In this section we introduce the important GNS construction and prove that every C*-algebra can be regarded as a C*-subalgebra of B(H) for some Hilbert space H. It is partly due to this concrete realisation of the C*-algebras that their theory is so accessible in comparison with more general Banach spaces.

A representation of a C*-algebra A is a pair (H,\varphi) where H is a Hilbert space and \varphi:A\rightarrow B(H) is a *-homomorphism. We say (H,\varphi) is faithful if \varphi is injective.

If \{H_\lambda,\varphi_\lambda\} is a family of representations of A, their direct sum is the representation (H,\varphi) got by setting setting H=\oplus_\lambda H_\lambda, and \varphi(a)((x_\lambda)_\lambda)=(\varphi_{\lambda}(a)(x_\lambda))_\lambda for all a\in A and (x_\lambda)_\lambda\in H. It is readily verified that (H,\varphi) is indeed a representation of A. If for each non-zero element a\in A there is an index \lambda such that \varphi_\lambda(a)\neq 0, then (H,\varphi) is faithful.

Recall now that if H is a pre-Hilbert space (not closed necessarily), then there is a unique inner product of the Banach space completion \bar{H} of H extending the inner product of H and having its associated norm the norm of \bar{H}. We call \bar{H} endowed with this inner product the Hilbert space completion of H.

With each positive linear functional, there is associated a representation. Suppose that \rho is a positive linear functional on a C*-algebra A. Setting

N_\rho=\{a\in A:\rho(a^*a)=0\},

it is easy to check (using Th.3.3.7 — closure is got by seeing that N_\rho is the kernel of the continuous map a\mapsto \rho(a^*a)=\rho\circ L_{a^*}) that N_\rho is a closed left ideal of A and that the map:

(A/N_\rho)^2\rightarrow \mathbb{C}; (a+N_\rho,b+N_\rho)\mapsto \rho(b^*a),

is a well defined inner product on A/N_\rho (all straightforward calculations). We denote by H_\rho the Hilbert space completion of A/N_\rho.

If a\in A, define an operator \varphi(a)\in B(A/N_\rho) by setting:

\varphi(a)(b+N_\rho)=ab+N_\rho.

The inequality \|\varphi(a)\|\leq \|a\| holds since we have

\|\varphi(a)(b+N_\rho)\|^2=\rho(b^*a^*ab)\leq \|a\|^2\rho(b^*b)=\|a\|^2\|b+N_\rho\|^2.

The latter inequality is by Th. 3.3.7. The operator \varphi(a) has a unique extension to a bounded operator \varphi_\rho(a) on H_\rho=\overline{A/N_\rho}. The map

\varphi_\rho:A\rightarrow B(H_\rho); a\mapsto \varphi_\rho(a),

is a *-homomorphism (an easy exercise — needed to show, where \langle,\rangle is the inner product on A/N_\rho:

\langle \varphi_\rho(a^*)(b+N_\rho),(c+N_\rho)\rangle=\langle (b+N_\rho),\varphi_\rho(a)(c+N_\rho)\rangle.

).

The representation (H_\rho,\varphi_\rho) of A is the Gelfand-Naimark-Segal representation (or GNS representation) associated to \rho.

If A is non-zero, we define its universal representation to be the direct sum of all the representations (H_\rho,\varphi_\rho), where \rho ranges over S(A).

Theorem 3.4.1 (Gelfand-Naimark)

If A is a C*-algebra, then it has a faithful representation. Specifically, its universal representation is faithful.

Proof

Let (H,\varphi) be the universal representation of A and suppose that a is an element of A such that \varphi(a)=0. By Th. 3.3.6 (if a is a normal element of A there exists a state \rho such that |\rho(a)|=\|a\|) there is a state \rho on A such that \|a^*a\|=\rho(a^*a). Hence, if b=(a^*a)^{1/4}, then \|a\|^2=\rho(a^*a)=\rho(b^4)=\|\varphi_\rho(b)(b+N_\rho)\|^2=0 (since \varphi_\rho(b^4)=\varphi_\rho(a^*a)=0, so \varphi_\rho(b)=0  The first thing to note is that b is positive. I can show that for a homomorphism \varphi in an algebra, b is nilpotent if and only if \varphi(b) is nilpotent.) Hence, a=0, and \varphi is injective \bullet

The Gelfand-Naimark theorem is one of those results that are used all of the time. For the present we give just two applications.

