Bishop theorem

One of the early generalizations of the famous Stone–Weierstrass theorem, which itself is a generalization of the celebrated Weierstrass theorem stating that every real-valued continuous function on a closed and bounded interval is a uniform limit (cf. also Uniform convergence) of a sequence of polynomials.

Let $X$ be a compact Hausdorff space, and let $C ( X )$ denote the set of all complex-valued continuous functions on $X$ equipped with the supremum norm, given by

\begin{equation*} \| f \| : = \{ \| f ( x ) \| : x \in X \}. \end{equation*}

Let $A$ be a non-empty subset of $C ( X )$ and let $K$ be a non-empty subset of $X$. One says that $K$ is a partially $A$-anti-symmetric set if $f \in A$ and $f | _ { K }$ (the restriction of $f$ to $K$) real imply that $f | _ { K }$ is a constant. A partially $A$-anti-symmetric set is called $A$-anti-symmetric if $f \in A$ and $f | _ { K }$ purely imaginary (that is, $\operatorname { Re } ( f | _ { K } ) = 0$) imply that $f | _ { K }$ is a constant.

It is easy to see that if $A$ is closed under multiplication by $i$ (that is, $f \in A$ implies $i f \in A$), then every partially $A$-anti-symmetric set is $A$-anti-symmetric. This is not true for arbitrary $A$. (See [a5] for an example.) Every partially $A$-anti-symmetric set is contained in a maximal partially $A$-anti-symmetric set. Every maximal partially $A$-anti-symmetric set is closed. Distinct maximal partially $A$-anti-symmetric sets are disjoint. Each singleton set is a partially $A$-anti-symmetric set. Thus, the family of all maximal partially $A$-anti-symmetric sets forms a partition of $X$. Proofs of these and many other interesting properties of partially $A$-anti-symmetric sets can be found in [a5]. All these statements are also true for $A$-anti-symmetric sets. (See [a5], [a3].)

Bishop's theorem.

1) Let $A$ be a uniformly closed real subalgebra of $C ( X )$ containing the constant function $1$ and let $f \in C ( X )$. If $f | _ { K } \in A | _ { K } : = \{ f | _ { K } : f \in A \}$ for every maximal partially $A$-anti-symmetric set $K$, then $f \in A$.

2) Let $\tau$ be a homeomorphism on $X$ such that $\tau \circ \tau$ is the identity mapping on $X$. Let

\begin{equation*} C ( X , \tau ) : = \{ f \in C ( X ) : f ( \tau ( x ) ) = \overline { f ( x ) } , \forall x \in X \}. \end{equation*}

Let $A$ be a uniformly closed real subalgebra of $C ( X , \tau )$ containing the constant function $1$ and let $f \in C ( X , \tau )$. If $f | _ { K } \in A | _ { K }$ for every maximal $A$-anti-symmetric set $K$, then $f \in A$.

In fact, 1) means that if a continuous function $f$ coincides with some function in $A$ for every maximal partially $A$-anti-symmetric set, then $f \in A$. In view of the comments preceding the statement of the theorem, if $A$ is a complex subalgebra of $C ( X )$, then in 1) above one can replace "partially A-anti-symmetric" by "A-anti-symmetric" . This was the classical statement by E. Bishop in 1961 [a2].

$A$ is said to separate the points of $X$, if for all $x , y \in X$, $x \neq y$, there is an $f \in A$ such that $f ( x ) \neq f ( y )$. A uniformly closed complex subalgebra $A$ of $C ( X )$ that contains $1$ and separates the points of $X$ is called a complex function algebra. Similarly, a uniformly closed real subalgebra $A$ of $C ( X , \tau )$ that contains $1$ and separates the points of $X$ is called a real function algebra. In view of this, 2) is called an analogue of Bishop's theorem for real function algebras. This was proved in [a6]. (See also [a5].) If a complex function algebra $A$ is closed under conjugation (that is, $f \in A$ implies $\overline { f } \in A$), then every maximal $A$-anti-symmetric set reduces to a singleton. (See [a5] for a proof.) Thus, the hypotheses of Bishop's theorem are trivially satisfied by every $f \in C ( X )$. Hence $A = C ( X )$. This is the classical Stone–Weierstrass theorem, which has permeated most of modern analysis and has many generalizations. Bishop's theorem is an essential tool in proving many of these generalizations. (See [a5], [a3].) Similarly, if a real function algebra $A$ is closed under conjugation, then $A = C ( X , \tau )$. This is an analogue of the Stone–Weierstrass theorem for real function algebras. (See [a5] for a proof.)

The proof of Bishop's theorem in [a2] uses many non-trivial tools from functional analysis, such as the Hahn–Banach theorem, the Krein–Mil'man theorem (cf. also Locally convex space) and the Riesz representation theorem (cf. Riesz theorem). This proof can also be found in [a3] and [a10]. J.B. Prolla extended this technique to the case of vector-valued functions [a8]. S. Machado formulated a quantitative version of Prolla's theorem and gave an elementary proof of it [a7]. A self-contained exposition of Machado's proof can be found in [a4]. In 1984, T.J. Ransford gave a very short, simple and elementary proof of Machado's version of Bishop's theorem [a9] (see also [a5]). This proof uses a technique also used in [a1].

References

[a1]	B. Brosowski, F. Deutsch, "An elementary proof of the Stone–Weierstrass theorem" Proc. Amer. Math. Soc. , 81 (1981) pp. 89–92
[a2]	E. Bishop, "A generalization of the Stone–Weierstrass theorem" Pacific J. Math. , 11 (1961) pp. 777–783
[a3]	R.B. Burckel, "Characterizations of $C ( X )$ among its subalgebras" , M. Dekker (1972)
[a4]	R.B. Burckel, "Bishop's Stone–Weierstrass theorem" Amer. Math. Monthly , 91 (1984) pp. 22–32
[a5]	S.H. Kulkarni, B.V. Limaye, "Real function algebras" , M. Dekker (1992)
[a6]	S.H. Kulkarni, N. Srinivasan, "An analogue of Bishop's theorem for real function algebras" Indian J. Pure Appl. Math. , 18 (1987) pp. 136–145
[a7]	S. Machado, "On Bishop's generalization of the Stone–Weierstrass theorem" Indag. Math. , 39 (1977) pp. 218–224 Nederl. Akad. Wetensch. Proc. Ser. A , 80 (1977)
[a8]	J.B. Prolla, "Bishop's generalized Stone–Weierstrass theorem for weighted spaces" Math. Ann. , 191 (1971) pp. 283–289
[a9]	T.J. Ransford, "A short elementary proof of the Bishop–Stone–Weierstrass theorem" Math. Proc. Cambridge Philos. Soc. , 96 (1984) pp. 309–311
[a10]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966)