If two operator algebras A and B are strongly Morita equivalent (in the sense of [5]), then their C*- envelopes C*(A) and C*(B) are strongly Morita equivalent (in the usual C*-algebraic sense due to Rieffel). Moreover, if Y is an equivalence bimodule for a (strong) Morita equivalence of A and B, then the operation, Y[otimes ]hA−, of tensoring with Y, gives a bijection between the boundary representations of C*(A) for A and the boundary representations of C*(B) for B. Thus the ‘noncommutative Choquet boundaries’ of Morita equivalent A and B are the same. Other important objects associated with an operator algebra are also shown to be preserved by Morita equivalence, such as boundary ideals, the Shilov boundary ideal, Arveson's property of admissability, and the lattice of C*-algebras generated by an operator algebra.