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A. Doering and J. Harding, Abelian Subalgebras and the Jordan

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A. Doering and J. Harding, Abelian Subalgebras and the Jordan Houston Journal of Mathematics c 2016 University of Houston Volume 42, No. 2, 2016 ABELIAN SUBALGEBRAS AND THE JORDAN STRUCTURE OF A VON NEUMANN ALGEBRA ANDREAS DORING¨ AND JOHN HARDING Communicated by Kenneth R. Davidson Abstract. For von Neumann algebras , not isomorphic to C C M N ⊕ and without type I2 summands, we show that for an order-isomorphism f : AbSub AbSub between the posets of abelian von Neumann sub- M ! N algebras of and ,thereisauniqueJordan -isomorphism g : M N ⇤ M ! N with the image g[ ]equaltof( )foreachabelianvonNeumannsubalgebra S S of .Theconversealsoholds.ThisshowstheJordanstructureofavon S M Neumann algebra not isomorphic to C C and without type I2 summands ⊕ is determined by the poset of its abelian subalgebras, and has implications in recent approaches to foundational issues in quantum mechanics. 1. Introduction We consider the question: given a von Neumann algebra ,howmuchin- M formation about is encoded in the order structure of its collection of unital M abelian von Neumann subalgebras? The set AbSub of such subalgebras, par- M tially ordered by set inclusion, becomes a complete meet semilattice in which every subset that is closed under finite joins has a join. The task is to reconstruct algebraic information about the algebra from the order-theoretic structure of M AbSub . More generally, we are interested in the interplay between these two M levels of algebraic structure. When is abelian, the projection lattice Proj forms a complete Boolean M M algebra, and one can show that the poset AbSub is isomorphic to the lattice of M complete Boolean subalgebras of Proj . Modifying a result of Sachs [26] that M every Boolean algebra is determined by its lattice of all subalgebras, to show each 2000 Mathematics Subject Classification. 46L10, 81P05, 03G12, 17C65, 18B25. Key words and phrases. von Neumann algebra; Jordan structure; abelian subalgebra; ortho- modular lattice. 559 560 ANDREAS DORING¨ AND JOHN HARDING complete Boolean algebra is determined by its lattice of complete subalgebras, one can then obtain that Proj is determined by AbSub . That is determined M M M by Proj is a consequence of the spectral theorem. M For a non-abelian von Neumann algebra, the situation is more complicated. Reconstruction of the non-commutative product in will not generally be pos- M sible as there are non-isomorphic von Neumann algebras having the same Jordan product, hence exactly the same posets of unital abelian subalgebras. However, we will show that the order structure of AbSub does determine as a Jor- M M dan algebra up to (Jordan) -isomorphism. This means that the poset AbSub ⇤ M encodes a substantial amount of algebraic information about . The proof goes M along the same lines as the abelian case, using a result of [16] that an orthomod- ular lattice is determined by its poset of Boolean subalgebras. In fact, our result is somewhat stronger than we described. Theorem. Suppose , are von Neumann algebras without type I summands M N 2 and f : AbSub AbSub is an order-isomorphism. Then there is a unique M ! N Jordan -isomorphism F : with f( ) equal to the image. ⇤ M ! N S This result is particularly interesting with respect to the so-called topos approach to the formulation of physical theories [6, 7, 8, 9, 10, 19], where a mathematical re- formulation of algebraic quantum theory is suggested. For a von Neumann algebra , one considers the poset AbSub of its abelian subalgebras and the topos of M M presheaves over this poset. The idea is that each abelian subalgebra represents a ‘classical perspective’ on the quantum system. By taking all classical perspectives together, one obtains a complete picture of the quantum system. Mathemati- cally, this corresponds to considering the poset AbSub and presheaves over M it. These presheaves form the topos associated with the quantum system. The so-called spectral presheaf ⌃M, whose components are the Gelfand spectra of the abelian von Neumann subalgebras of , plays a key role in the topos approach. M Physically, the spectral presheaf is interpreted as a generalized state space for the quantum system described by the algebra . Mathematically, ⌃M is a kind of M spectrum of the non-abelian von Neumann algebra . It becomes clear that, M from the perspective of the topos approach, it is very relevant to see how much information about the algebra can be extracted from the poset AbSub . M M Since the appearance of the draft of this manuscript on ArXiv [5], several related manuscripts and papers have arisen. In [14] a related task is undertaken for the poset of abelian subalgebras of a C⇤-algebra, and in [15] the matter is considered from the viewpoint of associative subalgebras of a Jordan algebra. In [4] applications to the topos approach to physical theories are considered further. ABELIAN SUBALGEBRAS 561 In particular, it is shown that if M,N are von Neumann algebras with no direct summands of type I , then there is a Jordan -isomorphism F : if and 2 ⇤ M ! N only there is an isomorphism Φ : ⌃N ⌃M between their spectral presheaves in ! the opposite direction. 