On the reflexivity of multivariable isometries

Let A ⊂ C ( K ) A \subset C(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K K in C n \mathbb {C}^n . We define the notion of an A A –isometry and show that, under a suitable regularity condition needed to apply Aleksandrov’s work on the inner function problem, every A A –isometry T ∈ L ( H ) n T \in L(\mathcal H)^n is reflexive. This result applies to commuting isometries, spherical isometries, and more generally, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain.