DITKIN CONDITIONS

2017 ◽  
Vol 60 (1) ◽  
pp. 153-163
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Commutative Banach Algebra ◽  
Algebraic Properties ◽  
The Relationship

AbstractThis paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$).

Functionals on Real C(S)

1978 ◽  
Vol 30 (03) ◽  
pp. 490-498 ◽  
Author(s):  
Nicholas Farnum ◽  
Robert Whitley
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Codimension One ◽  
Maximal Ideals ◽  
Invertible Elements ◽  
Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

Discontinuous homomorphisms from Banach *-algebras

1996 ◽  
Vol 120 (4) ◽  
pp. 703-708
Author(s):  
Volker Runde
Keyword(s):  
Banach Algebra ◽  
Open Problem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Commutative Banach Algebras ◽  
The Continuum

The long open problem raised by I. Kaplansky if, for an infinite compact Hausdorff space X, there is a discontinuous homomorphism from (X) into a Banach algebra was settled in the 1970s, independently, by H. G. Dales and J. Esterle. If the continuum hypothesis is assumed, then there is a discontinuous homomorphism from (X) (see [8] for a survey of both approaches and [9] for a unified exposition). The techniques developed by Dales and Esterle are powerful enough to yield discontinuous homomorphisms from commutative Banach algebras other than (X). In fact, every commutative Banach algebra with infinitely many characters is the domain of a discontinuous homomorphism ([7]).

scholarly journals OPERATORS ON BANACH ALGEBRA VALUED FUNCTION SPACES

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 ◽  
Locally Compact Hausdorff Space

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.

scholarly journals F-Rings of Continuous Functions I

1959 ◽  
Vol 11 ◽  
pp. 80-86 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
The Other ◽  
Topological Spaces ◽  
Closed Sets ◽  
Real Complex

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

scholarly journals Spatial numerical ranges of elements of subalgebras ofC0(X)

2000 ◽  
Vol 23 (12) ◽  
pp. 827-831
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Numerical Range ◽  
Commutative Banach Algebra ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued ◽  
Locally Compact Hausdorff Space

WhenAis a subalgebra of the commutative Banach algebraC0(X)of all continuous complex-valued functions on a locally compact Hausdorff spaceX, the spatial numerical range of element ofAcan be described in terms of positive measures.

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]):

Bourgain Algebras of Douglas Algebras

1992 ◽  
Vol 44 (4) ◽  
pp. 797-804 ◽  
Author(s):  
Pamela Gorkin ◽  
Keiji Izuchi ◽  
Raymond Mortini
Keyword(s):  
Banach Algebra ◽  
Unit Circle ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Closed Subspace ◽  
Closed Subalgebra ◽  
Douglas Algebras

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

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 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.

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