Vol. 62, No. 1, 1976

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Some mappings which do not admit an averaging operator

John W. Baker and R. C. Lacher

Vol. 62 (1976), No. 1, 43–48

The problem of determining for spaces X and Y necessary and sufficient conditions such that there exists a map ϕ of X onto Y which does not admit an averaging operator is considered. This corresponds to identifying the uncomplemented closed selfadjoint subalgebras of C(X) which contain 1X. Mappings ϕ of X onto Y are constructed which do not admit averaging operators, for example, when X is any uncountable compact metric space and Y is any countable product of intervals. Also, X can be any space containing an open set homeomorphic to a Banach space and Y = X. These resuts generalize earlier work by D. Amir and S. Ditor.

Mathematical Subject Classification 2000
Primary: 46E25
Secondary: 47B99
Received: 20 December 1973
Published: 1 January 1976
John W. Baker
R. C. Lacher