C Independence and W Independence of von Neumann Algebras

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1 Math. Nachr (2002, C Independence and W Independence of von Neumann Algebras By Jan Hamhalter of Prague (Received August 28, 2000; revised version July 20, 2001; accepted August 3, 2001 Abstract. It is shown that in any von Neumann algebra with infinite dimensional non abelian central part there are always finite dimensional subalgebras which are C independent but not W independent. On the other hand, it is proved that C independence is very close to W independence in the following sense: If A and B are C independent subalgebras of a von Neumann algebra M then the pairs (ϕ 1,ϕ 2, where ϕ 1 is a normal state on A and ϕ 2 is a normal state on B, which do admit a common normal extension over M form a norm dense set in the product of normal state spaces. Some other independence conditions (strict locality, logical independence are studied. 1. Introduction and preliminaries In the classical probability theory two probability spaces (A, A,µand(B, B,ν living in a common probability space are considered to be independent if the enveloping space can be organized into the product space (A B, A B,µ ν. There is a large variety of mutually nonequivalent concepts, generalizing this notion to the non commutative context of operator algebras. The study of them has led to many interesting results in the theory of operator algebras (see [30] for overview. For this reason the independence of operator algebras has been studied intensively recently [9, 10, 14, 16, 18, 25, 26, 27]. On the other hand, investigation of independence of algebras originated in quantum theory as an attempt to formalize various degrees of independence between local physical systems. For example, it is well known that a pair of von Neumann algebras generated by the spectral families of the momentum and the position operator, respectively, is manifestly nonindependent by violating all independence conditions hitherto known (see e. g. [27]. Even more important is the investigation of independence of operator algebras for the quantum field theory, where various concepts of indepen Mathematics Subject Classification. Primary: 46L10; Secondary: 46L30, 81T05. Keywords and phrases. Independence of von Neumann algebras, simultaneous extensions of states, axiomatic quantum field theory. hamhalte@math.feld.cvut.cz c WILEY-VCH Verlag Berlin GmbH, Berlin, X/02/ $ /0

2 Hamhalter, C Independence and W Independence 147 dence arise in the connection with describing relationship between spacelike separated regions [2, 3, 4, 11, 12, 18, 22, 27, 28, 30, 31]. The aim of this paper is to clarify the relationship between two major notions of independence known as C independence and W independence. These concepts embody the simultaneous extension property in the category of C algebras and W algebras as follows: Let A and B be von Neumann subalgebras of a von Neumann algebra M, each containing the unit of M. We say that the algebras A and B are C independent if for any state ϕ 1 on A and for any state ϕ 2 on B there is a state ϕ on M which extends both ϕ 1 and ϕ 2. Analogously, A and B are said to be W independent if for any normal state ϕ 1 on A and for any normal state ϕ 2 on B there is a normal state ϕ on M extending both ϕ 1 and ϕ 2. In physical interpretation, partial subsystems given by independent algebras do not interact (see e. g. [27]. Employing compactness of the state space and weak density of its normal part, it is easy to show that W independence always implies C independence. The reverse implication is far from being obvious. A deep recent result of Florig and Summers [10] says that if A and B are commuting subalgebras of σ finite von Neumann algebra, then A and B are W independent if and only if they are C independent. We show that this result does not hold for non commuting subalgebras. More specifically, we prove that if von Neumann algebra M has infinite dimensional non abelian central summand (and exactly in this case, we can always find finite dimensional subalgebras A and B of M which are C independent but not W independent. Nevertheless, despite the fact that the existence of C independent but not W independent algebras is generic in the setting of von Neumann algebras, we prove further that the moral of the result of Florig and Summers remains true even without the assumption of mutually commuting subalgebras. Namely, we show that C independence of subalgebras A and B is strong enough to force that arbitrarily chosen normal states ϕ 1 and ϕ 2 on A and B, respectively, can be approximated uniformly by restrictions of one normal state of the global algebra. Let us remark that the study of independence of mutually non commuting subalgebras can be motivated by the following reasons. Firstly, in the axiomatics of the quantum field theory based on quasilocal algebra the kinematical independence of spacelike separated regions (i. e. the requirement of subalgebras being commuting is often replaced by another independence conditions such as C independence (see e. g. [12]. Secondly, the relationship between non commuting operators can be described by the degree of independence of von Neumann algebras that they generate. Besides results stated above we provide a characterization of W independence in terms of the Jauch Piron states and relate the concepts of strict locality and logical independence of subalgebras. Next we recall a few notions and fix the notation. Throughout the paper M will denote a von Neumann algebra containing two von Neumann subalgebras A and B. All subalgebras will be assumed to contain the unit, usually denoted by 1 or 1 M,of larger algebra M. ThesymbolsM, M,(M +,(M + will be reserved for the dual of M, the predual of M, and their positive parts, correspondingly. By a state on M we mean a normalized positive functional. The set of all states and all normal states on M will be denoted by S(M ands n (M, respectively. It is well known that S(M is a convex compact set when endowed with the weak topology, while S n (M isanorm

