Dependencies

(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 \(\mathcal{A}\).

For all \(p,q \in \mathcal{A}\) projections such that \(q\le p\), \(p-q\) is a projection.

(Sak 1.7.9) Let \(H\) be a subset of \(M\) and \(\mathcal{A}\) be the \(C^{*}\)–subalgebra of \(M\) generated by \(H\), and let \(\overline{\mathcal{A}}\) be the \(\sigma (M,M_{*})\)–closure of \(\mathcal{A}\). Then \(\overline{\mathcal{A}}\) is a \(W^{*}\)–subalgebra of \(M\). \(\overline{\mathcal{A}}\) is called a \(W^{*}\)–subalgebra of \(M\) generated by \(H\). Moreover, \(\overline{\mathcal{A}}\) is commutative if \(\mathcal{A}\) is commutative.

The \(\sigma \)–topology is equivalent to the \(s\)–topology on the unit ball of \(M\), and hence on bounded sets.

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 \(0\).

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.

The locally convex topology on \(M\) generated by the seminorms \(p_{K}(x)=\sup _{\varphi \in K}|\varphi (x)|\), ranging over \(K\) the \(\sigma (M_{*},M)\)–compact subsets of \(M_{*}\), is called the Mackey topology on \(M\). We denote this topology by \(\tau (M,M_{*})\).

The weak-\(*\) topology \(\sigma (M,M_{*})\) on a \(W^{*}\)–algebra \(M\) is called the \(\sigma \)–topology.

The locally convex topology on \(M\) generated by the seminorms \(x\mapsto \varphi (x^{*}x)^{1/2}\) ranging over \(\varphi \in M_{*}\) is called the strong topology, or \(s\)–topology, on \(M\). We denote it by \(s(M,M_{*})\).

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 \(M\).

(Sak 1.8.10) A linear functional \(\rho \) on \(M\) is \(\sigma \)–continuous on the unit sphere (hence \(\sigma \)–continuous) if and only if it is \(s\)–continuous on the unit sphere (hence \(s\)–continuous).

Let \(K\) be a compact Hausdorff space, with \(U\subseteq K\) open, then the set of \(f\in C(K)\) supported on \(U\) such that \(0 \le f \le 1\) is a directed set w.r.t. the order on \(C(K)\).

If \(p\in M\) is a projection, \(pMp\) is also a \(W^{*}\)–algebra with identity \(p\).

(Sakai 1.7.6) Let \(p\) be any projection in \(M\). Then the subalgebra \(pMp\) is \(\sigma \)–closed and the mapping \(x\mapsto pxp\) is \(\sigma \)–continuous.

For every positive normal linear functional \(\varphi \) and nonzero \(p\in \mathcal{P}(M)\) there exists a positive \(\sigma (M,M_{*})\)–continuous linear functional \(\psi \) such that \(\varphi (p){\lt}\psi (p)\).

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.

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 \(n\to \infty \).

(Sak 1.3.1) Let \(K\) be a Stonean space. Then every positive self-adjoint element \(a\) in \(C(K)\) can be uniformly approximated by finite linear combinations of projections in \(C(K)\) having nonnegative coefficients.

Every continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

(Sak 1.7.7) If \(p\) is any projection of \(M\), then the mappings \(x\mapsto px\) and \(x\mapsto xp\) are \(\sigma \)–continuous.

For all positive linear functionals \(\varphi ,\psi \) with \(\varphi \) normal and \(\psi \) \(\sigma (M,M_{*})\)–continuous and every nonzero \(p\in \mathcal{P}(M)\) such that \(\varphi (p){\lt}\psi (p)\), there exists a nonzero \(p_1 \in \mathcal{P}(M)\) such that \(p_1\le p\) and for all nonzero \(q\in \mathcal{P}(M)\) with \(q\le p_1\), we have \(\varphi (q){\lt}\psi (q)\).

(Sak 1.7.2) For any self-adjoint element \(a\notin P\), there exists \(\varphi \in T\) such that \(\varphi (a){\lt}0\).

In a unital \(C^{*}\)–algebra \(\mathcal{A}\), the identity \(1\) is an extreme point of the closed unit sphere \(S\).

