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 1Let 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 2Let 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 3Let 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 4Let 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

(1)

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 5Let be a *-probability space and let be as in lemma 3. Then, there exists a unique satisfying

(2)

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 6Let be a bounded *-probability space with GNS representation . Then, .

The existence of the GNS representation follows from what we have shown so far.

Lemma 7Let be a bounded *-probability space, and assume the notation of lemmas 4 and 5. Then, satisfies the requirements of the GNS representation of theorem 1, and .

*Proof:* By definition, is a Hilbert space and is a *-homomorphism. Extend to an -isometry . By lemma 4, extends to a *-homomorphism satisfying (1). Then,

(3) |

Hence

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 8Let 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 9Let be a homomorphism of bounded *-probability spaces and , which have GNS representations and respectively. Then, there exists a unique isometric linear map satisfying

(4)

for all .

*Proof:* Let which, by definition, is a dense subspace of . Combining equations (4),

(5) |

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 is well-defined and an isometry. Hence, by continuous linear extension it uniquely extends to an isometry . Next, we show that (4) is satisfied. 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. ⬜

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