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AND the DOUBLE COMMUTANT THEOREM Recall Egbert Rijke Utrecht University [email protected] THE STRONG OPERATOR TOPOLOGY ON B(H) AND THE DOUBLE COMMUTANT THEOREM ABSTRACT. These are the notes for a presentation on the strong and weak operator topolo- gies on B(H) and on commutants of unital self-adjoint subalgebras of B(H) in the seminar on von Neumann algebras in Utrecht. The main goal for this talk was to prove the double commutant theorem of von Neumann. We will also give a proof of Vigiers theorem and we will work out several useful properties of the commutant. Recall that a seminorm on a vector space V is a map p : V ! [0;¥) with the properties that (i) p(lx) = jljp(x) for every vector x 2 V and every scalar l and (ii) p(x + y) ≤ p(x) + p(y) for every pair of vectors x;y 2 V. If P is a family of seminorms on V there is a topology generated by P of which the subbasis is defined by the sets fv 2 V : p(v − x) < eg; where e > 0, p 2 P and x 2 V. Hence a subset U of V is open if and only if for every x 2 U there exist p1;:::; pn 2 P, and e > 0 with the property that n \ fv 2 V : pi(v − x) < eg ⊂ U: i=1 A family P of seminorms on V is called separating if, for every non-zero vector x, there exists a seminorm p in P such that p(x) 6= 0. The topology generated by a separating family of seminors is always Hausdorff. Definition 1. Suppose that H is a Hilbert space. The weak operator topology on B(H) is the topology generated by collection fA 7! jhA(x);yij : x;y 2 Hg of seminorms. Lemma 2. For every net fAi : i 2 Ig in B(H) we have that fAig converges in the weak operator topology to A if and only if hAi(x);yi ! hA(x);yi for all x;y 2 H. Proof. Suppose that the net fAi : i 2 Ig converges weakly to A. Then, for every open set U there exists i 2 I such that A j 2 U whenever j ≥ i in I. In particular, for every x;y 2 H and e > 0 we can take Ux;y;e := fB 2 B(H) : jh(A − B)(x);yij < eg. Since for every e > 0 there exists an ie 2 I with the property that A j 2 Ux;y;e whenever j ≥ ie , we see that the net fh(A − Ai)(x);yij : i 2 Ig converges to 0. And hence hAi(x);yi ! hA(x);yi for all x;y 2 H. Suppose now that hAi(x);yi ! hA(x);yi for all x;y 2 H and suppose that U ⊂ B(H) is weakly open with A 2 U. Then there are x1;:::;xn;y1;:::;yn 2 H and e > 0 with the 1 2 THE STRONG OPERATOR TOPOLOGY ON B(H) AND THE DOUBLE COMMUTANT THEOREM property that n \ fB 2 B(H) : jh(A − B)(xk);ykij < eg ⊂ U k=1 By assumption there are i1;:::;in 2 I with the property that jh(A − A j)(xk);ykij < e for all j ≥ ik. This implies that A j 2 U for all j ≥ fi1;:::;ing. and hence that Ai ! A in the weak operator topology. Definition 3. The strong operator topology on B(H) is the topology generated by the collection fA 7! kA(x)k : x 2 Hg. The strong operator topology is the topology on B(H) in which convergence is equiv- alent to pointwise convergence: Lemma 4. A net fAi : i 2 Ig in B(H) converges to A in the strong operator topology if and only if Ai(x) ! A(x) for all x 2 H. Proof. Suppose that Ai ! A in the strong operator topology. Then, for every x 2 H and for every e > 0, there exists an i 2 I with the property that A j 2 fB 2 B(H) : k(A−A j)(x)k < eg for all j ≥ i in I, which shows that k(A − A j)(x)k ! 