scholarly journals Averaging operators on the ring of continuous functions on a compact space

1964 ◽  
Vol 4 (3) ◽  
pp. 293-298 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Compact Space ◽  
Linear Transformation ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Topological Product ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Ring Of Continuous Functions ◽  
Image Position

In this note we answer the following question: Given C(X) the latticeordered ring of real continuous functions on the compact Hausdorff space X and T an averaging operator on C(X), under what circumstances can X be decomposed into a topological product such that supports a measure m and Tf = h where By an averaging operator we mean a linear transformation T on C(X) such that: 1. T is positive, that is, if f>0 (f(x) ≧ 0 for all x ∈ and f(x) > 0 for some a ∈ X), then Tf>0. 2. T(fTg) = (Tf)(Tg). 3. T l = 1 where l(x) = 1 for all x ∈ X.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

scholarly journals Exhaustive operators and vector measures

1975 ◽  
Vol 19 (3) ◽  
pp. 291-300 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Vector Measures ◽  
Borel Functions ◽  
Image Position ◽  
Real Topological Vector Space ◽  
Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Semi-Groups of Maps in a Locally Compact Space

1967 ◽  
Vol 19 ◽  
pp. 688-696 ◽  
Author(s):  
J. R. Dorroh
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
List Type ◽  
Type Number ◽  
Semi Group ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Locally Compact Hausdorff Space

Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying(i)ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and(ii)ϕ0 = e, the identity map on S.A semi-group {ϕt} of maps in S is said to be(iii)of class (C0) if ϕt(x) → x as t → 0 for each x in S,(iv)separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and(v)doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.

An Extremum Result

1962 ◽  
Vol 14 ◽  
pp. 597-601 ◽  
Author(s):  
J. Kiefer
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Ad Hoc ◽  
Continuous Functions ◽  
Orthogonality Condition ◽  
Technical Detail ◽  
Real Numbers ◽  
Discrete Probability ◽  
Linearly Independent ◽  
Image Position

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

scholarly journals The Centre of the Spaces of Banach Lattice-Valued Continuous Functions on the Generalized Alexandroff Duplicate

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Faruk Polat
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Isolated Points ◽  
Discrete Functions

We characterize the centre of the Banach lattice of Banach lattice -valued continuous functions on the Alexandroff duplicate of a compact Hausdorff space in terms of the centre of , the space of -valued continuous functions on . We also identify the centre of whose elements are the sums of -valued continuous and discrete functions defined on a compact Hausdorff space without isolated points, which was given by Alpay and Ercan (2000).

scholarly journals A Choquet-Deny theorem for affine functions on a Choquet simplex

1969 ◽  
Vol 16 (4) ◽  
pp. 325-327 ◽  
Author(s):  
H. A. Priestley
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Present Note ◽  
Continuous Functions ◽  
Choquet Simplex ◽  
Affine Functions ◽  
Stone Theorem

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

The Measure Spectrum of a Uniform Algebra and Subharmonicity

1982 ◽  
Vol 34 (3) ◽  
pp. 673-685
Author(s):  
Donna Kumagai
Keyword(s):  
Probability Measure ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Uniform Algebra ◽  
Ideal Space ◽  
Representing Measure ◽  
Uniform Algebras ◽  
The Subject ◽  
Image Position

Let A be a uniform algebra on a compact Hausdorff space X. The spectrum, or the maximal ideal space, MA, of A is given byWe define the measure spectrum, SA, of A bySA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfyingThe concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.

A theorem of Stone-Weierstrass type

1966 ◽  
Vol 62 (4) ◽  
pp. 649-666 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Weierstrass Theorem ◽  
Necessary And Sufficient ◽  
Complex Valued

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

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