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.