3 148 Math. Nachr (2002 closed weak dense convex subset of S(M. A state ϱ on M is called singular if there is a system (p α of projections in M, such that α p α =1andϱ(p α = 0 for all α. If M acts on a Hilbert space H and ξ H, then the symbol ω ξ will represent a vector functional, ω ξ (x =(xξ, ξ, x M. For a Hilbert space H, B(H will stand for the algebra of all bounded operators on H. ThesymbolM 2 is reserved for the algebra of all 2 2 complex matrices. The notation W (a means the von Neumann subalgebra generated by a M. 2. Results As a starting point we prove a few characterizations of W independence complementing ones hitherto obtained ([10] which will be useful in the sequel and which are of some interest for the axiomatics of quantum theory [19, 23, 32]. The following proposition can be isolated from the proof of [10, Lemma 6]. Proposition 2.1. Let A and B be von Neumann subalgebras of von Neumann algebra M acting on a Hilbert space H. The following conditions are equivalent. (i A and B are W independent, (ii for any pair of normal states ϕ 1 and ϕ 2 on A and B, respectively, there is a normal, non zero, positive functional ϕ on M such that ϕ ϕ 1 on A and ϕ ϕ 2 on B, (iii for any pair of unit vectors ξ 1,ξ 2 H there is a non zero vector ξ H such that ω ξ ω ξ1 on A and ω ξ ω ξ2 on B. Proof. It is obvious that (i = (ii. Let us prove the implication (ii = (i. Fix two normal states ϕ 1 and ϕ 2 on A and B, respectively, and define the set S = { ϕ (M + ϕ A ϕ 1,ϕ B ϕ 2 }. The condition (ii guarantees that S contains a non zero element. Moreover, S has a maximal element with respect to the order given by the positive cone (M +. Indeed, let us consider an increasing chain (ϕ α ins. Since ϕ α 1, we can define a positive linear form ψ on M by Then ψ(x = sup α ϕ α (x, for all 0 x M. ψ ϕ α = (ψ ϕ α (1 0, implying that ψ is a normal fuctional. On applying Zorn s lemma, there is a maximal (non zero element, let us say τ, ofs. Weprovethatτ has to be a common extension of ϕ 1 and ϕ 2. Suppose, for a contradiction, that τ ϕ 1 on A and τ ϕ 2 on B. Then ϕ 1 τ = ϕ 2 τ =1 τ(1 > 0 and so by (ii there is a non zero normal positive functional τ on M with τ ϕ 1 τ on A and τ ϕ 2 τ on B. Thus,τ τ +τ S, which contradicts the maximality of τ. Hence, without loss of generality, we can assume that τ A = ϕ 1. Then immediately ϕ 2 τ B = ϕ 2 (1 τ(1 = 0,

