Let G be a locally compact
group, WAP(G) be the space of continuous weakly almost periodic functions on G
and C0(G) the space of continuous functions on G vanishing at infinity. We prove in
this paper, among other things, that if G is infinite and X is any subspace of
WAP(G) (or CB(G), the space of bounded continuous functions in case G is
nondiscrete) containing C0(G), then X is uncomplemented in L∞(G). If G is
non-compact, then WAP(G) is uncomplemented in LUC(G). Furthermore, AP(G),
the space of continuous almost periodic functions on G, is complemented in
LUC(G) if and only if G∕N is compact, where N is the intersection of the
kernels of all finite-dimensional continuous unitary representations of G.
We also prove that if A is any left translation invariant C∗-subalgebra of
C0(G), then A is the range of a continuous projection commuting with left
translations.