The first application is to matrix algebras. If A is an algebra, M_n(A) denotes the algebra of all n\times n matrices with entries in A (with the usual operations). If A is a *-algebra, so is M_n(A), where the involution is given by the conjugate transpose.

If \varphi:A\rightarrow B is a *-homomorphism between *-algebras, its inflation is the *-homomorphism

\varphi:M_n(A)\rightarrow M_n(B), a_{ij}\mapsto \varphi(a_{ij}).

If H is a Hilbert space, we write H^{(n)} for the orthogonal sum of n copies of H. If T\in M_n(B(H)), we define \varphi(T)\in B(H^{(n)}) by setting

\varphi(T)(x_1,\dots,x_n)=\left(\sum_{j=1}^nT_{1j}x_j,\dots,\sum_{j=1}^nT_{nj}x_j\right),

for all (x_1,\dots,x_n)\in H^{(n)}. It is readily verified that the map

\varphi:M_n(B(H))\rightarrow B(H^{(n)}), T\mapsto \varphi(T),

is a *-isomomorphism. We call \varphi the canonical *-isomorphism of M_n(B(H)) onto B(H^{(n)}), and useit to identify these two algebras. If S is an operator in B(H^{(n)}) such that S=\varphi(T) where T\in M_n(B(H)), we call T the operator matrix of S. We define a norm on M_n(B(H)) making it a C*-algebra by setting \|T\|=\|\varphi(T)\|. The following inequalities for T\in M_n(B(H)) are easy to verify and are often useful:

\|T_{ij}\|\leq\|T\|\leq \sum_{k,l=1}^n\|u_{kl}\|.

Theorem 3.4.2

If A is a C*-algebra, then there is a unique norm on M_n(A) making it a C*-algebra.

Proof

Let the pair (H,\varphi) be the universal representation of A, so the *-homomorphism \varphi:M_n(A)\rightarrow M_n(B(H)) is injective. We define a norm on M_n(A) making it a C*-algebra by setting \|a\|=\|\varphi(a)\| for all a\in\|\varphi(a)\| (completeness can be easily checked using the inequalities preceding this theorem). Uniqueness is given by Corollary 2.1.2 (the C*-algebra norm is unique) \bullet

Remark

If A is a C*-algebra and a\in M_n(A), then

\|a_{ij}\|\leq \|a\|\leq \sum_{k,l=1}^n\|a_{kl}\|.

These inequalities follow from the corresponding inequalities in M_n(B(H)).

Matrix algebras play a fundamental role in the K-theory of C*-algebras. The idea is to study not just the algebra A but simultaneously all of the matrix algebras M_n(A) over A also.

Whereas it seems that the only way of showing that matrix algebras over general C*-algebras are themselves normable as C*-algebras is to use the Gelfand-Naimark representation, for our second application of this representation alternative proofs exist, but the proof given here is very natural.

Theorem 3.4.3

Let a be a self-adjoint element of the C*-algebra A. Then a\in A^+ iff \rho(a)\geq 0 for all positive linear functionals \rho on A.

Proof

The forward implication is straightforward. Suppose conversely that \rho(a)\geq0 for all positive linear functionals \rho on A. Let (H,\varphi) be the universal representation of A, and let x\in H. Then the linear functional

\rho:A\rightarrow \mathbb{C}, b\mapsto\langle \varphi(b)x,x\rangle,

is positive, so \rho(a)\geq 0l that is, \langle \varphi(a)x,x\rangle\geq 0. Since this is true for all x\in H, and since \varphi(a) is self-adjoint, therefore \varphi(a) is a positive operator on H. Hence \varphi(a)\in\varphi(A)^+, so a\in A^+, because the map \varphi:A\rightarrow \varphi(A) is a *-isomorphism \bullet

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