\(P\) is a convex cone in \(M\).

(Sak 1.4.4) Let \(h \in \mathcal{A}\). TFAE:

1. \(h \ge 0\);

2. There exists \(x\in \mathcal{A}\) such that \(h=x^{*}x\).

(Sak 1.7.1) \(P\) and \(M^{s}\) are \(\sigma \)–closed.

For all projections \(p\in \mathcal{A}\), \(p \le 1\).

Let \(\mathcal{A}\) be a \(C^{*}\)–algebra and \(p,q \in \mathcal{A}\) be projections. Then \(p-q\ge 0\) iff \(qp = pq = q\).

If \(x\in \mathcal{A}\) is selfadjoint then \(\| x\| 1-x \ge 0\).

(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 \(x\in N\).

(Sak 1.7.4) Every uniformly bounded, increasing net in \(M^{s}\) converges to its least upper bound in the \(\sigma \)–topology. Further, if \(x=\sup _{\lambda }x_{\lambda }\) then \(a^{*}xa=\sup _{\lambda }a^{*}x_{\lambda }a\).

(Sak 1.7.5) If \(C\) is any maximal commutative \(C^*\)–subalgebra of the \(W^{*}\)–algebra \(M\), its spectrum space (maximal ideal space) is Stonean.

If \(h,a \in \mathcal{A}\) with \(h\ge 0\) then \(a^{*}ha\ge 0\)

(Sak 1.7.8) The mappings \(x\mapsto x^{*},ax\), and \(xa\) are \(\sigma \)–continuous for \(x,a \in M\).

(Sak 1.1.6) The involution on a \(C^{*}\)–algebra preserves the norm.

If \(a\in M\), and \(\psi (a)=0\) for every \(\psi \in T\), then \(a=0\).

A maximal abelian \(*\)–subalgebra \(\mathcal{A}\) of a \(W^{*}\)–algebra \(M\) is also a \(W^{*}\)–algebra.

Every \(W^{*}\)–algebra is unital.

Let \(\varphi \) be a normal positive linear functional on \(M\) and consider the predicate \(P:\mathcal{P}(M)\to \text{Prop}\) defined, for \(p\in \mathcal{P}(M)\), by “\(M\ni x \mapsto \varphi (xp)\) is \(\sigma (M,M_{*})\)–continuous”. If \((p_{\alpha })\) is a chain of projections in \(M\) such that \(P(p_{\alpha })\) is true for each \(\alpha \), then \(P(\sup (p_{\alpha }))\) is true. Hence by Zorn’s Lemma there is a maximal \(p_0\in \mathcal{P}(M)\) such that \(P(p_0)\) is true.

(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 \(\mathcal{A}\).

The set of \(\sigma (M,M_{*})\)–continuous linear functionals is precisely \(M_{*}\).

(Sakai 1.10.1) Let \(\mathcal{L}\) (resp. \(\mathcal{R}\)) be a left (resp. right) \(\sigma (M,M_{*})\)–closed ideal of a \(W^{*}\)–algebra \(M\). Then there exists a unique projection \(p\) (resp. \(q\)) in \(M\) so that \(\mathcal{L}=Mp\) (resp. \(\mathcal{R}=qM\)).

(Sakai 1.10.2) Let \(M\) be a \(W^{*}\)–algebra. Its set \(M^{p}\) of projections is a complete lattice with respect to \(\le \).

(Sakai 1.3.2) Let \(K\) be a compact Hausdorff space. Suppose every bounded increasing net of real valued, non-negative functions in \(C(K)\) has a least upper bound in \(C(K)\). Then \(K\) is Stonean.

(Banach-Alaoglu) The closed unit ball of the dual space of a normed vector space is compact in the weak-\(*\) topology.

(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.)

(Sakai 1.6.1) A \(C^{*}\)–algebra is unital if and only if its closed unit ball has an extreme point.

(Sak 1.13.2) Every positive normal linear functional \(\varphi \) on \(M\) is \(\sigma (M,M_{*})\)–continuous.

We have \(\sigma (M,M_{*})\le s(M,M_{*})\le \tau (M,M_{*})\).