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A GENTLE INTRODUCTION to VON NEUMANN ALGEBRAS for MODEL THEORISTS Contents 1. Topologies on B(H) and the Double Commutant Theore

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A GENTLE INTRODUCTION to VON NEUMANN ALGEBRAS for MODEL THEORISTS Contents 1. Topologies on B(H) and the Double Commutant Theore A GENTLE INTRODUCTION TO VON NEUMANN ALGEBRAS FOR MODEL THEORISTS ISAAC GOLDBRING Contents 1. Topologies on B(H) and the double commutant theorem 2 2. Examples of von Neumann algebras 5 2.1. Abelian von Neumann algebras 6 2.2. Group von Neumann algebras and representation theory 6 2.3. The Hyperfinite II1 factor R 9 3. Projections, Type Classification, and Traces 10 3.1. Projections and the spectral theorem 10 3.2. Type classification of factors 12 3.3. Traces 13 4. Tracial ultrapowers and the Connes Embedding Problem 15 4.1. The 2-norm 15 4.2. Tracial ultraproducts 16 4.3. The Connes Embedding Problem 17 5. Some Model Theory of tracial von Neumann algebras 17 5.1. Axiomatizability 18 5.2. The model theoretic version of CEP 20 5.3. Model companions and connection to CEP 20 5.4. Stability 22 References 23 These notes were part of my course on Continuous Model Theory at UIC in the fall of 2012. We spent several weeks on the model theory of von Neumann algebras and these notes served as an introduction to von Neumann algebras with the intended audience being model theorists who know a little bit of functional analysis. These notes borrow in great part from excellent notes of Vaughn Jones [10], Martino Lupini and Asger Tornquist [11], and Jacob Lurie [12]. Date: May 3, 2013. 1 2 ISAAC GOLDBRING 1. Topologies on B(H) and the double commutant theorem In this section, H denotes a (complex) Hilbert space and B(H) denotes the set of bounded operators on H: recall that the linear operator T : H ! H is bounded if T (B1(H)) is bounded, where B1(H) := fx 2 H : kxk ≤ 1g is the closed unit ball in H. It is straightforward to check that the bounded operators are precisely the continuous linear operators. For T 2 B(H), we set kT k := supfkT xk : kxk ≤ 1g, and call kT k the (operator) norm of T . It is easy to see that kT k is the radius of the smallest closed ball centered at 0 in H containing T (B1(H)). As the name indicates, k · k is a norm on B(H) and we call the resulting topology the norm topology on B(H). Exercise 1.1. For T 2 B(H) and v 2 H, we have kT vk ≤ kT k · kvk. Recall that for each T 2 B(H), there is a unique function T ∗ : H ! H satisfyiing hT x; yi = hx; T ∗yi for all x; y 2 H. T ∗ is called the adjoint of T and plays a crucial role in operator algebras. It is easy to see that T ∗ is once again in B(H) and that (T ∗)∗ = T . We now introduce two more topologies on B(H). First, the strong (opera- tor) topology on B(H) is defined to be the topology on B(H) whose subbasic open neighborhoods are of the form UT0;v, := fT 2 B(H): kT v − T0vk < g; >0 as T0, v and range over B(H), H, and R respectively. Exercise 1.2. The strong topology is the weakest topology on B(H) that makes, for each v 2 H, the map T 7! kT vk : B(H) ! R continuous. The weak (operator) topology on B(H) is the topology on B(H) whose subbasic open neighborhoods are of the form VT0;v;w, := fT 2 B(H)j : jhT v; wi − hT0v; wij < , >0 as T0, (v; w), and > 0 range over B(H), H × H, and R respectively. Exercise 1.3. The weak topology is the weakest topology on B(H) that makes, for each v; w 2 H, the map T 7! hT v; wi : B(H) ! R continuous. Let ONT, SOT, and WOT denote the operator norm, strong, and weak topologies respectively. Lemma 1.4. WOT ⊆ SOT ⊆ ONT. Proof. First suppose that O 2 WOT. Fix T0 2 O. Fix v1; : : : ; vn; w1; : : : ; wn 2 Tn H and > 0 such that i=1 VT0;vi;wi, ⊆ O. Without loss of general- ity, we may assume that wi 6= 0 for each i. By Cauchy-Schwarz, we have jh(T − T0)vi; wiij ≤ k(T − T0)vik · kwik; it follows that n n \ \ UT0;vi; ⊆ VT0;vi;wi, ⊆ O: kwik i=1 i=1 A GENTLE INTRODUCTION TO VON NEUMANN ALGEBRAS 3 Thus, O 2 SOT. Similarly, suppose that O 2 WOT and T0 2 O. Fix v1; : : : ; vn 2 H and Tn > 0 such that i=1 UT0;vi, ⊆ O. Without loss of generality, each vi 6= 0. Since k(T − T0)vik ≤ kT − T0kkvk, it follows that n \ Br(T0) ⊆ UT0;vi, ⊆ O; i=1 where r := min1≤i≤n and Br(T0) := fT 2 B(H): kT − T0k < rg. It kvik follows that O 2 ONT. Why consider other topologies on B(H) other than ONT? Consider the following example: Example 1.5. Let `2 be the separable Hilbert space, that is, 2 X 2 ` = fx = (xn) 2 CN : kxk2 := jxnj < 1g n P with inner product hx; yi := n xnyn. Let L 2 B(H) be the left-shift oper- ∗ ator, that is, L(x) = (0; x0; x1;:::). It is easy to see that L = R, the right- i shift operator given by R(x) := (x1; x2;:::). Notice that L ! 