2. Preliminaries For a complex Hilbert space H,let (H)betheC⇤-algebra of all bounded B operators on H. For a subset (H), the commutant 0 is the set of all S ✓ B S elements of (H) that commute with each member of . A von Neumann algebra B S is a subset (H)with = 00. For a von Neumann algebra ,weuse M ✓ B M M M Proj for the set of projections in . The following well-known result [22, M M pg. 69] will be used repeatedly. Proposition 2.1. For a von Neumann algebra, =(Proj )00. M M M For any von Neumann algebra the projections Proj form a complete or- M M thomodular lattice (abbreviated: oml). Our primary interest lies in subalgebras of von Neumann algebras, subalgebras of their projection lattices, and relation- ships between these and the original von Neumann algebra. We require several definitions. Definition 2.2. A von Neumann subalgebra of a von Neumann algebra is a M subset that is itself a von Neumann algebra. S ✓ M We will only consider von Neumann subalgebras such that the unit S ✓ M elements in and coincide. (In particular, we will not consider subalgebras S N of the form Pˆ Pˆ for a non-trivial projection Pˆ .) We remark that being M 2 M a von Neumann subalgebra is equivalent to being a unital C⇤-subalgebra that is closed in the σ-weak topology, equivalent to being a unital C⇤-subalgebra that is closed under monotone joins [1, pg. 101–110]. Definition 2.3. For a von Neumann algebra , we let Sub be the set of all M M von Neumann subalgebras of ordered by set inclusion; AbSub be the set of M M abelian von Neumann subalgebras of ordered by set inclusion; and F AbSub M M be the set of all abelian subalgebras of that contain only finitely many projec- M tions, ordered by set inclusion. We note that Sub is a complete lattice, with meets given by intersections. M The join of a family ( i)i I of subalgebras is the weak closure of the algebra S 2 generated by the algebras , i I. Analogously, AbSub is a complete meet Si 2 M semilattice where every subset that is closed under finite joins has a join, and 562 ANDREAS DORING¨ AND JOHN HARDING F AbSub is a complete meet semilattice where every meet is essentially finite. M Yet, neither AbSub nor F AbSub have a top element if is non-abelian, M M M so empty meets do not exist in these posets. Definition 2.4. For an oml L, we let Sub L be the set of all subalgebras of L; BSub L be the set of Boolean subalgebras of L,andFBSub L be the set of finite Boolean subalgebras of L, all partially ordered by set inclusion. If L is complete we let CSub L be the set of complete subalgebras of L, meaning subalgebras that are closed under arbitrary joins and meets from L,andCBSub L be the set of complete Boolean subalgebras of L. Again, these are considered as posets, partially ordered by set inclusion. For a von Neumann algebra we can use the associative, but not necessar- M ily commutative, product on to define a commutative, but not necessarily M associative product on , called the Jordan product, by setting ◦ M 1 a b = (ab + ba). ◦ 2 Suppose ' is a map between von Neumann algebras that is linear, bijective, and preserves the involution (adjoint) .Wesay' is a -isomorphism if it satisfies ⇤ ⇤ '(ab)='(a)'(b); a -antiisomorphism if it satisfies '(ab)='(b)'(a); and a ⇤ Jordan isomorphism if it satisfies '(a b)='(a) '(b). The following is well ◦ ◦ known [21, 27]. Proposition 2.5. Every Jordan isomorphism ⌘ : between von Neumann M ! N algebras , can be decomposed as the sum of a -isomorphism and a -anti- M N ⇤ ⇤ isomorphism. More concretely, there are central projections Pˆ , Pˆ and Qˆ , Qˆ such 1 2 2 M 1 2 2 N that and are unitarily equivalent to Pˆ Pˆ and Qˆ Qˆ ,respec- M N M 1 ⊕ M 2 N 1 ⊕ N 2 tively, and ⌘ ˆ : Pˆ1 Qˆ1 is a -isomorphism, while ⌘ ˆ : Pˆ2 Qˆ2 | P1 M ! N ⇤ | P2 M ! N is a -antiisomorphism.M M ⇤ It follows from [3] that there is a von Neumann algebra that is not -isomorphic ⇤ to its opposite, hence these two von Neumann algebras are Jordan isomorphic, but not -isomorphic. So there can be two di↵erent associative noncommutative prod- ⇤ ucts on a weakly closed set of operators, giving di↵erent von Neumann algebras, but the same Jordan structure.
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