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Von Neumann Algebras 1 VON NEUMANN ALGEBRAS ADRIAN IOANA These are lecture notes from a topics graduate class taught at UCSD in Winter 2019. 1. Review of functional analysis In this section we state the results that we will need from functional analysis. All of these are stated and proved in [Fo99, Chapters 4-7]. Convention. All vector spaces considered below are over C. 1.1. Normed vector spaces. Definition 1.1. A normed vector space is a vector space X over C together with a map k · k : X ! [0; 1) which is a norm, i.e., it satisfies that • kx + yk ≤ kxk + kyk, for al x; y 2 X, • kαxk = jαj kxk, for all x 2 X and α 2 C, and • kxk = 0 , x = 0, for all x 2 X. Definition 1.2. Let X be a normed vector space. (1) A map ' : X ! C is called a linear functional if it satisfies '(αx + βy) = α'(x) + β'(y), for all α; β 2 C and x; y 2 X. A linear functional ' : X ! C is called bounded if ∗ k'k := supkxk≤1 j'(x)j < 1. The dual of X, denoted X , is the normed vector space of all bounded linear functionals ' : X ! C. (2) A map T : X ! X is called linear if it satisfies T (αx+βy) = αT (x)+βT (y), for all α; β 2 C and x; y 2 X. A linear map T : X ! X is called bounded if kT k := supkxk≤1 kT (x)k < 1. A linear bounded map T is usually called a linear bounded operator, or simply a bounded operator. We denote by B(X) the normed vector space of all bounded operators T : X ! X. Definition 1.3. A Banach space is a normed vector space X which is complete in the norm metric: any Cauchy sequence fxng (i.e., such that limm;n!1 kxm − xnk = 0) is convergent (i.e., there exists x 2 X such that limn!1 kxn − xk = 0). Examples 1.4. (of Banach spaces): n p 2 2 • C , for n ≥ 1, with the Euclidean norm k(x1; :::; xn)k = x1 + ::: + xn, • Lp(Y ), for any measure space (Y; µ) and 1 ≤ p ≤ 1, with the Lp-norm: ( (R jfjp)1=p, if p is finite, kfkp = inffα > 0 j jf(y)j ≤ α, for µ-almost every y 2 Y g, if p = 1. ( P p 1=p p p ( i2I jf(i)j ) , if p is finite, • ` (I), for any set I and 1 ≤ p ≤ 1, with the ` -norm: kfkp = supi2I jf(i)j, if p = 1. • C(X) = ff : X ! C j f continuousg, for any compact Hausdorff topological space X, with the supremum norm kfk = supx2X jf(x)j. 1The author was supported in part by NSF Career Grant DMS 1253402 and NSF FRG Grant 1854074. 1 2 ADRIAN IOANA • B(X) = ff : X ! C j f bounded Borelg, for any compact Hausdorff topological space X, with the supremum norm kfk = supx2X jf(x)j. • The dual X∗ of any normed vector space X. (Recall that Lp(Y )∗ = Lq(Y ) and `p(I)∗ = `q(I), if 1 ≤ p < 1 and 1=p + 1=q = 1.) • B(X), for any Banach space X. Definition 1.5. A Hilbert space H is a Banach space whose norm comes from a scalar product, p i.e., there exists a scalar product h·; ·i : H × H ! C such that kxk = hx; xi, for all x 2 H. Examples 1.6. (of Hilbert spaces): n • C , for n ≥ 1, with the Euclidean scalar product h(x1; :::; xn); (y1; :::; yn)i = x1y¯1 +:::+xny¯n. • L2(Y ), for any measure space (Y; µ), with the scalar product hf; gi = R fg¯. 2 P • ` (I), for any set I, with the scalar product hf; gi = i2I f(i)g(i): • Recall that every Hilbert space H has an orthonormal basis, i.e., a set fξigi2I such that P hξi; ξji = δi;j, for all i; j 2 I, and ξ = i2I hξ; ξiiξi, for all ξ 2 H. This fact implies that every Hilbert space H is isomorphic to `2(I), for some set I. 1.2. Linear functionals. Theorem 1.7 (Hahn-Banach). Let X be a normed vector space, Y ⊂ X a subspace, and ' : Y ! C a linear functional such that j'(x)j ≤ kxk, for all x 2 Y . Then there exists '~ 2 X∗ such that j'~(x)j ≤ kxk, for all x 2 X, and '~jY = '. Definition 1.8. Let X be a normed vector space. The weak∗ topology on X∗ is the topology of pointwise convergence: 'i ! ' iff 'i(x) ! '(x), for all x 2 X. The weak topology on X is ∗ given by xi ! x