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
The GNS representation is constructed by taking a Hilbert space completion of under the semi-inner product. Rather than proving theorem 1 in one go, I will first show a few preliminary lemmas from which the full result will follow. Any triple satisfying the conclusion of theorem 1 will be called a (or, the) GNS representation of . First, assuming that the GNS representation exists, then it is called faithful if all satisfy only when . This occurs precisely when the the state is nondegenerate and, in this case, identifies with a *-subalgebra of .
Lemma 2 Let be a bounded *-probability space with GNS representation . Then,
- the map is an -isometry from to .
- is an -isometry, so that .
- has kernel .
- the representation is faithful if and only if is nondegenerate.
Proof: That is an -isometry follows from,
Next, as is cyclic for , the inequality
gives . Similarly,
gives , so is an -isometry. The second statement is immediate, as iff . The third statement is also immediate, as is faithful iff its kernel is and is nondegenerate iff whenever . ⬜
Now, consider a nonnegative linear map . As this does not have to be a state, and elements of might not be -bounded, the full GNS representation as described by theorem 1 need not exist. However, it is still possible to define the Hilbert space by taking the -completion of . Note that, if the GNS representation does exist, then satisfies the properties of the isometry defined by the following result.
Lemma 3 Let be a *-algebra and be a positive linear map. Then, there exists a Hilbert space and a linear isometry with dense image.
Proof: As previously explained, we make into a semi-inner product space by . Then, we take to be its completion. ⬜
If we introduce the condition that every is -bounded, then the *-homomorphism can be constructed.
Lemma 4 Let be a *-algebra and be a positive linear map such that every is -bounded. If is as in lemma 3 then there is a unique *-homomorphism satisfying
for all . Furthermore, .
Proof: As is bounded, is a bounded linear map on with operator norm . By continuous linear extension, there is a unique satisfying (1), and has operator norm . That is a *-homomorphism is immediate from the definitions. For example,
so that ⬜
Alternatively, if it is assumed that is a state or, equivalently, is a *-probability space, then the distinguished element can be constructed. To simplify matters, to handle the case where is not unitial, we use the fact that uniquely extends to a state on the unitial algebra by taking for . In fact, by lemma 10 of the post on states, is -dense in .
Lemma 5 Let be a *-probability space and let be as in lemma 3. Then, there exists a unique satisfying
for all . Furthermore, and, if is unitial, . More generally, uniquely extends to an -isometry , in which case .
Proof: Uniqueness of is immediate from (2) and the requirement that is dense in . When is unitial, then taking gives
In the non-unitial case, by existence and uniqueness of bounded linear extensions, uniquely extends to an isometry . Then, as above, and, hence,
In particular, if we have a GNS representation , then satisfies the requirements of lemma 5, and we see that is necessarily a unit vector.
Corollary 6 Let be a bounded *-probability space with GNS representation . Then, .
The existence of the GNS representation follows from what we have shown so far.
as required. Finally, (3) shows that , which is dense in . ⬜
To easily handle non-unitial algebras, we note that GNS representations of automatically extend to GNS representations of the unitial algebra .
Lemma 8 Let be a bounded *-probability space with GNS representation . Then, uniquely extends to a *-homomorphism , in which case is a GNS representation for .
Proof: Any extension satisfies
so, as is cyclic for , . Hence, is the unique extension of to a *-homomorphism from . As by corollary 6,
as required. ⬜
Next, the GNS representation is functorial. A homomorphism between *-probability spaces is a state preserving *-homomorphism of their *-algebras, and these canonically induce isometries of their GNS Hilbert spaces.
Lemma 9 Let be a homomorphism of bounded *-probability spaces and , which have GNS representations and respectively. Then, there exists a unique isometric linear map satisfying
for all .
Proof: Let which, by definition, is a dense subspace of . Combining equations (4),
which uniquely determines on and, by continuity, this uniquely determines . Conversely, we can use (5) to construct on . We need to show that this is an isometry and, to be well-defined, that the right-hand-side of (5) is zero whenever . Using
we see that the first identity of (4) holds on and, by continuity, holds on all of . Finally, by extending the constructions above to and , we can wlog assume that and are unitial. Then,
as required. ⬜
Finally, we put together the previous steps to complete the proof of theorem 1.
Proof of Theorem 1: The existence of the GNS representation was proven in lemma 7, so only uniqueness remains. Suppose that and are two GNS representations. Applying lemma 9 to the identity map on gives a unique isometry satisfying and . Similarly, there is a unique isometry satisfying and . Then, satisfies and, hence, is the identity map. Similarly, is the identity, so is invertible. ⬜