Algebras of measures on C-distinguished topological semigroups

1978 ◽  
Vol 84 (2) ◽  
pp. 323-336 ◽  
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.

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

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).

scholarly journals Generalized Numerical Index of Function Algebras

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.

scholarly journals The cardinality of the set of left invariant means on topological semigroups

1988 ◽  
Vol 37 (2) ◽  
pp. 247-262 ◽  
Author(s):  
Heneri A.M. Dzinotyiweyi
Keyword(s):  
Large Class ◽  
Topological Semigroup ◽  
Continuous Functions ◽  
Upper Bounds ◽  
Compact Sets ◽  
Lower And Upper Bounds ◽  
Topological Semigroups ◽  
Invariant Means ◽  
Uniformly Continuous

For a very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

Topological semigroups and their prequantale models

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6205-6210 ◽  
Author(s):  
Bin Zhao ◽  
Changchun Xia ◽  
Kaiyun Wang
Keyword(s):  
Topological Semigroup ◽  
Satisfying Condition ◽  
Locally Compact ◽  
Topological Semigroups

In this paper, we introduce a condition (?) on topological semigroups, and prove that every T1 topological semigroup satisfying condition (?) has a bounded complete algebraic prequantale model. On the basis of this result, we also show that every T0 topological semigroup satisfying condition (?) can be embedded into a compact and locally compact sober topological semigroup.

scholarly journals The Levi-Civita equation in function classes

2019 ◽  
Vol 94 (4) ◽  
pp. 689-701 ◽  
Author(s):  
Miklós Laczkovich
Keyword(s):  
Topological Semigroup ◽  
Classical Result ◽  
Continuous Functions ◽  
Exponential Polynomial ◽  
Additive Functions ◽  
Translation Invariant ◽  
Function Classes ◽  
Finite Dimensional ◽  
Special Cases ◽  
Complex Valued

Abstract Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form $$\sum _{i=1}^np_i \cdot m_i$$ ∑ i = 1 n p i · m i , where $$m_1 ,\ldots ,m_n$$ m 1 , … , m n are exponentials belonging to V, and $$p_1 ,\ldots ,p_n$$ p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra $${{\mathcal {A}}}$$ A of complex valued functions such that whenever an exponential m belongs to $${{\mathcal {A}}}$$ A , then $$m^{-1}\in {{\mathcal {A}}}$$ m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary $$\sigma $$ σ -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

scholarly journals Semi-algebras in C(T)

1974 ◽  
Vol 15 (1) ◽  
pp. 85-87
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Continuous Functions ◽  
Complex Numbers ◽  
Compact Set ◽  
Disc Algebra ◽  
Complex Valued

Let C(T) be the Banach algebra of all complex-valued continuous functions on the compact set T of all complex numbers with modulus one. As usual we may suppose that A is embedded in C(T), where A is the disc algebra, i.e., the algebra of all complex-valued functions f(λ) continuous for |λ| ≦ 1. and analytic for |λ| < 1. We set Mλ = {f ∈ A: f(λ) = 0} and Mλ = {f ∈ A: f(λ) ≧ 0}.

Convolutions as Bilinear and Linear Operators

1964 ◽  
Vol 16 ◽  
pp. 275-285 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Linear Operators ◽  
Locally Compact ◽  
Radon Measures

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

Isomorphisms of Multiplier Algebras

1968 ◽  
Vol 20 ◽  
pp. 1165-1172 ◽  
Author(s):  
G. I. Gaudry
Keyword(s):  
Haar Measure ◽  
Lebesgue Space ◽  
Dirac Measure ◽  
Locally Compact ◽  
Radon Measures ◽  
Continuous Complex ◽  
Complex Valued ◽  
Multiplier Algebras ◽  
Identity Elements

Suppose that G1 and G2 are two locally compact Hausdorff groups with identity elements e and e’ and with respective left Haar measures dx and dy. Let 1 ≦ p ≦ ∞, and Lp(Gi) be the usual Lebesgue space over Gi formed relative to left Haar measure on Gi. We denote by M(Gi) the space of Radon measures, and by Mbd(Gi) the space of bounded Radon measures on Gi. If a ϵ Gi we write ϵa for the Dirac measure at the point a. Cc(Gi) will denote the space of continuous, complex-valued functions on Gi with compact supports, whilst Cc+ (Gi) will denote that subset of Cc(Gi) consisting of those functions which are real-valued and non-negative.

Some results on the union of Helson sets

1971 ◽  
Vol 70 (2) ◽  
pp. 235-241
Author(s):  
E. Galanis
Keyword(s):  
Abelian Group ◽  
Compact Subset ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Radon Measures ◽  
Bounded Complex ◽  
Complex Valued

Let G be a compact abelian group and let Ĝ be its dual group. We shall denote by M(G) the set of all bounded complex valued Radon measures on G and by M(E) the elements of M(G) with support in a compact subset E of G.

scholarly journals An approximative property of spaces of continuous functions

1974 ◽  
Vol 15 (1) ◽  
pp. 48-53 ◽  
Author(s):  
R. B. Holmes ◽  
J. D. Ward
Keyword(s):  
Banach Space ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Compact Set ◽  
Canonical Embedding ◽  
Locally Compact ◽  
Nearest Point ◽  
Spaces Of Continuous Functions ◽  
Approximative Property ◽  
Canonical Image

A Banach space X is said to have property (PROXBID) if the canonical image of X in its bidual X** is proximal. In other words, if J: X → X** is the canonical embedding, then it is required that every element of X** have at least one best approximation (i.e., nearest point) from the closed subspace J(X). We show below that, if X is the space of (real or complex) continuous functions on a compact set, or the space of (real or complex) continuous functions that vanish at infinity on a locally compact set, then X has property (PROXBID). At this point we should mention the existence of a variety of examples [2, 8] of Banach spaces which lack property (PROXBID).

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