1.1 Basic Definitions and Results
In this section we collect basic definitions and results about \(C^{*}\)–algebras and \(W^{*}\)–algebras.
An associative algebra \(\mathcal{A}\) over the complex numbers is called a normed algebra if there is a map assigning for every \(x\in \mathcal{A}\) a real number \(\| x\| \), and this map satisfies:
\(\| x\| \ge 0\) and \(\| x\| =0 \iff x=0\);
\(\| x+y\| \le \| x\| +\| y\| \);
\(\| \lambda x\| = |\lambda |\| x\| \);
\(\| xy\| \le \| x\| \| y\| \).
If \(\mathcal{A}\) is also complete with respect to the norm, it is called a Banach algebra. A mapping \(*: \mathcal{A}\to \mathcal{A}\) is called an involution if it satisfies:
\((x^{*})^{*}=x\);
\((x+y)^{*}=x^{*}+y^{*}\);
\((xy)^{*}=y^{*}x^{*}\);
\((\lambda x)^{*}=\overline{\lambda }x^{*}\).
where \(\lambda \) above is a complex number.
A Banach algebra \(\mathcal{A}\) with an involution is a \(\mathbf{C^{*}}\)–algebra if the relation \(\| x^{*}x\| =\| x\| ^{2}\) holds for every \(x\in \mathcal{A}\). A \(C^{*}\)–algebra is unital if it has a multiplicative identity.
(Sak 1.1.6) The involution on a \(C^{*}\)–algebra preserves the norm.
Note \(\| x\| ^{2}=\| x^{*}x\| \le \| x^{*}\| \| x\| \), and \(\| x^{*}\| ^{2}\le \| xx^{*}\| \le \| x\| \| x^{*}\| \).
(Sak 1.2.1) Every commutative, unital \(C^{*}\)–algebra \(\mathcal{A}\) is \(*\)–isomorphic to \(C(K)\), the algebra of complex-valued continuous functions on \(K\), where \(K\) is the compact Hausdorff space of maximal ideals of \(\mathcal{A}\). (Note \(C(K)\) is itself a \(C^{*}\)–algebra with pointwise complex conjugation as \(*\) and the uniform norm.)
By Banach algebra theory, the Gelfand transform provides a homomorphism of \(\mathcal{A}\) into \(C(K)\). Standard calculation using \(C^{*}\)–identity gets that it is isometric. Mapping a self-adjoint element \(h\) to the unit circle (complex exponentiation) one can show that the spectrum lands on the unit circle and so the spectrum of \(h\) is in the reals. The spectrum of an element in \(C(K)\) is its range, from which \(*\)– preservation follows. (I should have just written this proof out...will do later...)
(Sak 1.2.2) Let \(\mathcal{A}\) be a commutative \(C^{*}\)–algebra without identity. Then \(\mathcal{A}\) is \(*\)–isomorphic to the algebra \(C_{0}(\Omega )\) of continuous functions vanishing at infinity on a locally compact Hausdorff space \(\Omega \). The space \(\Omega \) is called the spectrum space of \(\mathcal{A}\).
Let \(\mathcal{A}\) be a commutative \(C^{*}\)–algebra generated by a single self-adjoint element \(a\) and let \(\Omega \) be the spectrum space of \(\mathcal{A}\). Then \(\Omega \cup \{ \infty \} \) is homeomorphic to \(\sigma (a)\cup \{ 0\} \) and \(\mathcal{A}=C_{0}(\sigma (a)\cup \{ 0\} )\), the algebra of complex-valued continuous functions on \(a\)’s spectrum that vanish at \(0\).
The function \(\xi \) sending a point \(t\in \Omega \) to \(a(t)\in \sigma (a)\) and \(\infty \) to \(0\), where “\(a(\cdot )\)” is the image of \(a\) under the Gelfand transform. Since \(\mathcal{A}\) is generated by \(a\), the map is bijective. The map is continuous, since \(a\) is a continuous function. We have a continuous bijection from a compact space (one-point compactification of \(\Omega \)) into a Hausdorff space \(\sigma (a)\cup \{ 0\} \), and by Lemma 43 this is a homeomorphism. (Need to clarify a few details here...)
