On identifying the maximal ideals in Banach algebras

AbstractWe consider the problem of identifying the maximal ideals of a Banach algebra S that is contained in a larger Banach algebra B. If S is dense in B in an appropriate sense and if the spectral radii in S and B are the same, then S and B have the same maximal ideals. The result is illustrated by two examples of Banach algebras S in which the density and spectral radius conditions are easily shown to be valid with respect to a larger algebra B whose maximal ideals are known. The first example is a convolution algebra on a group where the functions in the algebra have specified rate of decay. The second example is a generalized version of an algebra introduced by I. Hirschman. Two applications of the generalized Hirschman algebra are presented: these relate to filtering and prediction of stationary random sequences and concern the asymptotic behavior of the errors incurred by the finite memory predictor and by the finite lag filter