iff '(xi) ! '(x), for all ' 2 X . Theorem 1.9 (Alaoglu). The closed unit ball of X∗, B := f' 2 X∗ j k'k ≤ 1g, is compact in the weak∗ topology. Q Proof. Put Dx = fz 2 C j jzj ≤ kxkg for x 2 X. Then P := x2X Dx is compact by Tychonoff's ∗ theorem. Define θ : B ! P by letting θ(') = ('(x))x2X . Then 'i ! ' is the weak topology ∗ iff θ('i) ! θ(') in the product topology. Thus, in order to show that B is compact in the weak ∗ topology (equivalently, that every net ('i) 2 B has a weak convergent subnet) it suffices to prove that θ(B) is compact in the product topology. As it can be easily seen that θ(B) is closed, and thus compact, in the product topology, the conclusion follows. Theorem 1.10. Let H be a Hilbert space. If ' 2 H∗, then there exists y 2 H such that '(x) = hx; yi; for all x 2 H. Proof. Let K = fx 2 H j '(x) = 0g. Since ' is bounded, K is a closed subspace of H. If K = H, then ' ≡ 0 and y = 0 works. If K is a proper subspace of H, we can find z 2 H such that z ? K and kzk = 1 (see [Fo99, Theorem 5.24]). Since '('(z) · x − '(x) · z) = 0, we have '(z) · x − '(x) · z 2 K, hence h'(z) · x; zi = h'(x) · z; zi, thus '(x) = hx; '(z)zi, for all x 2 H. Definition 1.11. Let X be a compact Hausdorff topological space. A Borel measure on X is a measure defined on the σ-algebra BX of all Borel subsets of X. A finite Borel measure µ on X is called regular if µ(A) = inffµ(U) j U ⊃ A openg = supfµ(K) j K ⊂ A compactg, for every Borel set A ⊂ X.A complex Borel regular measure on X is a map µ : BX ! C of the form µ = (µ1 − µ2) + i(µ3 − µ4), where µ1; :::; µ4 are finite Borel regular measures on X. Theorem 1.12 (Riesz's Representation Theorem). Let X be a compact Hausdorff topological space. If ' 2 C(X)∗, then there exists a complex Borel regular measure µ on X such that '(f) = R f dµ. Pn n Moreover, we have that k'k = kµk := supf i=1 jµ(Ai)j j fAigi=1 Borel partition of Xg. VON NEUMANN ALGEBRAS 3 Theorem 1.13 (Stone-Weierstrass). Let X be a compact Hausdorff topological space. Let A be a closed subalgebra of C(X) such that • 1X 2 A, • if f 2 A then f¯ 2 A, and • A separates points: for all x 6= y 2 X, there is f 2 A such that f(x) 6= f(y). Then A = C(X). 1.3. The adjoint operation and topologies on B(H). Let H be a Hilbert space. ∗ Exercise 1.14. Let T 2 B(H). Prove that there exists a unique T 2 B(H), called the adjoint of T , such that hT x; yi = hx; T ∗yi, for all x; y 2 H. Prove that kT ∗k = kT k and kT ∗T k = kT k2. Definition 1.15. An operator T 2 B(H) is called: • self-adjoint (or, hermitian) if T ∗ = T , • a projection if T = T ∗ = T 2, ∗ ∗ • a unitary if T T = TT = IdH , ∗ • an isometry if kT xk = kxk, for all x 2 H, or equivalently T T = IdH , • normal if T ∗T = TT ∗, • compact if the closure of fT (x) j kxk ≤ 1g in H is compact. Remark 1.16. The set of unitary operators T 2 B(H) is group which is denoted by U(H). Exercise 1.17. Let T 2 B(H) be a projection. Prove that there exists a closed subspace K ⊂ H such that T is the orthogonal projection onto K (see [Fo99, page 177, Exercise 58]). Definition 1.18. There are three topologies on B(H) that we will consider: • the norm topology: Ti ! T iff kTi − T k ! 0. • the strong operator topology (SOT): Ti ! T iff kTi(ξ) − T (ξ)k ! 0, for all ξ 2 H. • the weak operator topology (WOT): Ti ! T iff hTi(ξ); ηi ! hT (ξ); ηi, for all ξ; η 2 H. Exercise 1.19. Let (Ti)i2I ⊂ B(H) be a net such that Ti ! T (WOT), for some T 2 B(H). (1) Assume that (Ti)i2I and T are projections. Prove that Ti ! T (SOT). (2) Assume that (Ti)i2I and T are unitaries. Prove that Ti ! T (SOT). Exercise 1.20. Assume that H is infinite dimensional (() any orthonormal basis of H is infinite). (1) Given an example of a net of projections (Ti)i2I converging in the WOT but not the SOT.
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