0 as i ! 1 2 i P1 2 in the strong topology as, for x 2 ` , we have kL xk = n=i jxij ! 0 as i ! 1 since v 2 `2. On the other hand, (Li)∗ = Ri 6! 0 as i ! 1 as, for x = (1; 0; 0;:::), we have kRixk = 1 for all i. Thus, we see that the map T 7! T ∗ : B(`2) !B(`2) is not strongly continuous, that is, not continuous with respect to SOT. Contrast the previous example with the following lemma: Lemma 1.6. The map T 7! T ∗ : B(H) !B(H) is weakly continuous, that is, is continuous with respect to WOT. Proof. The inverse image of U under the adjoint map is U ∗ . T0;v;w, T0 ;w;v, Exercise 1.7. For any S0;T0 2 B(H), the maps T 7! S0T;S 7! ST0 : B(H) !B(H) are both weakly continuous; that is, composition in B(H) is separately weakly continuous. (Extra credit: Show that composition itself need not be jointly weakly continuous in general.) wk st For B ⊆ B(H), let B (resp. B ) denote the closure of B with respect to WOT (resp. SOT). We call B ⊆ B(H) a subalgebra of B(H) if B is closed under scalar multi- plication, addition, and composition. If B is also closed under taking adjoint, we call B a ∗-subalgebra of B(H). If the identity operator I belongs to the subalgebra B, we say that B is a unital subalgebra of B(H). Lemma 1.6 and Exercise 1.7 are enough to prove the following important result: 4 ISAAC GOLDBRING wk Proposition 1.8. If A ⊆ B(H) is a ∗-subalgebra of B(H), then so is A . For X ⊆ B(H), set X0 := fT 2 B(H): ST = TS for all S 2 Xg. X0 is called the commutant of X. We set X00 := (X0)0, the double commutant of X. Exercise 1.9. Suppose that X ⊆ B(H). (1) X0 is a weakly closed subalgebra of B(H). (2) If X is closed under adjoints, then X0 is a weakly closed ∗-subalgebra of B(H). wk (3) X ⊆ X00. (4) X0 = X000 := (X0)00. The following theorem can be considered the beginning of von Neumann algebra theory. Theorem 1.10 (von Neumann double commutant theorem). Suppose that A ⊆ B(H) is a unital ∗-subalgebra of B(H). Then A is strongly (and hence weakly) dense in A00. Before we prove Theorem 1.10, let us state the following important corol- laries. st wk Corollary 1.11. For A as in Theorem 1.10, we have A = A = A00. Proof. By Exercise 1.9(3) (and the definition of the topologies), we have st wk A ⊆ A ⊆ A00: 00 st By Theorem 1.10, we have A ⊆ A . Corollary 1.12. For A ⊆ B(H) a unital ∗-subalgebra of B(H), the following are equivalent: (1) A = X0 for some X ⊆ B(H); (2) A = A00; (3) A is weakly closed; (4) A is strongly closed. Proof. (1) ) (2) follows from Exercise 1.9(4). (2) ) (3) ) (4) ) (1) follows from the previous corollary. Definition 1.13. A unital ∗-subalgebra of B(H) satisfying any of the equiv- alent conditions of Corollary 1.12 is called a von Neumann algebra. What makes von Neumann algebras such a robust notion is the equiva- lence of the algebraic conditions in (1) and (2) of Corollary 1.12 with the topological conditions (3) and (4). 00 Proof of Theorem 1.10. Fix T0 2 A ; we must show that any basic strongly open neighborhood of T0 intersects A. We first deal with the special case of a subbasic strongly open neighborhood of T ; the general case of a basic strongly open neighborhood will follow from an “amplification” trick. A GENTLE INTRODUCTION TO VON NEUMANN ALGEBRAS 5 We thus fix v 2 H and > 0; we aim to prove that UT0;v, \ A 6= ;. Set A · v := fT v : T 2 Ag and let P : H ! H denote the orthogonal projection onto the closed subspace A · v of H. Notice that v 2 A · v (since A is unital) and that A · v is A-invariant, that is, T (A · v) ⊆ A · v for each T 2 A.
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