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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.

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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.

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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_my_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.

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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 LR be linear mappings from A to itself such that for all a,b\in A,

L(ab)=L(a)bR(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 Ac^*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

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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)=0E(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}).

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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 <e_j\otimes e_k>, (j,k=1,\dots,n) (*), and therefore is finite dimensional.

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