The Gelfand-Naimark Theorem

Attention

All the theorems and properties in this page hold for non-unital C*-algebras as well, but we will only consider the unital case for simplicity.

Positive Linear Functional

Positive Linear Functional

Let be a (unital) C*-algebra. A linear map is said to be positive if for any positive element .

e.g.

1. For , the trace map is a positive linear functional.
2. For , the map is positive (but it is not a character).
3. If is commutative and is a character, then is positive. For any , .

Proposition

Let be a positive linear map, then it is order-preserving. That is, if with , then .

Properties of Positive Maps

Let be a positive linear map. Then:

1. for all . In particular, if is self-adjoint.
2. (Cauchy-Schwarz Inequality) for all .
3. is a bounded linear functional, and .
   

Proof

1. For a self-adjoint element , we have . Since is positive, . This implies is real. For any , we can write . Then .
2. The proof for the Cauchy-Schwarz inequality is omitted here but is a standard result from the theory.
3. A special case of the Cauchy-Schwarz inequality is . Since , we have . Combining these gives , so . This shows is bounded and . Since , we conclude .

Lemma

Let be a linear functional such that . Then is positive if and only if .

Proof The forward direction follows from the previous lemma. For the converse, assume . Let be a positive element with . We need to show . Since is positive and , the spectrum . This implies . By our assumption, . Since , we have . This implies lies in the closed disk of radius 1 centered at 1 in the complex plane. We also know that since is self-adjoint, must be real. Thus, . For any positive element , we can consider (if ), which gives , so .

States and the GNS Construction

State

A state on a unital C*-algebra is a positive linear functional with norm 1. The set of all states on is denoted by . By the previous lemmas, this is equivalent to .

Lemma

If is a self-adjoint element, then there exists a state such that .

Proof Let be the abelian C*-algebra generated by and . By the Gelfand representation for abelian C*-algebras, there exists a character such that . Since is a character on a unital algebra, and . By the Hahn-Banach theorem, we can extend to a linear functional on all of such that and . Since and , is a state.

The Gelfand-Naimark-Segal (GNS) Construction

For any state , there exists a Hilbert space , a -homomorphism (i.e., -representation) such that:

Proof Let us first define a positive semi-definite sesquilinear form by . Let . By the Cauchy-Schwarz inequality, is a closed left ideal of . Consider the vector space . Define an inner product on by , where denotes the equivalence class of in . This is well-defined and gives the structure of an inner product space. Let be the completion of with respect to this inner product, so that is a Hilbert space. For each , define a linear map by . This is well-defined because is a left ideal. This map is also bounded: Since , we have . By positivity of , . Thus . Since is a bounded linear operator on a dense subspace of , it extends uniquely to a bounded operator with . The map is a -homomorphism.

The Gelfand-Naimark Theorem

Direct Sum of Hilbert Spaces

Let be a collection of Hilbert spaces. The direct sum is the Hilbert space of elements where and . If for each , we have an operator such that , we can define the direct sum of operators by . Then .

Gelfand-Naimark Theorem

Any unital C*-algebra is isometrically -isomorphic to a subalgebra of for some Hilbert space .

Proof For each state , perform the GNS construction to get a representation . Define the Hilbert space , and the universal representation by . This is a well-defined -homomorphism because . We now show that is an isometry. For any , since , from the GNS construction, for any state , for any . On the other hand, by the lemma above, for the self-adjoint element , there exists a state such that . So, , and thus , is an isometry. An isometric -homomorphism is injective, so is isometrically -isomorphic to its image .