Maximal Ideals and the Numerical Range in Banach Algebras: A Survey

We survey the relationship between the maximal ideal space of a commutative Banach algebra and the numerical range of its elements and of operators on Banach spaces. The Gelfand transform identifies the spectrum of an element with the range of its transform on the character space, and since every character is a state, the spectrum lies inside the algebra numerical range. Starting from this inclusion we describe how the Šilov and Choquet boundaries, Hermitian elements and the Vidav–Palmer theorem, and the ideal and M-ideal structure of an algebra interact with numerical ranges. We then review results obtained since 1990 on the Bochner–Schoenberg–Eberlein property, numerical radius inequalities, closedness of the numerical range of compact operators, non-commutative boundaries of operator systems, and the Crouzeix–Palencia theorem that the numerical range is a (1+√2)-spectral set. Every result is stated with its hypotheses and with a source, several standard claims are illustrated by explicit examples and counterexamples, and the paper closes with open problems that arise from this literature.