On the reflexivity of multivariable isometries

Let $A \subset C(K)$ be a unital closed subalgebra of the algebra of all continuous functions on a compact set $K$ in $\mathbb{C}^n$. We define the notion of an $A$-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov's work on the inner function problem, every $A$-isometry $T \in L(\mathcal H)^n$ 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.