4 Hamhalter, C Independence and W Independence 149 which concludes the proof of (ii = (i. The equivalence of (ii and (iii follows immediately from the fact that any normal state is a σ convex combination of vector states. Corollary 2.2. Let M be a von Neumann algebra containing von Neumann subalgebras A and B. The algebras A and B are W independent if and only if for any normal state ϕ 1 on A and any normal state ϕ 2 on B there is a non singular state ϕ on M with ϕ A = ϕ 1 and ϕ B = ϕ 2. Proof. Since any non singular state majorizes a non zero normal functional, Corollary 2.2 follows from condition (ii of Proposition 2.1. It turns out, as a consequence of Corollary 2.2, that W independence is, in the most important cases, equivalent to a weaker extension property formulated in terms of so called Jauch Piron states which play an important role in axiomatic foundations of quantum theory [19]. A state ϱ on a von Neumann algebra M is called Jauch Piron if ϱ(p q = 1 whenever p and q are projections in M with ϱ(p =ϱ(q= 1. (By p q we denote the infimum in the projection lattice. For the physical motivation of this concept we refer to [19, 20], for a subsequent mathematical analysis to [1, 5, 6, 7, 8, 13, 29]. Any normal state is Jauch Piron (see e. g. [8], the converse is far from being true. Proposition 2.3. Let M be a von Neumann algebra satisfying (a or (b below; (a M is σ finite and properly infinite. (b M is a σ finite factor. Suppose that A and B are von Neumann subalgebras of M. The following conditions are equivalent: (i A and B are W independent. (ii For any normal state ϕ 1 on A and for any normal state ϕ 2 on B there is a Jauch Piron state ϕ on M such that ϕ A = ϕ 1 and ϕ B = ϕ 2. Proof. The analysis of the continuity properties of Jauch Piron states carried out in [5] yields that if M is of type (a or (b, then any Jauch Piron state on M is automatically non singular and the Proposition follows now from Corollary 2.2. If, in addition to the hypothesis in Proposition 2.3, also A and B are of the type (a or (b above, then A and B are W independent exactly when for any Jauch Piron state on A and for any Jauch Piron state on B there is a common Jauch Piron state extension on M. This holds, for instance, if A and B are properly infinite, which is a situation typical in the quantum field theory. Now we focus on the relationship between C independence and W independence. We recall a recent remarkable result of Florig and Summers. Theorem 2.4. ( [10, Proposition 8, Theorem 11]. Let M be a σ finite von Neumann algebra and A and B be commuting subalgebras of M. ThenA and B are C independent if and only if they are W independent. It is worth noting that there are many non commuting W independent algebras (see e. g. [9, Examples 2.15, 2.16], [24], [30, p. 205, 206]. The following theorem

5 150 Math. Nachr (2002 shows that the assumption of A and B being commuting in Theorem 2.4 is essential. Moreover, we show that in any von Neumann algebra which is essentially non abelian, we can always find finite dimensional subalgebras which are C independent but not W independent. According to the structure theory of von Neumann algebras (see e. g. [21] for any von Neumann algebra M there is the largest central projection z in M (possibly zero such that zm is abelian. Then M = zm (1 zm and the summand (1 zm is either zero or it does not contain any non zero central abelian projection. In the sequel we shall call the subalgebra (1 zm the non abelian central summand of M. Theorem 2.5. Let M be a von Neumann algebra. The following conditions are equivalent: (i There are finite dimensional von Neumann subalgebras A and B of M which are C independent but not W independent. (ii M has an infinite dimensional non abelian central summand. Proof. (ii = (i. Supposing that the non abelian central summand of M is infinite dimensional, we are going to construct a pair of finite dimensional subalgebras of M which are C independent but not W independent. As a preparation for this we shall prove that M contains a copy of the algebra ( l M 2 C C C. At first step we show that there is a copy of l M 2 contained, as a subalgebra, in some corner of M. According to the assumption the non abelian Type I part or the non Type I part of M has infinite dimension. Let us consider the first case, meaning that M has an infinite dimensional direct summand consisting of Type I nk parts, each n k 2. Suppose that one of these homogeneous Type I summands is infinite dimensional. In that case there is an infinite dimensional central summand of M isomorphic to R B(K, where R is abelian and dim K 2, which implies that R is infinite dimensional or K is infinite dimensional. In the former case R contains a copy of l and so R B(K contains a copy of l M 2. In the latter case B(K B(L M 2,dimL =, and we can find a copy of l in B(L andso acopyofl M 2 in M. On the other hand, if each homogeneous Type I nk direct summand has finite dimension, then M has to contain infinitely many non zero direct summands of this kind and so there is a copy of M 2 in M, which is an algebra isomorphic to l M 2. Let us now discuss the case of M having non zero central summand G which is of Type II or III. In both cases there is a projection r in G such that r 1 r and dim rgr =. So, again, rgr contains a subalgebra isomorphic to l and G rgr M 2 contains a copy of l M 2. In summary, we have shown that M admits a projection p such that pmp has a subalgebra isomorphic to l M 2. Inthesequelweshallalwaysidentifyl M 2 with M 2 via the map (a n b a n b, (a n l, b M 2.

