Vol. 27, No. 3, 1968

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Extensions of the maximal ideal space of a function algebra

Jan-Erik Björk

Vol. 27 (1968), No. 3, 453–462

Let A be a function algebra with its maximal ideal space MA. Let B be a function algebra such that A B C(MA). What can be said about MB? We prove that MA = MB if every point x MA has a fundamental neighborhood system {W} such that the topological boundary bW of each W is contained in the Choquet boundary of A or if A is a normal function algebra. The first condition is satisfied if MA is a one dimensional topological space. Let H(A) be the function algebra on MA generated by all functions which are locally approximable in A. We prove that MH(A) = MA and then we try to generalize this result. If f C(MA) is such that f is locally approximable in A at every point where f is different from zero then MA is the maximal ideal space of the function algebra generated by A and f. We also look at closed subsets F of MA such that MH(F) = F where H(F) is the function algebra generated by restricting to F all functions that are defined and locally approximable in A in some neighborhood of F. These sets are called natural sets. We prove that there exists a smallest natural set B(F) containing a closed set F in MA and that the Silov boundary of H(B(F)) is contained in F. We also find conditions that guarantee that a closed set in MA is a natural set.

Mathematical Subject Classification
Primary: 46.55
Received: 30 June 1967
Revised: 18 December 1967
Published: 1 December 1968
Jan-Erik Björk