As is well known, the space of bounded linear operators on any Hilbert space forms a *-algebra, and (pure) states on this algebra are defined by unit vectors. Considering a Hilbert space , the space of bounded linear operators is denoted as . This forms an algebra under the usual pointwise addition and scalar multiplication operators, and involution of the algebra is given by the operator adjoint,
for any and all . A unit vector defines a state by .
The Gelfand-Naimark–Segal (GNS) representation allows us to go in the opposite direction and, starting from a state on an abstract *-algebra, realises this as a pure state on a *-subalgebra of for some Hilbert space .
Consider a *-algebra and positive linear map . Recall that this defines a semi-inner product on the *-algebra , given by . The associated seminorm is denoted by , which we refer to as the -seminorm. Also, every defines a linear operator on by left-multiplication, . We use to denote its operator norm, and refer to this as the -seminorm. An element is bounded if is finite, and we say that is bounded if every is bounded.
Theorem 1 Let be a bounded *-probability space. Then, there exists a triple where,
- is a Hilbert space.
- is a *-homomorphism.
- satisfies for all .
- is cyclic for , so that is dense in .
Furthermore, this representation is unique up to isomorphism: if is any other such triple, then there exists a unique invertible linear isometry of Hilbert spaces such that