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.

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.

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).

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.

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 .

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.

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.

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.

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.

Isometries of self-adjoint complex function spaces

1989 ◽  
Vol 105 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complex Function ◽  
Supremum Norm ◽  
Affine Functions ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued ◽  
Uniformly Closed

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-spaceendowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

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