1. The Banach algebra A. Let L2(S) be the set of all complexvalued measurable functions, on a measure space S, whose magnitudes are square summable. We consider L2(S) as a Hilbert space, and assume the existence of a countable complete orthonormal set {+k} in L2(S), which has the additional property that OkELO(S) nL1(S) for all k. For the given {ckk} let {vk} be any sequence of nonzero complex numbers satisfying the condition