M-Ideals and Function Algebras

1993 ◽  
Vol 36 (1) ◽  
pp. 123-128
Author(s):  
K. Seddighi ◽  
H. Zahedani
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Uniform Algebra ◽  
Singular Measures ◽  
Borel Measures ◽  
Function Algebras ◽  
Continuous Complex ◽  
And Function ◽  
Complex Valued

AbstractLet C(X) be the space of all continuous complex-valued functions defined on the compact Hausdorff space X. We characterize the M-ideals in a uniform algebra A of C(X) in terms of singular measures. For a Banach function algebra B of C(X) we determine the connection between strong hulls for B and its peak sets. We also show that M(X) the space of complex regular Borel measures on X has no M-ideal.

Ideals and Subalgebras of a Function Algebra

1974 ◽  
Vol 26 (02) ◽  
pp. 405-411 ◽  
Author(s):  
Bruce Lund
Keyword(s):  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Ideal Space ◽  
Shilov Boundary ◽  
Closed Point ◽  
Continuous Complex ◽  
Complex Valued ◽  
Uniformly Closed

Let X be a compact Hausdorff space and C(X) the set of all continuous complex-valued functions on X. A function algebra A on X is a uniformly closed, point separating subalgebra of C(X) which contains the constants. Equipped with the sup-norm, A becomes a Banach algebra. We let MA denote the maximal ideal space and SA the Shilov boundary.

On the Boundary and Tensor Product of Function Algebras

1966 ◽  
Vol 9 (1) ◽  
pp. 103-106
Author(s):  
A. S. Fox
Keyword(s):  
Tensor Product ◽  
Algebraic Structure ◽  
Compact Hausdorff Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Maximum Modulus ◽  
Function Algebras ◽  
The Family ◽  
Continuous Complex ◽  
Complex Valued

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .

Weak and norm sequential convergence in M(S)

1973 ◽  
Vol 15 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. Tong ◽  
D. Wilken
Keyword(s):  
Uniform Convergence ◽  
Compact Hausdorff Space ◽  
Dual Space ◽  
Hausdorff Space ◽  
Compact Operators ◽  
Borel Measures ◽  
Sequential Convergence ◽  
Continuous Complex ◽  
Complex Valued ◽  
Convergent Sequences

Let S be a compact Hausdorff space; let C(S) be the algebra of all continuous complex valued functions on S; and let M(S) be the dual space of (S) (the space of all regular Borel measures on S). In [2] Grothendieck gave a description of weak sequential convergence in M(S) in terms of uniform convergence on sequences of disjoint open sets in S. In this note we give a condition on the carriers of measures to guarantee that weak zero convergent sequences are norm zero convergent. While this condition is interesting in its own right, it can also be used to obtain immediately some well-known results about compact operators from C(S) to c0.

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).

scholarly journals REPRESENTATIONS OF REAL BANACH ALGEBRAS

2010 ◽  
Vol 88 (3) ◽  
pp. 289-300 ◽  
Author(s):  
F. ALBIAC ◽  
E. BRIEM
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Ideal Space ◽  
Gelfand Transform ◽  
Continuous Complex ◽  
Unital Banach Algebra ◽  
Complex Valued

AbstractA commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

A short elementary proof of the Bishop–Stone–Weierstrass theorem

1984 ◽  
Vol 96 (2) ◽  
pp. 309-311 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Elementary Proof ◽  
Uniform Norm ◽  
Weierstrass Theorem ◽  
Continuous Complex ◽  
Empty Subset ◽  
Complex Valued

Fix the following notation. Let X be a compact Hausdorff space, and denote by C(X) the vector space of continuous complex-valued functions on X, equipped with the uniform norm ∥·∥x. Let A be a unital subalgebra of C(X). A non-empty subset S of X is said to be A-antisymmetric if whenever h ∈ A and h is real-valued on S then h is constant on S.

On Unitary Equivalence of Matrices over the Ring of Continuous Complex-Valued Functions on a Stonian Space

1963 ◽  
Vol 15 ◽  
pp. 323-331 ◽  
Author(s):  
Carl Pearcy
Keyword(s):  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Unitary Equivalence ◽  
Continuous Complex ◽  
Complex Valued

This paper is a continuation of the earlier papers (1, 5) in which the author studied matrices with entries from the algebra C() of all continuous, complex-valued functions on an extremely disconnected, compact Hausdorff space . (Such spaces are sometimes called Stonian, after M. H. Stone, who first considered them in (8). They arise naturally as maximal ideal spaces of abelian W*-algebras.) In this note, three theorems are proved.

Commuting Dilations and Uniform Algebras

1990 ◽  
Vol 42 (5) ◽  
pp. 776-789 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Hardy Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Uniform Algebra ◽  
Representing Measure ◽  
Uniform Algebras ◽  
Weak Closure ◽  
Abstract Hardy Space ◽  
Complex Valued

Let X be a compact Hausdorff space, let C(X) be the algebra of complex-valued continuous functions on X, and let A be a uniform algebra on X. Fix a nonzero complex homomorphism τ on A and a representing measure m for τ on X. The abstract Hardy space Hp = Hp(m), 1 ≤ p ≤ ∞, determined by A is defined to the closure of Lp = Lp(m) when p is finite and to be the weak*-closure of A in L∞ = L∞(m) p = ∞.

scholarly journals BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 419-426 ◽  
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Function Algebra ◽  
Shilov Boundary ◽  
Function Algebras ◽  
Peak Points ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

scholarly journals Isomorphisms of Function Algebras and Algebras of Analytic Functions

1978 ◽  
Vol 21 (1) ◽  
pp. 61-71
Author(s):  
Bruce Lund
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Finite Codimension ◽  
Open Riemann Surface ◽  
Analytic Boundary ◽  
Function Algebras

AbstractLet R be a finite open Riemann surface with analytic boundary Γ. Set and define is analytic on R}. Conditions are given on a function algebra A on a compact Hausdorff space X which imply that A is isomorphic to a subalgebra of A(R) of finite codimension.

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