A Morita Type Equivalence for Dual Operator Algebras
Abstract
We generalize the main theorem of Rieffel for Morita equivalence of W*-algebras to the case of unital dual operator algebras: two unital dual operator algebras A and B have completely isometric normal representations alpha, beta such that alpha(A) is the w*-closed span of M*beta(B)M and beta(B) is the w*-closed span of Malpha(A)M* for a ternary ring of operators M (i.e. a linear space M such that MM*M \subset M if and only if there exists an equivalence functor
F
:
A
M
→
B
M
which "extends" to a *-functor implementing an equivalence between the categories
A
DM
and
B
DM.
By
A
M
we denote the category of normal representations of A and by
A
DM
the category with the same objects as
A
M
and
Δ(A)
-module maps as morphisms (
Δ(A)=A∩
A
∗
). We prove that this functor is equivalent to a functor "generated" by a B, A bimodule, that it is normal and completely isometric.