on this subject based on the ideXlab platform.
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on the structure of Commutative Banach Algebras generated by toeplitz operators on the unit ball quasi elliptic case ii gelfand theory
Complex Analysis and Operator Theory, 2015Co-Authors: Wolfram Bauer, Nikolai VasilevskiAbstract:Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956–2990, 2013) the present paper gives a detailed structural analysis of a class of Commutative Banach Algebras \(\mathcal {B}_k(h)\) generated by Toeplitz operators on the standard weighted Bergman spaces \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) over the complex unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). In the most general situation we explicitly determine the set of maximal ideals of \(\mathcal {B}_k(h)\) and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of Algebras \(\mathcal {B}_k(h)\) in the full algebra of bounded operators on \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) for certain choices of \(h\). Moreover, it is remarked that \(\mathcal {B}_k(h)\) is not semi-simple. In the case of \(k=(n)\) we explicitly describe the radical \(\hbox {Rad}\, \mathcal {B}_n(h)\) of the algebra \(\mathcal {B}_n(h)\). This result generalizes and simplifies the characterization of \(\hbox {Rad}\,\mathcal {B}_2(1)\), which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199–231, 2012).
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on the structure of Commutative Banach Algebras generated by toeplitz operators on the unit ball quasi elliptic case i generating subAlgebras
Journal of Functional Analysis, 2013Co-Authors: Wolfram Bauer, Nikolai VasilevskiAbstract:Abstract Extending recent results in [3] to the higher dimensional setting n ⩾ 3 we provide a further step in the structural analysis of a class of Commutative Banach Algebras generated by Toeplitz operators on the standard weighted Bergman space over the n -dimensional complex unit ball. The Algebras B k ( h ) under study are subordinated to the quasi-elliptic group of automorphisms of B n and in terms of their generators they were described in [23] . We show that B k ( h ) is generated in fact by an essentially smaller set of operators, i.e., the Toeplitz operators with k -quasi-radial symbols and a finite set of Toeplitz operators with “elementary” k -quasi-homogeneous symbols. Then we analyze the structure of the Commutative subAlgebras corresponding to these two types of generating symbols. In particular, we describe spectra, joint spectra, maximal ideal spaces and the Gelfand transform.
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on the structure of a Commutative Banach algebra generated by toeplitz operators with quasi radial quasi homogeneous symbols
Integral Equations and Operator Theory, 2012Co-Authors: Wolfram Bauer, Nikolai VasilevskiAbstract:Let \({\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}\) denote the standard weighted Bergman space over the unit ball \({\mathbb{B}^n}\) in \({\mathbb{C}^n}\) . New classes of Commutative Banach Algebras \({\mathcal{T}(\lambda)}\) which are generated by Toeplitz operators on \({\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}\) have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141–152, 2010). These Algebras are induced by the action of the quasi-elliptic group of biholomorphisms of \({\mathbb{B}^n}\) . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n = 2. We explicitly describe the maximal ideal space and the Gelfand map of \({\mathcal{T}(\lambda)}\) . Since \({\mathcal{T}(\lambda)}\) is not invariant under the *-operation of \({\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}\) its inverse closedness is not obvious and is proved. We remark that the algebra \({\mathcal{T}(\lambda)}\) is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in \({\mathcal{T}(\lambda)}\) is always connected.
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quasi radial quasi homogeneous symbols and Commutative Banach Algebras of toeplitz operators
Integral Equations and Operator Theory, 2010Co-Authors: Nikolai VasilevskiAbstract:We present here a quite unexpected result: Apart from already known Commutative C*-Algebras generated by Toeplitz operators on the unit ball, there are many other Banach Algebras generated by Toeplitz operators which are Commutative on each weighted Bergman space. These last Algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.