scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

A representation theorem for partially ordered Banach algebras

1968 ◽  
Vol 64 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Kung-Fu Ng
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Positive Cone ◽  
Partially Ordered ◽  
Real Algebra ◽  
Empty Subset ◽  
Image Position ◽  
Ordered Banach Algebra

Let be a real algebra which is also a Banach space. Then is called a partially ordered Banach algebra if there is specified a non-empty subset of , called the positive cone, such thatand(A 5) is 1-normal and closed. (We recall that is said to be 1-normal if

Nonstandard Ideals in Radical Convolution Algebras on a Half-Line

1987 ◽  
Vol 39 (2) ◽  
pp. 309-321 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Half Line ◽  
Positive Sequence ◽  
Convolution Product ◽  
Weight Sequence ◽  
Convolution Algebras ◽  
Image Position

This note is about the interplay between two classes of radical Banach algebras, and we begin by describing the algebras in question.A weight sequence is a positive sequence w = (wn) defined on Z+ (the non-negative integers) and satisfying w0 = 1 and wm+n ≦ wmwn for all m and n in Z+. For such a sequence w, the Banach spaceis a Banach algebra with respect to the convolution product, defined by

Jordan algebras spanned by Hermitian elements of a Banach algebra

1977 ◽  
Vol 81 (1) ◽  
pp. 3-13 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Jordan Algebras ◽  
The Self ◽  
Image Position ◽  
Adjoint Operators ◽  
Unital Banach Algebra ◽  
Algebra Of Operators

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that ifthen B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

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.

scholarly journals On well-bounded operators of class Г

1983 ◽  
Vol 24 (1) ◽  
pp. 1-5
Author(s):  
Adnan A. S. Jibril
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Operators ◽  
Strong Limit ◽  
Riemann Sums ◽  
Image Position

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the formwhere {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

On Banach algebra elements of thin numerical range

1979 ◽  
Vol 86 (1) ◽  
pp. 71-83 ◽  
Author(s):  
R. R. Smith
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Natural Generalization ◽  
The Real ◽  
C Algebra ◽  
Image Position ◽  
Unital Banach Algebra ◽  
Real Subspace

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.

scholarly journals The evaluation functionals associated with an algebra of bounded operators

1969 ◽  
Vol 10 (1) ◽  
pp. 73-76 ◽  
Author(s):  
J. Duncan
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Image Position

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given byGiven a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

scholarly journals Stable perturbation in Banach algebras

2007 ◽  
Vol 83 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Yifeng Xue
Keyword(s):  
Banach Algebra ◽  
Error Estimates ◽  
Generalized Inverse ◽  
Invertible Element ◽  
Banach Algebras ◽  
Gap Function ◽  
Drazin Inverse ◽  
C Algebras ◽  
Stable Perturbation ◽  
Unital Banach Algebra

AbstractLet be a unital Banach algebra. Assume that a has a generalized inverse a+. Then is said to be a stable perturbation of a if . In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation is closely related to the gap function . These results will be applied to error estimates for perturbations of the Moore-Penrose inverse in C*–algebras and the Drazin inverse in Banach algebras.

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