LOGARITHMIC RESIDUES, GENERALIZED IDEMPOTENTS, AND SUMS OF IDEMPOTENTS IN BANACH-ALGEBRAS

In a commutative Banach algebra B the set of logarithmic residues (i.e ., the elements that can be written as a contour integral of the logar ithmic derivative of an analytic B-valued function), the set of genera lized idempotents !i.e., the elements that are annihilated by a polyno mial with non-negative integer simple zeros), and the set of sums of i dempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition a nd multiplication. Counterexamples show that the commutativity conditi on on B is essential. The results extend to logarithmic residues of me romorphic B-valued functions.