A Generalization of Strassen’s Theorem on Preordered Semirings

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.

Homomorphisms of near-rings of continuous real-valued functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017159 ◽

1996 ◽

Vol 53 (3) ◽

pp. 401-411

Author(s):

K.D. Magill

Keyword(s):

Compact Hausdorff Space ◽

Topological Structure ◽

Hausdorff Space ◽

Additive Group ◽

Continuous Functions ◽

Endomorphism Semigroup ◽

Real Numbers ◽

Extensive Class ◽

Ring Of Functions

For any topological near-ring (which is not a ring) whose additive group is the additive group of real numbers, we investigate the near-ring of all continuous functions, under the pointwise operations, from a compact Hausdorff space into that near-ring. Specifically, we determine all the homomorphisms from one such near-ring of functions to another and we show that within a rather extensive class of spaces, the endomorphism semigroup of the near-ring of functions completely determines the topological structure of the space.

Distinguishedness of weighted Fréchet spaces of continuous functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500005538 ◽

1992 ◽

Vol 35 (2) ◽

pp. 271-283 ◽

Cited By ~ 6

Author(s):

Françoise Bastin

Keyword(s):

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Fréchet Space ◽

Frechet Space ◽

Fréchet Spaces ◽

Density Condition ◽

Locally Compact ◽

Spaces Of Continuous Functions ◽

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2019/7146073 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7

Author(s):

Manuel Felipe Cerpa-Torres ◽

Michael A. Rincón-Villamizar

Keyword(s):

Banach Space ◽

Compact Hausdorff Space ◽

Geometrical Parameter ◽

Hausdorff Space ◽

Continuous Functions ◽

Continuous Image ◽

Topological Properties ◽

Locally Compact ◽

Regular Subspace ◽

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

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

Journal of the Australian Mathematical Society ◽

10.1017/s144678870002406x ◽

1964 ◽

Vol 4 (3) ◽

pp. 293-298 ◽

Cited By ~ 3

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.

OPERATORS ON BANACH ALGEBRA VALUED FUNCTION SPACES

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.23.1992.4546 ◽

1992 ◽

Vol 23 (3) ◽

pp. 233-238

Author(s):

JOR-TING CHAN

Keyword(s):

Banach Algebra ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Function Spaces ◽

Linear Operators ◽

Locally Compact ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Let $S$ be a locally compact Hausdorff space and let $A$ be a Banach algebra. Denote by $C_0(S, A)$ the Banach algebra of all $A$-valued continuous functions vanishing at infinity on $S$. Properties of bounded linear operators on $C_0(S,A)$, like multiplicativity, are characterized by Choy in terms of their representing measures. We study these theorems and give sharper results in certain cases.

The Size of the Unit Sphere

10.4153/cjm-1968-041-1 ◽

1968 ◽

Vol 20 ◽

pp. 450-455 ◽

Cited By ~ 6

Author(s):

Robert Whitley

Keyword(s):

Banach Spaces ◽

Unit Sphere ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Locally Compact ◽

Spaces Of Continuous Functions ◽

Banach (1, pp. 242-243) defines, for two Banach spaces X and Y, a number (X, Y) = inf (log (‖L‖ ‖L-1‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if (X, Y) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.

A Characterization of the Algebra of Functions Vanishing at Infinity

10.4153/cjm-1969-085-1 ◽

1969 ◽

Vol 21 ◽

pp. 751-754 ◽

Cited By ~ 1

Author(s):

Robert E. Mullins

Keyword(s):

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Locally Compact ◽

Algebra Of Functions ◽

Continuous Complex ◽

Complex Valued ◽

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

Semi-Groups of Maps in a Locally Compact Space

10.4153/cjm-1967-063-3 ◽

1967 ◽

Vol 19 ◽

pp. 688-696 ◽

Cited By ~ 7

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 ◽

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.

On Additive Operators

10.4153/cjm-1971-050-7 ◽

1971 ◽

Vol 23 (3) ◽

pp. 468-480 ◽

Cited By ~ 8

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]):

On the theorem of Kishi for a continuous function-kernel

Nagoya Mathematical Journal ◽

10.1017/s0027763000016081 ◽

1974 ◽

Vol 53 ◽

pp. 127-135 ◽

Cited By ~ 4

Author(s):

Isao Higuchi ◽

Masayuki Itô

Keyword(s):

Continuous Function ◽

Potential Theory ◽

Compact Hausdorff Space ◽

Symmetric Function ◽

Hausdorff Space ◽

Locally Compact ◽

Continuity Principle ◽

Lower Semi ◽

In the potential theory with respect to a non-symmetric function-kernel, the following theorem is obtained by M. Kishi ([3]).Let X be a locally compact Hausdorff space and G be a lower semi-continuous function-kernel on X. Assume that G(x, x)>0 for any x in X and that G and the adjoint kernel Ğ of G satisfy “the continuity principle”.