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Note on Operator Algebras Note on Operator Algebras Takahiro Sagawa Department of Physics, The University of Tokyo 15 December 2010 Contents 1 General Topology 2 2 Hilbert Spaces and Operator Algebras 5 2.1 Hilbert Space . 5 2.2 Bounded Operators . 6 2.3 Trace Class Operators . 8 2.4 von Neumann Algebras . 10 2.5 Maps on von Neumann Algebras . 12 3 Abstract Operator Algebras 13 3.1 C∗-Algebras . 13 3.2 W ∗-algebras . 14 1 Chapter 1 General Topology Topology is an abstract structure that can be built on the set theory. We start with introducing the topological structure by open stets, which is the most standard way. A topological space is a set Ω together with O, a collection of subsets of Ω, satisfying the following properties: ∙ 휙 2 O and Ω 2 O. ∙ If O1 2 O and O2 2 O, then O1 \ O2 2 O. ∙ If O훼 2 O (훼 2 I) for arbitrary set of suffixes, then [훼2I O훼 2 O. An element of O is called an open set. In general, a set may have several topologies. If two topologies satisfy O1 ⊂ O2, then O1 is called weaker than O2, or smaller than O2. Topological structure can be generated by a subset of open spaces. Let B be a collection of subsets of a set Ω. The weakest topology O such that B ⊂ O is called generated by B. We note that such O does not always exist for an arbitrary B. Figure 1.1: An open set and a compact set. We review some important concepts in topological spaces: 2 ∙ If O is an open set, then Ω n O is called a closed set. ∙ The closure of S ⊂ Ω is the smallest closed set containing S. ∙ S ⊂ Ω is called compact if an arbitrary open cover of S has a finite subcover. Ex- plicitly, for every arbitrary collection of open sets fU훼g훼2I ⊂ O with I being the 0 set of suffixes such that S ⊂ [훼2I U훼, there exists a finite subset I of I such that S ⊂ [훼2I0 U훼. In particular, if Ω is compact, then the topological space is called compact. Example: In the usual topology in Rn, a set S ⊂ Rn is compact if and only if it is closed and bounded. The followings are important properties that will be used in the proofs of the main argument. ◜ ◝ ∙ A closed subset of a compact space is compact. ∙ If fSpgp2N be a family of compact subsets of a topological space such that Sp+1 ⊂ Sp, then \pSp 6= 휙. ◟ ◞ An important example of topological spaces is a Banach space. A Banach spaces V is a complete normed vector space. Here, a normed vector space is a vector space with a norm k ⋅ k satisfying: ∙ kxk ≥ 0 for all x 2 V , where kxk = 0 if and only if x = 0. ∙ kaxk = jajkxk with a 2 C. ∙ kx + yk ≤ kxk + kyk. A normed vector space is complete if every Cauchy sequence in V converges to an element of V . Explicitly, fxngn2N ⊂ V is called a Cauchy sequence, if for every " > 0 there exists N 2 N such that kxn − xmk < " for every n; m > N. ◜ ◝ P n n 2 C is a Banach space with the standard norm kxk = i=1 jxij with x = (x1; x2; ⋅ ⋅ ⋅ ; xn). ◟ ◞ We next discuss continuous maps between two topological spaces Ω and Ω0. Map f;Ω ! Ω0 is continuous if, for every open set O0 ⊂ Ω0, f −1(O0) is an open set in Ω. We can also define the continuity of map f;Ω ! Ω0 at a single point x 2 Ω. In usual Rn, such a continuity can be defined in terms of the convergence of sequences. However, this definition is not enough in general. Instead, we need the concept of net. 3 A set is called net if it is labeled by a directed set I as fxigi2I . Here, I is a directed set if it has \≤" that satisfies the following properties: ∙ a ≤ a ∙ If a ≤ b and b ≤ c, then a ≤ c. ∙ For arbitrary a; b 2 I, there exists c 2 I such that a ≤ c and b ≤ c. We note that a sequence is a special net with I = N. A net fxigi2I ⊂ Ω converges to x 2 Ω if, for every open set O with x 2 O, there exists i 2 I such that xk 2 O for all k with i ≤ k. We write this as limi2I xi = x. The following proposition is important. 