scholarly journals On the Homology Theory of Operator Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Alaa Hassan Noreldeen
Keyword(s):  
Banach Algebra ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Free Resolution ◽  
Homology Theory ◽  
Cyclic Homology ◽  
Commutative Banach Algebras ◽  
Unital Banach Algebra

We investigate the cyclic homology and free resolution effect of a commutative unital Banach algebra. Using the free resolution operator, we define the relative cyclic homology of commutative Banach algebras. Lemmas and theorems of this investigation are studied and proved. Finally, the relation between cyclic homology and relative cyclic homology of Banach algebra is deduced.

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.

ONE FUNCTIONAL OPERATOR INVERSION FORMULA

2001 ◽  
Vol 6 (1) ◽  
pp. 138-146 ◽  
Author(s):  
P. Plaschinsky
Keyword(s):  
Banach Algebra ◽  
Explicit Form ◽  
Maximal Ideal ◽  
Inversion Formula ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Functional Operator ◽  
Inversion Operator ◽  
Commutative Banach Algebras ◽  
Algebra Theory

Some results about inversion formula of functional operator with generalized dilation are given. By means of commutative Banach algebra theory the explicit form of inversion operator is expressed. Some commutative Banach algebras with countable generator systems are constructed, their maximal ideal spaces are investigated.

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

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.

Compact semigroups in commutative Banach algebras

1969 ◽  
Vol 66 (2) ◽  
pp. 265-274 ◽  
Author(s):  
M. A. Kaashoek ◽  
T. T. West
Keyword(s):  
Banach Algebra ◽  
Topological Semigroup ◽  
Spectral Properties ◽  
Banach Algebras ◽  
Structure Theory ◽  
Compact Semigroups ◽  
Commutative Banach Algebras ◽  
Continuous Multiplication ◽  
Natural Way

A monothetic semigroup is a topological semigroup with jointly continuous multiplication which contains a dense cyclic subsemigroup. These semi-groups arise in a natural way in the study of semi-algebras. In (4) we showed that a compact monothetic semigroup in a Banach algebra can be characterized in terms of the spectral properties of a generating element. In this paper these spectral theorems are linked with the well-known structure theory of compact semigroups.

scholarly journals Homology and cohomology groups of commutative Banach algebras and analytic polydiscs

2000 ◽  
Vol 42 (1) ◽  
pp. 15-24
Author(s):  
L. I. Pugach ◽  
M. C. White
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Deformation Theory ◽  
Banach Algebras ◽  
Subject Classification ◽  
Analytic Structure ◽  
Homological Dimension ◽  
Cohomology Groups ◽  
Main Method ◽  
Commutative Banach Algebras

In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.1991 Mathematics Subject Classification. 46J20, 46M20.

scholarly journals REPRESENTATIONS OF REAL BANACH ALGEBRAS

2010 ◽  
Vol 88 (3) ◽  
pp. 289-300 ◽  
Author(s):  
F. ALBIAC ◽  
E. BRIEM
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Ideal Space ◽  
Gelfand Transform ◽  
Continuous Complex ◽  
Unital Banach Algebra ◽  
Complex Valued

AbstractA commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations

2020 ◽  
Vol 70 (4) ◽  
pp. 1003-1011
Author(s):  
Behrooz Fadaee ◽  
Kamal Fallahi ◽  
Hoger Ghahramani
Keyword(s):  
Banach Algebra ◽  
Complex Variable ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Jordan Derivation ◽  
Continuous Linear ◽  
Linear Map ◽  
Standard Operator ◽  
Standard Operator Algebras

AbstractLet 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

scholarly journals Similarité entre l'algèbre de Volterra et un quotient d'algèbre uniforme

1991 ◽  
Vol 34 (3) ◽  
pp. 383-391 ◽  
Author(s):  
Konin Koua
Keyword(s):  
Banach Algebra ◽  
Half Plane ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Compact Mapping ◽  
Dense Ideal ◽  
Right Hand ◽  
Commutative Banach Algebras ◽  
One To One

Two commutative Banach algebras A and B are said to be similar if there exists a Banach algebra D such that [xD]− = D for some x in D, and two one-to-one continuous homomorphisms φ:D→A and ψ:D→B such that φ(D) is a dense ideal of A and ψ(D) a dense ideal of B.We prove in this paper that the Volterra algebra is similar to A0/e-z A0 where A0 is the commutative uniform, separable Banach algebra of all continuous functions on the closed right-hand half plane , analytic on H and vanishing at infinity. We deduce from this result that multiplication by an element of A0/e-z A0 is a compact mapping.

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