scholarly journals Prime ideal structure of rings of bounded continuous functions

1968 ◽  
Vol 19 (6) ◽  
pp. 1432-1432 ◽  
Author(s):  
Mark Mandelker
Keyword(s):  
Prime Ideal ◽  
Continuous Functions ◽  
Ideal Structure ◽  
Bounded Continuous Functions

Some Characterizations of The Projection Property in Archimedean Riesz Spaces

1972 ◽  
Vol 24 (2) ◽  
pp. 306-311 ◽  
Author(s):  
K. K. Kutty ◽  
J. Quinn
Keyword(s):  
Band Structure ◽  
Prime Ideal ◽  
Vector Lattice ◽  
Strong Relationship ◽  
Ideal Structure ◽  
Projection Property ◽  
Dedekind Completion ◽  
Riesz Spaces

In this paper we give some new characterizations of the projection property in Archimedean Riesz spaces. Our approach primarily explores the interrelationships between such things as the band structure or the prime ideal structure of an Archimedean vector lattice and corresponding structures of its Dedekind completion. Our results show that, in general, there is a ‘strong“ relationship if and only if the original vector lattice has the projection property. The main result of this paper is Theorem 2.6 which both summarizes and extends all of the results we obtain prior to it.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals Near-rings of continuous functions on disconnected groups

1979 ◽  
Vol 28 (4) ◽  
pp. 433-451 ◽  
Author(s):  
Robert D. Hofer
Keyword(s):  
Topological Group ◽  
Identity Element ◽  
Continuous Functions ◽  
Ideal Structure ◽  
Rings Of Continuous Functions ◽  
The Ideal

AbstractN(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

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 Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

scholarly journals On the prime ideal structure of symbolic Rees algebras

2012 ◽  
Vol 4 (3) ◽  
pp. 327-343
Author(s):  
S. Bouchiba ◽  
S. Kabbaj
Keyword(s):  
Prime Ideal ◽  
Ideal Structure ◽  
Rees Algebras

scholarly journals Local invertibility in subrings of C*(X)

1992 ◽  
Vol 46 (3) ◽  
pp. 449-458 ◽  
Author(s):  
H. Linda Byun ◽  
Lothar Redlin ◽  
Saleem Watson
Keyword(s):  
Continuous Functions ◽  
Closed Sets ◽  
Complete Ring ◽  
Maximal Ideals ◽  
One To One ◽  
Local Invertibility ◽  
Bounded Continuous Functions ◽  
Ring Of Functions ◽  
Uniformly Closed

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

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