6 Hamhalter, C Independence and W Independence 151 Assume that 1 p 0. Then S = C1 M2 C 1 M2 M 2 is a subalgebra of M 2.OntakingS C (1 p, we get the desired subalgebra. If p = 1 we can, in the same way, take the subalgebra C 1 M2 C 1 M2 C 1 M2 M 2 in M 2 to obtain the subalgebra required. In the rest of the proof we shall assume that M = ( l M 2 C C C. Let x, x 1,x 2,..., be a sequence of distinct unit vectors in the two dimensional Hilbert space such that x n x. Denotebye and f n,, 2,..., the atomic projections in M 2 (identified with B(H, dim H = 2 with ranges spanned by the vectors x, x 1,x 2,..., correspondingly. We define two projections g and h in M by letting ( g = e 1 0 0, h = n=3 n=4 ( f n Put A = W (g = sp {g, 1 g}, B = W (h = sp {h, 1 h}. It is clear that g h = 0, while ( (2.1 (1 g h , (2.2 (2.3 g (1 h (1 g (1 h ( , ( As x n x, weinferthat ef n e 1. Whence, ghg = sup efn e = 1. n The element ghg being positive there is a state ϕ on M such that ϕ(ghg = 1. The inequalityghg g entailsϕ(g = 1. By employing the Cauchy Schwarz inequality we obtain ϕ(h = ϕ(ghg = 1.

7 152 Math. Nachr (2002 Summing it up, there is a state on the global algebra taking value 1 at both g and h. The order relations (2.1, (2.2, (2.3 imply that the same holds for the pairs (1 g, h, (g, 1 h, (1 g, 1 h. Since all states on A and B are convex combinations of states concentrated at atomic projections, we infer that A and B are C independent. However, these algebras are not W independent for the following reason: If there were a normal state ϱ on M with ϱ(g =ϱ(h = 1, then by the Jauch Piron property of normal states ϱ(g h = 1, which is absurd because g h =0. (i = (ii. We shall prove that if M has finite dimensional non abelian central summand then all C independent finite dimensional subalgebras of M are W independent. For this take a C independent pair of finite dimensional subalgebras A and B. It is enough to verify that for any pure state ϕ 1 on A concentrated at projection p atomic in A and for any pure state ϕ 2 on B concentrated at atomic projection q in B there is a common normal state extension on M. (See e. g. Proposition 2.1. By the C independence we can always find a state ϱ (perhaps non normal on M with ϱ(p =ϱ(q = 1. Since all states on M are Jauch Piron [5], ϱ(p q = 1 giving p q 0. Sothereisanormalstateϕ on M with ϕ(p =ϕ(q = 1 which has to coincide with ϕ 1 and ϕ 2 on the corresponding subalgebras. By inspecting the proof of the previous theorem we can see that in any algebra with infinite dimensional non abelian central summand there are two dimensional (i. e. the smallest possible abelian subalgebras which are C independent but not W independent. In other words, one can find two strictly positive operators a and b in M, each having two point spectrum, such that W (a andw (b arec independent but not W independent. Let us remark that typical examples of such operators are given by the spins of two electrons. We have seen that W independence and C independence differ for all algebras interesting from the point of view of non commutative infinite dimensional analysis as well as quantum physics. However, we show that the C independence is, perhaps surprisingly, strong enough to entail that nearly all pairs of normal states extend to a normal state on a global algebra. To prove it we need some preparation. At first we prove a W version of [16, Theorem 2.1], where it was shown that two C algebras A and B are C independent if and only if, given positive norm one elements a A and b B, thereisstateϕ of the global algebra with ϕ(a =ϕ(b =1. Proposition 2.6. Two von Neumann subalgebras A and B of a von Neumann algebra M are C independent if and only if for all self adjoint elements a A, b B, and for any ε>0, there is a normal state ϱ on M with ϱ(a > a ε, and ϱ(b > b ε. Proof. Suppose that A and B are C independent and fix self adjoint elements a A, b B, andε>0. Then there are states ϕ 1 on A and ϕ 2 on B such that ϕ 1 (a = a and ϕ 2 (b = b. There is common state extension of ϕ 1 and ϕ 2 over M. By the weak density of normal states in the state space we can always find a normal state ϱ on M with ϱ(a ϕ 1 (a < ε and ϱ(b ϕ 2 (b < ε. Therefore

