Characterization of generalized Haar spaces

We say that a subset G of C-0(T, R-k) is rotation-invariant if (Qg: g is an element of G) = G for any k x k orthogonal matrix e. Let G be a rotation-invariant finite-dimensional subspace of C-0(T, R-k) on a connected, locally compact, metric space T. We prove that G is a generalized Haar subspace if and only if P-G(f) is strongly unique of order 2 whenever P-G(f) is a singleton. (C) 1998 Academic Press.