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.