0 and hence that A j(x) ! A(x). On the other hand, suppose that Ai(x) ! A(x) for all x 2 H and let U be a strongly open subset of B(H) which contains A. Then there are x1;:::;xn 2 H and e > 0 such that n \ fB 2 B(H) : k(A − B)(xk)k < eg ⊂ U: k=1 By assumption, there are i1;:::;ik 2 I with the property that k(A − A j)(xk)k < e whenever j ≥ ik. Hence for j ≥ fi1;:::;ikg it follows that A j 2 U and we conclude that Ai ! A in the strong operator topology. Lemma 5. The weak operator topology is weaker than the strong operator topology and the strong operator topology is weaker than the uniform topology on B(H). Proof. Suppose first that U weakly open. Then we can find for every operator A in U elements x1;:::;xn;y1;:::;yn 2 H and e > 0 such that n \ fB 2 B(H) : jhA(xi) − B(xi);yiij < eg ⊂ U: i=1 Without loss of generality we can assume that each yi is non-zero. Since jhA(xi)−B(xi)ij ≤ kA(xi) − B(xi)kkyik it follows that e fB 2 B(H) : kA(xi) − B(xi)k ≤ g = fB 2 B(H) : kA(xi) − B(xi)kkyik ≤ eg kyik ⊂ fB 2 B(H) : jhA(xi) − B(xi);yiij < eg Hence if we take d := minf e : 1 ≤ i ≤ ng we see that kyik n \ fB 2 B(H) : kA(xi) − B(xi)k ≤ dg ⊂ U i=1 and hence that U is open in the strong operator topology. The same trick works to show that the strong operator topology is weaker than the uniform topology on B(H). Indeed, we have the inequality kA(xi)−B(xi)k ≤ kA−Bkkxik and therefore we have the inclusion fB 2 B(H) : kA − Bkkxk < eg ⊂ fB 2 B(H) : kA(x) − B(x)k < eg THE STRONG OPERATOR TOPOLOGY ON B(H) AND THE DOUBLE COMMUTANT THEOREM 3 for all x 2 H and e > 0. If U is open in the strong operator topology we can find x1;:::;xn 2 H and e > 0 for each bounded operator A, with the property that n \ fB 2 B(H) : kA(xi) − B(xi)k < eg ⊂ U: i=1 e We can safely assume that each xi is non-zero. Taking d = minf : 1 ≤ i ≤ ng it follows kxik Tn from the mentioned inclusion that i=1fB 2 B(H) : kA − Bk < dg ⊂ U. The following theorem basically says that an increasing net of positive operators has a least upper bound and converges to it whenever it has an upper bound: Theorem 6 (Vigiers theorem). Suppose that fAl : l 2 Ig is a net of self-adjoint operators on a Hilbert space H. If Ak − Al is positive for all l ≤ k and if there is an M 2 R such that kAl k ≤ M for all l 2 I, then fAl g is strongly convergent. Proof. Note that we can always pick l0 2 I and look at the net fAl −Al0 : l ≥ l0 2 Ig, so without loss of generality we can assume that the net fAl g consists of positive operators. 2 Hence the net fhAl (x);xi : l 2 Ig is increasing and bounded above by Mkxk and therefore the net fhAl (x);xig is convergent for each x 2 H. For any x;y 2 H the polarisation identity 3 k k k hAl (x);yi = ∑ i hAl (x + i y);x + i yi k=0 gives us that the net fhAl (x);yig is also convergent. Denote its limit by s(x;y); one can verify that s : H × H ! H defines a sesquilinear form on H which is bounded by js(x;y)j ≤ Mkxkkyk. Hence there is a bounded operator A for which s(x;y) = hA(x);yi and kAk = ksk. A is a positive operator, larger than any Al , since for every l 2 I and for every e > 0 there is a l0 ≥ l 2 I with the property that jh(A − Ak )(x);xij < e whenever k ≥ l0 and hA(x);xi − hAl (x);xi ≥ hAk (x);xi − hAl (x);xi − e: Since hAk (x);xi ≥ hAl (x);xi ≥ 0 it follows, by taking e smaller and smaller, that A is positive and Al ≤ A for all l. In particular we can take the positive square root of A − Al for all l 2 I. So we have 2 1 1 2 kA(x) − Al (x)k = k(A − Al ) 2 (A − Al ) 2 (x)k 1 2 ≤ kA − Al kk(A − Al ) 2 (x)k ≤ 2Mh(A − Al )(x);xi ! 0 We conclude that Ai converges pointwise (hence strongly) to A. Definition 7. The commutant S0 of a subset S of an algebra A is the set fa 2 A : as − sa = 0 for all s 2 Sg: Since every element of S commutes with every element of S0 we have the inclusion S ⊂ S00 (and also S0 ⊂ S000). Also, it is easy to see that if S ⊂ T then T 0 ⊂ S0 and hence we have S000 ⊂ S0. Therefore we have the identity (1) S0 = S000 for every subset S of an algebra A .
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