8 Hamhalter, C Independence and W Independence 153 ϱ(a > a ε and ϱ(b > b ε. On the other hand, suppose that the condition of the Proposition 2.6 is fulfilled. Then for all norm one positive elements a A and b B, there is a net (ϱ n ofnormal states on M with ϱ n (a > 1 1 n and ϱ n(b > 1 1 n. Employing the compactness of the state space we can assume that the sequence (ϱ n has a limit ϱ in the weak topology. Then ϱ(a =ϱ(b = 1 and an application of [16, Theorem 2.1] concludes the proof. Lemma 2.7. Let A and B be C independent von Neumann subalgebras of a von Neumann algebra M. For a fixed non negative element b B and ε>0 the set S ε = {ϕ A ϕ S n (M with ϕ(b > b ε} is norm dense in S n (A. Proof. The set S ε is non empty and convex. Therefore the norm closure S ε is a closed convex subset of S n (A. Suppose that S ε S n (A and take a normal state ψ S n (A \ S ε. Using the Hahn Banach theorem for the real Banach space of all self adjoint normal functionals on A the dual of which is the self adjoint part of A, we can find a real number α and a self adjoint element c A such that ψ(c > α > ψ (c for all ψ S ε. By adding a suitable multiple of identity to c and modifying α accordingly, we can assume that c 0. We have, c ψ(c > α > ψ (c for all ψ S ε. However, by Proposition 2.6 there is a state ψ S ε such that It leads to a contradiction ψ(c > c ε. ψ(c > c ε > α > ψ(c. S ε,whereε =min ( ε, c α 2 So we have S ε = S n (A. Now we are ready to prove one of the main results of this note. Theorem 2.8. Let A and B be C independent von Neumann subalgebras of a von Neumann algebra M. Then for any ε>0 and for all normal states ϕ 1 on A and ϕ 2 on B there is a normal state ϕ on M such that ϕ A ϕ 1 < ε and ϕ B ϕ 2 < ε.

