2-Local isometries between Banach algebras of continuous functions with involution

2019 ◽  
Vol 14 (1) ◽  
pp. 203-227
Author(s):  
Davood Alimohammadi ◽  
Hadis Pazandeh
Keyword(s):  
Banach Algebras ◽  
Continuous Functions

scholarly journals On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

2019 ◽  
Vol 21 (4) ◽  
pp. 430-442
Author(s):  
Oleg E. Galkin ◽  
Svetlana Y. Galkina
Keyword(s):  
Differential Equations ◽  
Banach Algebras ◽  
Foundations Of Quantum Mechanics ◽  
Continuous Functions ◽  
Heisenberg Equation ◽  
Complex Numbers ◽  
Bounded Operators ◽  
First Order ◽  
First Case ◽  
Families Of Operators

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

scholarly journals Some Inequalities arising from a Banach algebra norm

1983 ◽  
Vol 34 (2) ◽  
pp. 199-202 ◽  
Author(s):  
A. M. Russell
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Norm Inequality ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWe derive some specific inequalities involving absolutely continuous functions and relate them to a norm inequality arising from Banach algebras of functions having bounded k th variation.

scholarly journals THE KERNELS AND CONTINUITY IDEALS OF HOMOMORPHISMS FROM 𝒞0(Ω)

2010 ◽  
Vol 88 (1) ◽  
pp. 103-130 ◽  
Author(s):  
HUNG LE PHAM
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces

AbstractWe give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions.

scholarly journals Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative C⁎-Algebras

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Osamu Hatori
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
C Algebras ◽  
Lipschitz Maps ◽  
Spaces Of Continuous Functions ◽  
Continuous Maps

We study isometries on algebras of the Lipschitz maps and the continuously differentiable maps with the values in a commutative unital C⁎-algebra. A precise proof of a theorem of Jarosz concerning isometries on spaces of continuous functions is exhibited.

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.

scholarly journals Multiplicative spectrum of ultrametric Banach algebras of continuous functions

2010 ◽  
Vol 157 (16) ◽  
pp. 2505-2515 ◽  
Author(s):  
Alain Escassut ◽  
Nicolas Maïnetti
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
Multiplicative Spectrum

Banach Algebras of Continuous Functions

1951 ◽  
Vol 73 (1) ◽  
pp. 30 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebras ◽  
Continuous Functions

scholarly journals Jointly separating maps between vector-valued function spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 707-718
Author(s):  
Ziba Pourghobadi ◽  
Masoumeh Najafi Tavani ◽  
Fereshteh Sady
Keyword(s):  
Banach Algebras ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Hausdorff Spaces ◽  
Absolutely Continuous Functions ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet X and Y be compact Hausdorff spaces, E be a real or complex Banach space and F be a real or complex locally convex topological vector space. In this paper we study a pair of linear operators S, T : A(X, E) → C(Y, F) from a subspace A(X, E) of C(X, E) to C(Y, F), which are jointly separating, in the sense that Tf and Sg have disjoint cozeros whenever f and g have disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pair T : A(X) → C(Y) and S : A(X, E) → C(Y, F) of linear operators, where A(X) is a regular Banach function algebra on X, such that f ⋅ g = 0 implies Tf ⋅ Sg = 0, for all f ∈ A(X) and g ∈ A(X, E). If T and S are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between X and Y and, furthermore, T−1 and S−1 are also jointly separating maps.

scholarly journals A Survey and New Results on Banach Algebras of Ultrametric Continuous Functions

2020 ◽  
Vol 12 (3) ◽  
pp. 185-202
Author(s):  
Monique Chicourrat ◽  
Bertin Diarra ◽  
Alain Escassut
Keyword(s):  
Banach Algebras ◽  
Continuous Functions
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