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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.

For abelian C*-algebras we were able to completely determine the structure of the algebra in terms of the character space, that is, in terms of the one-dimensional representations. For the non-abelian case this is quite inadequate, and we have to look at representations of arbitrary dimension. There is a deep inter-relationship between the representations and the positive linear functionals of  a C*-algebra. Representations will be defined and some aspects of this inter-relationship investigated in the next section. In this section we establish the basis properties of positive linear functionals.

I’m getting the impression that the bra-ket notation is more useful for linear ON THE LEFT!

An approximate unit for a C*-algebra is an increasing net $\{u_\lambda\}_{\lambda\in\Lambda}$ of positive elements in the closed unit ball of $A$ such that $a= \lim_{\lambda }au_\lambda=\lim_\lambda u_\lambda a$ for all $a\in A$.

## Example

Let $H$ be a Hilbert space with infinite orthonormal basis $\{e_n\}$. The C*-algebra $K(H)$ is now non-unital. If $P_n$ is the projection onto $\langle e_1,\dots,e_n\rangle$, then the increasing sequence $\{P_n\}\subset K(H)$ is an approximate unit for $K(H)$. It will suffice to show that $T=\lim_np_nT$ if $T\in F(H)$, since $F(H)$ is dense in $K(H)$. Now if  $T\in F(H)$, there exist $x_1,\dots,x_m$$y_1,\dots,y_m\in H$ such that:

$T=\sum_{k=1}^m|x_k\rangle\langle y_k|$.

Hence,

$P_nT=\sum_{k=1}^m|P_nx_k\rangle\langle y_k|$.

Since $\lim_n P_nx=x$ for all $x\in H$, therefore for each $k$:

$\lim_{n\rightarrow \infty}\||P_nx_k\rangle\langle y_k-|x_k\rangle\langle y_k|\|=\lim_{n\rightarrow \infty} \|P_nx_k-x_k\|\|y_k\|=0$.

Hence, $\lim_{n}P_nT=T$.

# Question 1

Let $A$ be a Banach algebra such that for all $a\in A$ the implication

$Aa=0$ or $aA=0$ $\Rightarrow a=0$

holds. Let $L$$R$ be linear mappings from $A$ to itself such that for all $a,b\in A$,

$L(ab)=L(a)b$$R(ab)=aR(b)$, and $R(a)b=aL(b)$.

Show that $L$ and $R$ are necessarily continuous.

# Question 2

Let $A$ be a unital C*-algebra.

## (a)

If $a,b$ are positive elements of $A$, show that $\sigma(ab)\subset \mathbb{R}^+$.

### Solution (Wills)

For elements $a,b$ of a unital algebra $A$:

$\sigma(ab)\cup\{0\}=\sigma(ba)\cup\{0\}$

If $a\in A^+$ then $a^{1/2}\in A^+$ so that

$\sigma(ab)\cup\{0\}=\sigma(a^{1/2}(a^{1/2}b))\cup\{0\}=\sigma(a^{1/2}ba^{1/2})\cup\{0\}$

Now if $b\in A^+$, for any $c\in A$$c^*bc\in A^+$. Hence $\sigma(a^{1/2}ba^{1/2})\subset \mathbb{R^+}$ and the result follows (note that $ab$ need not be hermitian) $\bullet$

Let $X$ be a compact Hausdorff space and $H$ a Hilbert Space. A spectral measure $E$ relative to $(X,H)$ is a map from the $\sigma$-algebra of all Borel sets of $X$ to the set of projections in $B(H)$ such that

1. $E(\emptyset)=0$$E(X)=1$;
2. $E(S_1\cap S_2)=E(S_1)E(S_2)$ for all Borel sets $S_1,\,S_2$ of $X$;
3. for all $x,y\in H$, the function $E_{x,y}:S\mapsto \langle E(S)x,y\rangle$, is a regular Borel complex measure on $X$.

A Borel measure $\mu$ is a measure defined on Borel sets. If every Borel set in $X$ is both outer and inner regular, then $\mu$ is called regular. A measurable $A\subset X$ is inner and outer regular if

$\mu(A)=\sup\left\{\mu(F):\text{ closed }F\subset A\right\}$, and

$\mu(A)=\inf\left\{\mu(G):A\subset G\text{ open }\right\}$

Denote by $M(X)$ the Banach space of all regular Borel complex measures on $X$, and by $B_\infty(X)$ the C*-algebra of all bounded Borel-measurable complex-valued functions on $X$ (I assume with respect to the Borel $\sigma$-algebra on $\mathbb{C}$).

This starts on P.58 rather than p.53. I underline my own extra explanations, calculations, etc. The notation varies on occasion from Murphy to notation which I prefer.

$H$ is always a Hilbert space. $H^1$ is the set of unit vectors.

If $P$ is a finite-rank projection on $H$, then the C*-algebra $A=PB(H)P$ is finite dimensional. To see this, write $P=\sum_{j=1}^ne_j\otimes e_j$, where $e_1,\dots,e_n\in H$. If $T\in B(H)$, then

$PTP=\sum_{j,k=1}^n(e_j\otimes e_j)T(e_k\otimes e_k)=\sum_{j,k=1}^n\langle Te_k,e_j\rangle (e_j\otimes e_k)$

Hence, $A\subset $, ($j,k=1,\dots,n$) (*), and therefore is finite dimensional.