 
Summary: NOTES ON THE UNIQUE EXTENSION PROPERTY
WILLIAM ARVESON
Abstract. In a recent paper, Dritschel and McCullough established the ex
istence of completely positive maps of operator algebras that have a unique
extension property. In this expository note we give a more explicit rendering
of that result geared to operator systems, and discuss consequences.
1. Maximality
An operator system is a selfadjoint linear subspace S of a unital C
algebra
that contains the unit; we usually require that the C
algebra be generated by S,
and express that by writing S C
(S). We consider unital completely positive
(UCP) maps : S B(H), that is, completely positive maps that carry the unit
of S to the identity operator of B(H). Such maps satisfy (x
) = (x)
, x S.
A linear map : S B(H) that preserves the unit is completely positive iff it is
completely contractive. If S is a linear subspace of C
(S) containing 1, then every
