  We consider algebras in a modular tensor category C. If the trace pairing of
an algebra A in C is non-degenerate we associate to A a commutative algebra
Z(A), called the full centre, in a doubled version of the category C. We prove
that two simple algebras with non-degenerate trace pairing are
Morita-equivalent if and only if their full centres are isomorphic as algebras.
This result has an interesting interpretation in two-dimensional rational
conformal field theory; it implies that there cannot be several incompatible
sets of boundary conditions for a given bulk theory.
