For algebras in NP_∞, compact scattered spectra imply the algebra is isometrically C(Δ(A)) via Gelfand transform, while any compact space with a Cantor subset arises as spectrum of some A in NP_∞ that is not C(Δ(A)).
Katznelson, An introduction to harmonic analysis , third edition, Cambridge Mathematical Library, Cambridge Univ
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Nevanlinna--Pick norms: towards a scattered--Cantor dichotomy for spectra of commutative Banach algebras
For algebras in NP_∞, compact scattered spectra imply the algebra is isometrically C(Δ(A)) via Gelfand transform, while any compact space with a Cantor subset arises as spectrum of some A in NP_∞ that is not C(Δ(A)).