scholarly journals Spatial numerical ranges of elements of Banach algebras

1989 ◽  
Vol 12 (4) ◽  
pp. 633-640 ◽  
Author(s):  
A. K. Gaur ◽  
T. Husain
Keyword(s):  
Convex Hull ◽  
Banach Algebra ◽  
Numerical Range ◽  
Banach Algebras ◽  
Hermitian Element ◽  
Regular Norm ◽  
The Relationship

In this paper, the notion of spatial numerical range of elements of Banach algebras without identity is studied. Specifically, the relationship between spatial numerical ranges, numerical ranges and spectra is investigated. Among other results, it is shown that the closure of the spatial numerical range of an element of a Banach algebra without Identity but wlth regular norm is exactly its numerical range as an element of the unitized algebra. Futhermore, the closure of the spatial numerical range of a hermitian element coincides with the convex hull of its spectrum. In particular, spatial numerical ranges of the elements of the Banach algebraC0(X)are described.

On the Relationship between Interpolation of Banach Algebras and Interpolation of Bilinear Operators

2010 ◽  
Vol 53 (1) ◽  
pp. 51-57 ◽  
Author(s):  
Fernando Cobos ◽  
Luz M. Fernández-Cabrera
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Interpolation Theorem ◽  
Algebra Structure ◽  
Bilinear Interpolation ◽  
Bilinear Operators ◽  
Real Method ◽  
The Relationship

AbstractWe show that if the general real method (· , ·)Γ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for (· , ·)Γ.

scholarly journals Polynomials in a hermitian element

1988 ◽  
Vol 30 (2) ◽  
pp. 171-176 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Hermitian Element ◽  
Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

scholarly journals A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges

1986 ◽  
Vol 28 (2) ◽  
pp. 121-137 ◽  
Author(s):  
C. Aparicio ◽  
F. Ocaña ◽  
R. Payá ◽  
A. Rodríguez
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Banach Algebras ◽  
Continuous States ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Smooth Extension ◽  
Unital Banach Algebra

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

scholarly journals Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

1988 ◽  
Vol 30 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Volker Wrobel
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
Continuous Linear Operators

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

scholarly journals Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups

1985 ◽  
Vol 28 (1) ◽  
pp. 91-95
Author(s):  
J. Martinez-Moreno ◽  
A. Rodriguez-Palacios
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Compact Convex ◽  
Banach Algebras ◽  
Point Of View ◽  
Holomorphic Semigroup ◽  
Compact Convex Set ◽  
Basic Ideas ◽  
Unital Banach Algebra ◽  
First Time

If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

scholarly journals The spectral theorem in Banach algebras

1972 ◽  
Vol 13 (1) ◽  
pp. 49-55
Author(s):  
Stephen Plafker
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Hermitian Operator ◽  
Spectral Theorem ◽  
Continuous Linear ◽  
Hermitian Element ◽  
Definition Of ◽  
Original Norm ◽  
Original Theorem

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

scholarly journals TRIVIALITY OF THE GENERALISED LAU PRODUCT ASSOCIATED TO A BANACH ALGEBRA HOMOMORPHISM

2016 ◽  
Vol 94 (2) ◽  
pp. 286-289 ◽  
Author(s):  
YEMON CHOI
Keyword(s):  
Harmonic Analysis ◽  
Banach Algebra ◽  
Direct Product ◽  
Banach Algebras ◽  
Short Note ◽  
Studia Math ◽  
Algebra Homomorphism ◽  
The Relationship

Several papers have, as their raison d’être, the exploration of the generalised Lau product associated to a homomorphism $T:B\rightarrow A$ of Banach algebras. In this short note, we demonstrate that the generalised Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalised Lau product and an older construction of Monfared [‘On certain products of Banach algebras with applications to harmonic analysis’, Studia Math. 178(3) (2007), 277–294].

scholarly journals On complemented and annihilator algebras

1969 ◽  
Vol 10 (1) ◽  
pp. 38-45 ◽  
Author(s):  
Freda E. Alexander
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Semisimple Algebra ◽  
The Relationship ◽  
Left Annihilator

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

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