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.
