scholarly journals Linear Isometries between Real Banach Algebras of Continuous Complex-Valued Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Davood Alimohammadi ◽  
Hadis Pazandeh
Keyword(s):  
Composition Operators ◽  
Banach Algebras ◽  
Extreme Points ◽  
Function Algebra ◽  
Hausdorff Spaces ◽  
Uniform Function ◽  
Weighted Composition ◽  
Continuous Complex ◽  
Complex Valued ◽  
Linear Isometries

Let and be compact Hausdorff spaces, and let and be topological involutions on and , respectively. In 1991, Kulkarni and Arundhathi characterized linear isometries from a real uniform function algebra on (, ) onto a real uniform function algebra on (, ) applying their Choquet boundaries and showed that these mappings are weighted composition operators. In this paper, we characterize all onto linear isometries and certain into linear isometries between and applying the extreme points in the unit balls of and .

scholarly journals Algebraic Reflexivity of Non-Canonical Isometries on Lipschitz Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1635
Author(s):  
Antonio Jiménez-Vargas ◽  
María Isabel Ramírez
Keyword(s):  
Isometry Group ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Integral Operators ◽  
Supremum Norm ◽  
Weighted Composition Operators ◽  
Local Isometry ◽  
Weighted Composition ◽  
Complex Valued ◽  
Linear Isometries

Let Lip([0,1]) be the Banach space of all Lipschitz complex-valued functions f on [0,1], equipped with one of the norms: fσ=|f(0)|+f′L∞ or fm=max|f(0)|,f′L∞, where ·L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0,1]). Namely, we prove that every approximate local isometry of Lip([0,1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0,1]). Additionally, some complete characterizations of such reflections and projections are stated.

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).

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.

Ideals and Subalgebras of a Function Algebra

1974 ◽  
Vol 26 (02) ◽  
pp. 405-411 ◽  
Author(s):  
Bruce Lund
Keyword(s):  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Ideal Space ◽  
Shilov Boundary ◽  
Closed Point ◽  
Continuous Complex ◽  
Complex Valued ◽  
Uniformly Closed

Let X be a compact Hausdorff space and C(X) the set of all continuous complex-valued functions on X. A function algebra A on X is a uniformly closed, point separating subalgebra of C(X) which contains the constants. Equipped with the sup-norm, A becomes a Banach algebra. We let MA denote the maximal ideal space and SA the Shilov boundary.

On The Center Of Quasi-Central Banach Algebras With bounded Approximate Identity

1981 ◽  
Vol 33 (1) ◽  
pp. 68-90
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Completely Regular ◽  
Continuous Complex ◽  
Complex Valued ◽  
Canonical Image

Let A be a quasi-central complex Banach algebra with a bounded approximate identity and Prim A the structure space of A. In [15], we have shown that every central double centralizer T on A can be represented as a bounded continuous complex-valued function ΦT on Prim A such that Tx + P = ΦT(P)(x + P) for all x ∈ A and P ∈ Primal when the center Z(A) of A is completely regular. Here x + P for P ∈ Prim A denotes the canonical image of x in A/P. In particular, in the case of quasi-central C*-algebras, this result is equivalent to the Dixmier's representation theorem of central double centralizers on C*-algebras (see [3, Section 2] and [9, Theorem 5]).In this paper, it is shown that if Z(A) is completely regular then the space Prim A is locally quasi-compact and for each element z of Z(A), ΦLz vanishes at infinity, where Lz for z ∊ Z(i) is the central double centralizer on A defined by Lz(x) = zx for all x ∊ A.

Separating maps on weighted function algebras on topological groups

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Saeid Maghsoudi ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Weight Functions ◽  
Compact Groups ◽  
Topological Isomorphism ◽  
Topological Groups ◽  
Locally Compact ◽  
Weighted Function ◽  
Function Algebras ◽  
Weighted Composition ◽  
Continuous Complex ◽  
Complex Valued

AbstractLet G 1 and G 2 be locally compact groups and let ω 1 and ω 2 be weight functions on G 1 and G 2, respectively. For i = 1, 2, let also C 0(G i, 1/ω i) be the algebra of all continuous complex-valued functions f on G i such that f/ω i vanish at infinity, and let H: C 0(G 1, 1/ω 1) → C 0(G 2, 1/ω 2) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C 0(G 1, 1/ω 1) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C 0(G 1, 1/ω 1), where φ: G 2 → ℂ is a non-vanishing continuous function and h: G 2 → G 1 is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.

Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions

1968 ◽  
Vol 20 ◽  
pp. 1150-1164 ◽  
Author(s):  
Ashoke K. Roy
Keyword(s):  
Banach Space ◽  
Lipschitz Condition ◽  
Metric Space ◽  
Lipschitz Constant ◽  
Extreme Points ◽  
Lipschitz Functions ◽  
Compact Metric Space ◽  
Complex Valued ◽  
Linear Isometries

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such thatThe smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

M-Ideals and Function Algebras

1993 ◽  
Vol 36 (1) ◽  
pp. 123-128
Author(s):  
K. Seddighi ◽  
H. Zahedani
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Uniform Algebra ◽  
Singular Measures ◽  
Borel Measures ◽  
Function Algebras ◽  
Continuous Complex ◽  
And Function ◽  
Complex Valued

AbstractLet C(X) be the space of all continuous complex-valued functions defined on the compact Hausdorff space X. We characterize the M-ideals in a uniform algebra A of C(X) in terms of singular measures. For a Banach function algebra B of C(X) we determine the connection between strong hulls for B and its peak sets. We also show that M(X) the space of complex regular Borel measures on X has no M-ideal.

scholarly journals On some classes of weighted composition operators

1990 ◽  
Vol 32 (1) ◽  
pp. 87-94 ◽  
Author(s):  
James T. Campbell ◽  
James E. Jamison
Keyword(s):  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Point Transformation ◽  
Weighted Composition Operators ◽  
Absolutely Continuous ◽  
Finite Measure Space ◽  
Weighted Composition ◽  
Image Position ◽  
Complex Valued

Let (X, Σμ) denote a complete a-finite measure space and T: X → X a measurable (T-1A ε Σ each A ε Σ) point transformation from X into itself with the property that the measure μ°T-1 is absolutely continuous with respect to μ. Given any measurable, complex-valued function w(x) on X, and a function f in L2(μ), define WTf(x) via the equation

Central Double Centralizers on Quasi-Central Banach Algebras with Bounded Approximate Identity

1983 ◽  
Vol 35 (2) ◽  
pp. 373-384
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Continuous Complex ◽  
Image Position ◽  
Centralizer Algebra ◽  
Complex Valued ◽  
Canonical Image

We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such thatHere x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given byfor each T ∊ Z(M(A)).

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