Banach Algebra of Bounded Complex-Valued Functionals

Let G be a locally compact Abelian group, written adoptively, with Haar measure m, L1(G) the group algebra of G, and M(G) the Banach algebra of all bounded, complex-valued, regular, countably additive measures on G. For a general account of L1(G) and M(G) see Rudin (7).

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A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-059-2 ◽

1971 ◽

Vol 23 (3) ◽

pp. 544-549

Author(s):

G. E. Peterson

Keyword(s):

Hausdorff Spaces ◽

Locally Compact ◽

Compact Spaces ◽

Bounded Complex ◽

Hausdorff Method ◽

Borel Functions ◽

Bounded Functions ◽

Abelian Theorem ◽

Complex Valued ◽

Locally Compact Spaces

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

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Generalized Numerical Index of Function Algebras

Journal of Function Spaces ◽

10.1155/2019/9080867 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6

Author(s):

Han Ju Lee

Keyword(s):

Banach Space ◽

Function Algebra ◽

Complex Banach Space ◽

Continuous Functions ◽

Function Algebras ◽

Bounded Complex ◽

Closed Subalgebra ◽

Bounded Continuous Functions ◽

Complex Valued ◽

Numerical Index

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

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Algebras of measures on C-distinguished topological semigroups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055146 ◽

1978 ◽

Vol 84 (2) ◽

pp. 323-336 ◽

Cited By ~ 3

Author(s):

H. A. M. Dzinotyiweyi

Keyword(s):

Total Variation ◽

Topological Semigroup ◽

Continuous Mapping ◽

Continuous Functions ◽

Compact Set ◽

Locally Compact ◽

Radon Measures ◽

Topological Semigroups ◽

Bounded Complex ◽

Complex Valued

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.

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The algebras of bounded and essentially bounded Lebesgue measurable functions

Demonstratio Mathematica ◽

10.1515/dema-2017-0010 ◽

2017 ◽

Vol 50 (1) ◽

pp. 94-99

Author(s):

Raymond Mortini ◽

Rudolf Rupp

Keyword(s):

Positive Lebesgue Measure ◽

Discrete Topology ◽

Measurable Functions ◽

Almost Everywhere ◽

Bounded Complex ◽

Elementary Methods ◽

Point Evaluations ◽

Complex Valued ◽

Totally Disconnected ◽

The Ideal

Abstract Let X be a set in ℝn with positive Lebesgue measure. It is well known that the spectrum of the algebra L∞(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒb(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τdis),where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒb(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L∞(X))), of the ideal of all functions in ℒb(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒb(X, ℂ).

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Functionals on Real C(S)

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-044-8 ◽

1978 ◽

Vol 30 (03) ◽

pp. 490-498 ◽

Cited By ~ 2

Author(s):

Nicholas Farnum ◽

Robert Whitley

Keyword(s):

Banach Algebra ◽

Compact Hausdorff Space ◽

Linear Functional ◽

Hausdorff Space ◽

Function Algebra ◽

Continuous Functions ◽

Codimension One ◽

Maximal Ideals ◽

Invertible Elements ◽

Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

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REPRESENTATIONS OF REAL BANACH ALGEBRAS

Journal of the Australian Mathematical Society ◽

10.1017/s144678871000011x ◽

2010 ◽

Vol 88 (3) ◽

pp. 289-300 ◽

Cited By ~ 3

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).

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Linear equations in B(ℤ)*

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050002967x ◽

1993 ◽

Vol 123 (6) ◽

pp. 1001-1009 ◽

Cited By ~ 1

Author(s):

M. Filali

Keyword(s):

Linear Equations ◽

Adjoint Operator ◽

Translation Operator ◽

Complex Numbers ◽

Bounded Complex ◽

Image Position ◽

Complex Valued

SynopsisLet B(ℤ)* be the Banach dual of the space of all bounded complex-valued functions on ℤ. For each n ε ℤ, let Ln be the translation operator on B(ℤ) and Tn be its adjoint operator on B(ℤ)*. This paper concerns itself with equations of the formwhere (an)nεℤ is a sequence of complex numbers.

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Functional Space C(ω), C0(ω)

Formalized Mathematics ◽

10.2478/v10037-012-0003-3 ◽

2012 ◽

Vol 20 (1) ◽

pp. 15-22

Author(s):

Katuhiko Kanazashi ◽

Hiroyuki Okazaki ◽

Yasunari Shidama

Keyword(s):

Topological Space ◽

Banach Algebra ◽

Function Space ◽

Normed Space ◽

Functional Space ◽

Continuous Functions ◽

Bounded Support ◽

Compact Topological Space ◽

Definition Of ◽

Complex Valued

Functional Space C(ω), C0(ω) In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-008-0 ◽

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.

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