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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.

Nonanalytic functional calculi and spectral maximal spaces