Vol. 50, No. 1, 1974

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The algebra of bounded continuous functions into a nonarchimedean field

Richard Staum

Vol. 50 (1974), No. 1, 169–185

Let S be a topological space, F a complete nonarchimedean rank 1 valued field, and C(S,F) the Banach algebra of bounded, continuous, F-valued functions on [S. Various topological conditions on S and/or F are shown to be equivalent, respectively, to each of the following: every maximal ideal of C(S,F) is fixed; the only quotient field of C(S,F) is F itself; every homomorphism of C(S,F) into F is an evaluation at a point of S; the Stone-Weierstrass theorem holds for C(S,F). It is also shown that a certain topological space derived from S may be embedded in the space of maximal ideals of C(S,F) with Gelfand topology, or in the space of homomorphisms of C(|S,F) into F.

Mathematical Subject Classification 2000
Primary: 46E25
Secondary: 54C40, 54D20
Received: 4 August 1972
Published: 1 January 1974
Richard Staum