Algebras of Holomorphic Functions in Ringed Spaces, I

1969 ◽  
Vol 21 ◽  
pp. 1281-1292
Author(s):  
Maxwell E. Shauck
Keyword(s):  
Open Subset ◽  
Hausdorff Space ◽  
Holomorphic Functions ◽  
Continuous Functions ◽  
Open Neighbourhood ◽  
Continuous Complex ◽  
Algebras Of Holomorphic Functions ◽  
Complex Valued

A pair () is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to by . If , then implies that there exists some open neighbourhood V of u, V ⊂ U, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by and is called the -holomorphic functions on U.THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.

A Characterization of the Algebra of Functions Vanishing at Infinity

1969 ◽  
Vol 21 ◽  
pp. 751-754 ◽  
Author(s):  
Robert E. Mullins
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Compact ◽  
Algebra Of Functions ◽  
Continuous Complex ◽  
Complex Valued ◽  
Locally Compact Hausdorff Space

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.

Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra

2007 ◽  
Vol 50 (1) ◽  
pp. 3-10
Author(s):  
Richard F. Basener
Keyword(s):  
Continuous Functions ◽  
Uniform Algebra ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Spaces Of Continuous Functions ◽  
The Family ◽  
Continuous Complex ◽  
Finite Dimensional Case ◽  
Complex Valued ◽  
Shed Light

AbstractIn this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the “finite dimensional” case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy–Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

A theorem of Stone-Weierstrass type

1966 ◽  
Vol 62 (4) ◽  
pp. 649-666 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Weierstrass Theorem ◽  
Necessary And Sufficient ◽  
Complex Valued

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

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).

Montel Algebras on the Plane

1970 ◽  
Vol 22 (1) ◽  
pp. 116-122 ◽  
Author(s):  
W. E. Meyers
Keyword(s):  
Topological Algebra ◽  
Maximum Modulus ◽  
Continuous Homomorphism ◽  
Continuous Functions ◽  
Maximum Modulus Principle ◽  
Compact Open Topology ◽  
Bounded Functions ◽  
Continuous Complex ◽  
Complex Valued

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

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.

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 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 = ∞.

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