9 154 Math. Nachr (2002 Proof. Let us consider the topological cartesian product S n (A S n (B ofthe topological spaces (S n (A, and(s n (B,. We have to show that the set L = {(τ A, τ B τ S n (M} is dense in S n (A S n (B. Fix a state ϕ S n (A, ε>0, and define the set K ε = {τ B τ S n (M, with τ A ϕ <ε}. K ε is a convex non empty set. We shall show that K ε = S n (B. Suppose the contrary and take ψ S n (B \ K ε. Employing the Hahn Banach theorem as in the previous lemma, we can find a non negative c B and a real α such that It follows ψ(c > α > ψ (c for all ψ K ε. c ψ(c > α > ψ (c for all ψ K ε. But by Lemma 2.7 there is a state τ S n (M such that τ(c > c c α 2 = c + α 2 > α and τ A ϕ < ε. So τ B K ε, which implies τ(c >α>τ(c a contradiction. Therefore, K ε = S n (B for any ε>0, proving that the set L is dense in S n (A S n (B. By the previous Theorem we can view C independence as approximate W independence. At the end of this note we briefly discuss, in the light of the preceding results, the relationship between some other independence conditions. The ordered pair (A, B of von Neumann subalgebras of von Neumann algebra M is strictly local if for any non zero projection p A and any normal state ϕ on B there is a normal state τ on M such that τ(p =1andτ B = ϕ [22, 30]. Subalgebras A and B are logically independent if for any non zero projection p A and any non zero projection q B the intersection p q is nonzero in M (see [27] for the meaning of this concept in the semantic approach to quantum theory. By [10, Proposition 9] strict locality is equivalent to C independence provided that the algebras A and B commute. In contrast to this it follows from the construction in the proof of Theorem 2.5 that for non commuting subalgebras the strict locality is nearly always stronger than the C independence. As far as the position of logical independence and extension type independence conditions is concerned, it has been proved in [16] that logical independence is stronger than C independence. However, if one of the subalgebras in question is discrete, then strict locality coincides with the logical independence. (We call an algebra M discrete if every projection in M is a sum of atomic projections. Proposition 2.9. Let A and B be von Neumann subalgebras of a von Neumann algebra M. Suppose that B is discrete. Then A and B are logically independent if and only if (A, B is strictly local.

10 Hamhalter, C Independence and W Independence 155 Proof. Let p be a non zero projection in A and ϕ be a normal state on B. AsB is discrete, ϕ is a mixture of pure normal states on B, i.e. ϕ = i=1 λ i ϕ i, 0 λ i 1, λ i =1, where each ϕ i is a pure normal state concentrated at atomic projection p i B. If A and B are logically independent then p i p 0foralli. Therefore there is always a normal state ϱ i on M such that ϱ i (p =ϱ i (p i = 1. By setting, ϱ = λ i ϱ i i=1 we get a normal state on M with ϱ(p =1andϱ B = ϕ. As a consequence of Proposition 2.9 we have, for example, that the pair (A, W (b, where b is an operator with the discrete spectrum, is strictly local if and only if it is logically independent. Acknowledgements The author would like to thank to the Alexander von Humboldt Foundation, Bonn, the Grant Agency of the Czech Republic Grant No. 201/00/0331, and the Czech Technical University Grant No. MSM for the support of the research the result of which are presented in this paper. He is also grateful to Dietrich Kölzow for his hospitality, helpful discussion and encouragement during author s von Humboldt stay in the Mathematical Institute, Universität Erlangen Nürnberg, in summer i=1 References [1] Amann, A.: Jauch Piron States in W Algebraic Quantum Mechanics, Journal of Mathematical Physics 28 (1989, [2] Araki, H.: Local Quantum Field Theory I. In: Local Quantum Theory, ed. R. Jost, pp , Academic Press, New York, 1969 [3] Baumgärtel, H., and Wollenberg, M.: Casual Nets of Operator Algebras, Mathematical Aspects of Quantum Field Theory, Mathematische Monographien 80, Berlin, Akademie Verlag, 1992 [4] Baumgärtel, H.: Operatoralgebraic Methods in Quantum Field Theory, Akademie Verlag, Berlin, 1995 [5] Bunce, L. J., andhamhalter, J.: Jauch Piron States on von Neumann Algebras, Mathematische Zeitschrift 215 (1994, [6] Bunce, L. J., andhamhalter, J.: Extension of Jauch Piron States on Jordan Algebras, Mathematical Proceedings of the Cambridge Philosophical Society 119 (1996, [7] Bunce, L. J., andhamhalter, J.: Jauch Piron States and σ Additivity, Reviews in Mathematical Physics Vol. 12, No. 6 (2000, [8] Bunce, L. J., Navara, M., Pták, P.,andWright, J.D.M.: Quantum Logics with Jauch Piron States, Quarterly Journal of Oxford 36 (2 (1985, [9] Goldstein, S., Luczak, A., andwilde, I.: Independence in Operator Algebras, Foundations of Physics Vol. 29, Iss. 1 (1999, 79 89

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