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 $\rho(a)\geq 0$l 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 $\bullet$