scholarly journals The three kernels of a compact semiring

1969 ◽  
Vol 10 (3-4) ◽  
pp. 299-319 ◽  
Author(s):  
K. R. Pearson
Keyword(s):  
Hausdorff Space ◽  
Topological Semigroups ◽  
Binary Operations ◽  
Image Position

A topological semiring is a system (S, +, ⋅) where S is a Hausdorff space, (S, +) and (S, ⋅) are topological semigroups (i.e., + and ⋅ are continuous associative binary operations on S) and the distributive laws , hold for all x, y, z in S. The operations + and ⋅ are called addition and multiplication respectively.

scholarly journals Compact Semirings Which are Multiplicatively 0-Simple

1969 ◽  
Vol 10 (3-4) ◽  
pp. 320-329
Author(s):  
K. R. Pearson
Keyword(s):  
Topological Semigroups ◽  
Image Position

A topological semiring is a system (S, +, ⋅) where (S, +) and (S, ⋅) are topological semigroups and the distributive laws , hold for all x, y, z in S; + and ⋅ are called addition and multiplication respectively.

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]):

The Multiplicative Groups of Quasifields

1987 ◽  
Vol 39 (4) ◽  
pp. 784-793 ◽  
Author(s):  
Michael J. Kallaher
Keyword(s):  
Zero Element ◽  
Binary Operations ◽  
Definition Of ◽  
Image Position ◽  
Multiplicative Groups ◽  
Multiplicative Mapping

Let (Q, +, ·) be a finite quasifield of dimension d over its kernel K = GF(q), where q = pk with p a prime and k ≧ 1. (See p. 18-22 and p. 74 of [7] or Section 5 of [9] for the definition of quasifield.) For the remainder of this article we will follow standard conventions and omit, whenever possible, the binary operations + and · in discussing a quasifield. For example, the notation Q will be used in place of the triple (Q, +, ·) and Q* will be used to represent the multiplicative loop (Q − {0}, ·).Let m be a non-zero element of the quasifield Q; the right multiplicative mapping ρm:Q → Q is defined by1

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.

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.

scholarly journals Embedding semirings in semirings with multiplicative unit

1968 ◽  
Vol 8 (2) ◽  
pp. 183-191
Author(s):  
K. R. Pearson
Keyword(s):  
Topological Semigroups ◽  
Image Position

A topological semiring is a system (S, +, ·) where (S, +) and (S, ·) are topological semigroups and · distributes across + as in a ring; that is, for all x, y, z in S, The operations + and · are called addition and multiplication respectively.

scholarly journals On weak approximation and convexification in weighted spaces of vector-valued continuous functions

1989 ◽  
Vol 31 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Marek Nawrocki
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Weak Approximation ◽  
Completely Regular ◽  
Hausdorff Topological Vector Space ◽  
The Family ◽  
Image Position ◽  
Positive Functions ◽  
Vector Valued

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

Characterization of some classes of operators on spaces of vector-valued continuous functions

1985 ◽  
Vol 97 (1) ◽  
pp. 137-146 ◽  
Author(s):  
Fernando Bombal ◽  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Representing Measure ◽  
Bounded Linear ◽  
Second Dual ◽  
Image Position ◽  
Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

scholarly journals Double ideals in compact semirings

1973 ◽  
Vol 15 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Martha Bertman ◽  
John Selden
Keyword(s):  
Hausdorff Space ◽  
The Other ◽  
Image Position

By a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that for all x, y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.

scholarly journals Averaging distances in certain Banach spaces

1997 ◽  
Vol 55 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Finite Dimension ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Empty Interval ◽  
Positive Real ◽  
Complete Discussion ◽  
Image Position

Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such thatIt follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

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