Nonseparating function algebras
Larry Q. Eifler · Bulletin of the American Mathematical Society · 1972
Let A be a function algebra on X (compact).We say A is a separating algebra on X if for each closed subset S of X and for each x G X\S there exists ƒ in A such that ƒ (x) = 0 and ƒ does not vanish on S. We say that A is essential on X if for each open subset U of X there is a continuous function ƒ £ A such that ƒ vanishes on X/U.Csordas and Reiter asked [2] if there exists a nonseparating, essential algebra A on a (connected) space X for which X is the maximal ideal space of A and also the Silov boundary of A. We give an example of such an algebra and simple examples of nonseparating algebras.Given a compact subset K of C, let P(K) denote the uniform closure in C(K) of the polynomials in z l5 ..., z".An easy application of Hurwitz' theorem [1, p. 176] shows that the first three of the following algebras are nonseparating.EXAMPLE 1.Let A = {z:|z| ^ 1}.Then P(A x A) is nonseparating since /(A x A) = /({(z, w):|z| = 1 or |w| = 1}) for each ƒ in P(A x A).Also, A x A is the maximal ideal space of P(A x A).EXAMPLE 2. The algebra P([0,1] x A) is nonseparating since /([0,1] x A) = ƒ({(*, z):t = 0 or \z\ = 1}).Also, [0,1] x A is the maximal ideal space of P([0,1] x A).EXAMPLE 3. If A is a separating algebra on X = M{A) and if B is a function algebra on X containing A, then B is separating on X but not necessarily separating on M(B).Let B denote the uniform closure in C(A) of polynomials in z and |z|.Then P(A) çBç C(A).One can embed B into P([0,1] x A) by setting F(r, z) = /(rz) for 0 ^ r ^ 1 and |z| g 1.Now one can see that the maximal ideal space of B is [0,1] x A/{0} x A and so B is nonseparating on its maximal ideal space.EXAMPLE 4. Let S 2 = Cu {oo} and let £> = {z: |z| 0, define T r (0) = g(re* 0 ).All of the curves T r are homotopic AM S 1970 subject classifications.Primary 46J10.