We are concerned with the
algebraic representation of the spectral maximal spaces for certain classes of
decomposable operators on Banach spaces. The main emphasis will be on those
operators which admit a functional calculus on a suitable algebra of functions such
as, for instance, differentiable functions, Lipschitz functions, or functions with
absolutely convergent Fourier series. Using appropriate partitions of unity, we shall
give a unified approach to various previous results in this area as well as to certain
extensions and variants thereof. In the case of spectral operators, we shall
obtain the optimal results without any assumption on the underlying Banach
space or the Boolean algebra of projections corresponding to the spectral
measure.