In a unital \(C^{*}\)–algebra \(\mathcal{A}\), the identity \(1\) is an extreme point of the closed unit sphere \(S\).
Suppose \(a,b\in S\) and \(1=\frac{a+b}{2}\). Then, since \(1=1^{*}\), we have \(1=\frac{a^{*}+b^{*}}{2}\) and one checks that \(1=\frac{c+d}{2}\), where \(c=\frac{a+a^{*}}{2}\) and \(d=\frac{b+b^{*}}{2}\). Now, \(d=2-c\) and so \(d\) and \(c\) commute and each are self-adjoint (hence normal) and therefore \(C^{*}\{ 1,d,c\} \) is commutative and can be identified with \(C(K)\) for a compact Hausdorff space \(K\). The self-adjointness of \(c\) and \(d\), the triangle inequality and Lemma 2 imply that the range of the associated functions \(f,g\) are in \([-1,1]\). In order for \(f(t)+g(t)=2\) for all \(t\in K\) it must be that \(f(t)\) and \(g(t)\) are always \(1\), and thus \(c=d=1\). Now \(2=2c=a+a^{*}\) and so \(a=2-a^{*}\) and so \(a\) is normal and so the unital \(C^{*}\)–subalgebra generated by \(a\) is commutative. Again represent the unital \(C^{*}\)–algebra as \(C(K)\) for some compact Hausdorff \(K\). Let \(f\) be the function corresponding to \(a\). We have \(\Re (f)(t)=1\) for all \(t\), and hence \(\Re (f)(t)^2=|f(t)|=1\) for all \(t\) and \(\Im (f)(t)=0\) for all \(t\) and therefore \(a=1\).
Let \(x\) be an extreme point of the norm unit ball of a \(C^{*}\)–algebra \(\mathcal{A}\), so that the \(C^{*}\)–subalgebra generated by \(x^{*}x\) is identified with \(C_{0}(\Omega )\) as in Lemma 4. There exists a sequence \((y_n)\) of positive elements in \(C_{0}(\Omega )\) with \(\| y_{n}\| \le 1\) for all \(n\), such that \(\| (x^{*}x)y_n-x^{*}x\| \to 0\) and \(\| (x^{*}x)y_n^{2}-x^{*}x\| \to 0\) as \(n\to \infty \).
If \(x\) is an extreme point of the norm unit ball of a \(C^{*}\)–algebra \(\mathcal{A}\), then \(x^{*}x\) is a projection. I.e. \(x\) is a partial isometry.
(Sakai 1.6.1) A \(C^{*}\)–algebra is unital if and only if its closed unit ball has an extreme point.
(Sakai 1.6.2) Let \(P\) be the set of positive elements in a \(C^{*}\)–algebra \(\mathcal{A}\) with unit sphere \(S\). Then the extreme points of \(P\cap S\) are the projections of \(\mathcal{A}\).
(Sakai 1.6.3) Let \(\mathcal{A}^{s}\) be the set of self-adjoint elements in a \(C^{*}\)–algebra \(\mathcal{A}\) with unit sphere \(S\). Then the extreme points of \(\mathcal{A}^{s}\cap S\) is the set of all self-adjoint unitary elements of \(\mathcal{A}\).
A \(C^{*}\)–algebra \(M\) is called a \(\mathbf{W^{*}}\)–algebra if it is a dual Banach space, i.e. if there exists a Banach space \(M_{*}\) so that \((M_{*})^{*}\) is isometrically isomorphic to \(M\). We call \(M_{*}\) a predual of \(M\).
Note: It is premature to fix a notation like \(M_{*}\) in the above, but the aim of this project is to prove that any two preduals of \(M\) are isomorphic as Banach spaces, so we take the liberty with Sakai to “fix a reference predual” at the outset.
(Sakai 1.1.4) A self-adjoint, \(\sigma \)–closed \(*\)–subalgebra \(N\) of a \(W^{*}\)–algebra \(M\) is also a \(W^{*}\)–algebra, since \((M_{*}/N^{0})^{*}=N\), where \(N^{0}\) is the polar (annihilator) of \(N\) in \(M_{*}\), i.e. the set of \(\varphi \in M_{*}\) such that \(\varphi (x)=0\) for all \(x\in N\).