0 f;Ω ! Ω is continuous at x 2 Ω if limi2I f(xi) = f(x) holds for for every net fxigi2I that converges to x. ◜ ◝ f;Ω ! Ω0 is continuous if and only if it is continuous at all a 2 Ω. ◟ ◞ We note that the concept of net is not necessary for first-countable spaces.1 If the topological space is first-countable, every \net" above can be replaced by \sequence". Usual Rn, every Hilbert space, and every Banach space are all first-countable in their norm topologies. Finally, we discuss a way to create a topological space from another topological space. Let Ω be a topological space and Ω0 be its subset. Then we can define a topology on Ω0 as follows: O0 ⊂ Ω0 is an open set if and only if there exists an open set O ⊂ Ω such that O0 = O \ Ω0. This topology on Ω0 is called relative topology. Figure 1.2: Relative topology. 1Here we only note that every metric space is first-countable. 4 Chapter 2 Hilbert Spaces and Operator Algebras We shortly review some basic concepts of Hilbert spaces and operator algebras. 2.1 Hilbert Space A Hilbert space H is a complex vector space with an inner product (⋅; ⋅) that is complete in terms of the norm k'k2 ≡ ('; '). By definition, a Hilbert space is a Banach space. Cn with the standard inner product is a n-dimensional Hilbert space and vice versa. We then generally introduce the orthonormal bases which can be applied even to \non-countable- dimensional" Hilbert spaces. Let I be a set of suffixes. A set of vectors f'훼g훼2I ⊂ H is an orthonormal basis of H , if it satisfies that: ∙ ('i;'j) = 훿i;j. ∙ For any 2 H , there exists a countable subset I0 ⊂ I such that XN lim k − ('i ; )'i k = 0; (2.1) N!1 n n n=1 0 where fi1; i2; ⋅ ⋅ ⋅ g = I . Then we can show that: ◜ ◝ Every Hilbert space has orthonormal bases, and their cardinalities are equal. ◟ ◞ The above proposition leads to the dimension of Hilbert spaces. Let f'igi2I be an orthonormal basis of H . The cardinality of I is called the dimension of H . If I is a countable set, then H is called separable. 5 In the following, we only consider separable Hilbert spaces. A typical example of a separable infinite-dimensional Hilbert space is L2(Rn), which is defined as the set of all Lebesgue measurable functions f; Rn ! R satisfying Z jf(x)j2dx < 1: (2.2) Rn In the case of infinite-dimensional Hilbert spaces, we can define two types of convergences. One is the ordinary convergence in terms of the standard norm, and the other topology is weaker than it. (Strong) convergence: A sequence of points f'ngn2N ⊂ H is said to converge strongly (or simply \converge") to a point ' 2 H if k'n − 'k ! 0. Weak convergence: A sequence of points f'ngn2N ⊂ H is said to converge weakly to a point ' 2 H if ( ; 'n) ! ( ; ') for all 2 H . These two convergences are equivalent only in the case of a finite-dimensional Hilbert space. In general, if a sequence f'ng strongly converges, then it weakly converges, because j( ; ') − ( ; 'n)j = j( ; ' − 'n)j ≤ k kk' − 'nk: (2.3) However, the inverse of this does not always hold true for infinite-dimensional Hilbert spaces. For example, let us consider an orthonormal basis f'ngn2N of a separable Hilbert space. This sequence weakly converge to 0 2 H , because limn!1('n; ) = 0 for all 2 H . However, f'ng does not converge to 0, because k'n − 0k = 1 for all n. 2.2 Bounded Operators We will focus on operator algebras on a separable Hilbert space H . An important class of linear operators on a Hilbert space is the set of bounded operators, which corresponds to the set of observables (and its linear combinations over C). A bounded operator x is a linear operator on H satisfying kxk ≡ sup kx k < 1; (2.4) 2H ;k k=1 where k⋅k is the norm of bounded operators, which actually satisfies the properties of norm. We write as B(H ) the set of all bounded operators on H . We next introduce the concept of adjoint. 6 ◜ ◝ It can be shown that for any x 2 B(H ), there exists a unique operator x∗ 2 B(H ) such that (x ; 휙) = ( ; x∗휙) (2.5) for all ; 휙 2 H . ◟ ◞ ∙ We call x∗ 2 B(H ) the adjoint of x 2 B(H ). ∙ A bounded operator x 2 B(H ) is self-adjoint if x = x∗. The topology on B(H ) defined by norm k ⋅ k is called the uniform topology or the norm topology. The following property plays an important role in the theory of operator algebras: ◜ ◝ B(H ) is complete in terms of the norm k ⋅ k, in other words, B(H ) is a Banach space. ◟ ◞ Besides the uniform topology, B(H ) has several important topologies. They can be under- stood in terms of locally convex topologies that are defined in terms of seminorms.
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