PATH SPACES, CONTINUOUS TENSOR PRODUCTS, AND E0-SEMIGROUPS William Arveson Department of Mathematics University of California Berkeley CA 94720, USA Abstract. We classify all continuous tensor product systems of Hilbert spaces which are “infinitely divisible” in the sense that they have an associated logarithmic struc- ture. These results are applied to the theory of E0-semigroups to deduce that every E0-semigroup possessing sufficiently many “decomposable” operators must be cocy- cle conjugate to a CCR flow. A path space is an abstraction of the set of paths in a topological space, on which there is given an associative rule of concatenation. A metric path space is a pair 2 (P, g) consisting of a path space P and a function g : P → C which behaves as if it were the logarithm of a multiplicative inner product. The logarithmic structures associated with infinitely divisible product systems are such objects. The preceding results are based on a classification of metric path spaces. Contents Introduction Part I. Path Spaces 1. Definitions and examples 2. Additive forms and multiplicative forms 3. Exactness of cocycles 4. Classification of additive forms Part II. Continuous Tensor Products 5. Continuity of the modulus 6. Inner products of decomposable vectors 7. Continuity and normalization 8. Continuous logarithms 9. Infinite divisibility 10. Existence of measurable propagators Part III. Applications 11. Decomposable continuous tensor products 12. Decomposable E0-semigroups 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. von Neumann algebras, semigroups, automorphism groups. This research was supported in part by NSF grant DMS92-43893 Typeset by AMS-TEX 1 2 WILLIAM ARVESON Introduction. Let α = {αt : t > 0} be an E0-semigroup. That is, each αt is a normal unit-preserving ∗-endomorphism of B(H) such that αs ◦ αt = αs+t, and which is continuous in the sense that for all A ∈ B(H), ξ, η ∈ H, hαt(A)ξ, ηi is continuous in t. A unit for α is a strongly continuous semigroup U = {U(t): t ≥ 0} of bounded operators on H satisfying αt(A)U(t) = U(t)A, for every t ≥ 0, A ∈ B(H). α is called completely spatial if there is a t > 0 such that H is spanned by the ranges of all operator products of the form U1(t1)U2(t2) ...Un(tn) where U1,U2,...,Un are units, t1, t2, . , tn are nonnegative reals summing to t, and n is an arbitrary positive integer. Completely spatial E0-semigroups are those which have “sufficiently many” units. In [2, section 7], completely spatial E0-semigroups are shown to be completely classified up to cocycle conjugacy by their numerical index. This is established at the level of continuous tensor product systems. In somewhat more detail, a product system is a measurable family of separable Hilbert spaces E = {Et : t > 0} which is endowed with an associative multiplication that “acts like tensoring” in the sense 0 0 0 0 that for any choices x, x ∈ Es, y, y ∈ Et, the products xy, x y both belong to Es+t and we have hxy, x0y0i = hx, x0i hy, y0i , and Es+t = spanEsEt. In addition, there is a third key axiom which plays the role of local triviality for Hermitian vector bundles. The precise definition of product systems can be found in [2, Definition 1.4]. The intuition is that Et resembles a continuous tensor product O (I.1) Et = Hs,Hs = H 0<s<t of copies of a single Hilbert space H. However, this heuristic picture of continuous tensor products cannot be pushed too far. While the formula (I.1) can be made precise for certain standard examples, it is also known that there are many product systems for which the “germ” H fails to exist. th Every E0-semigroup α determines a product system Eα. The t Hilbert space Eα(t) is the linear space of operators Eα(t) = {T ∈ B(H): αt(A)T = T A, ∀A ∈ B(H)}, with inner product defined by ∗ hS, T i 1 = T S, S, T ∈ Eα(t), and with multiplication given by ordinary operator multiplication. It is known that two E0-semigroups are cocycle conjugate iff their product systems Eα, Eβ PATH SPACES, CONTINUOUS TENSOR PRODUCTS, AND E0-SEMIGROUPS 3 are isomorphic [2, Corollary of Theorem 3.18]; moreover, every abstract product system is associated with some E0-semigroup [5, Corollary 5.17]. These remarks show that, up to cocycle conjugacy, the theory of E0-semigroups is equivalent to the theory of continuous tensor product systems. Consequently, a central component of our approach to E0-semigroups has been to develop the theory of product systems. The classification of E0-semigroups described above was obtained by classifying product systems which possess a property that corresponds to complete spatiality of their product systems (this was called divisibility in [2]). In this paper, we extend that result so as to include certain product systems which do not appear to contain any units a priori, but which do contain sufficiently many vectors that resemble “elementary tensors”. Such product systems are called decomposable. The corresponding property of E0-semigroups is described as follows. Fix t > 0. An operator T ∈ Eα(t) is called decomposable if, for every 0 < s < t there are operators A ∈ Eα(s), B ∈ Eα(t − s) such that T = AB. Let D(t) denote the set of all decomposable operators in Eα(t). It is easy to see that if H is spanned by vectors of the form D(t0)H = {T ξ : T ∈ D(t0), ξ ∈ H} for some particular t0 > 0, then we have (I.2) H = [D(t)H] for every t > 0. α is called decomposable if (I.2) is valid for some t > 0. Notice that if U1,U2,...,Un are units for α and t1, t2, . , tn are positive numbers summing to t, then every operator product of the form U1(t1)U2(t2) ...Un(tn) is a decomposable operator in Eα(t) because each Uj is a semigroup. Thus, every completely spatial E0-semigroup is decomposable. On the other hand, decomposable E0-semigroups are not required to contain any units a priori. The results of this paper imply that decomposable E0-semigroups are completely spatial, and are therefore classified to cocycle conjugacy by their nu- merical index. In particular, decomposable E0-semigroups necessarily have plenty of units. Decomposability translates into an important property of abstract product sys- tems, and we want to discuss this property and its role in structural issues. Let E = {E(t): t > 0} be an abstract product system and fix t > 0. A nonzero vector x ∈ E(t) is called decomposable if for every s ∈ (0, t) there are vectors a ∈ E(s), b ∈ E(t − s) such that x = ab. Let D(t) be the set of all decomposable vectors in E(t). D(t) can be empty. But if it is large enough that it spans E(t) for some particular t, then it is easy to show that (I.3) E(t) = spanD(t) for every positive t. A product system satisfying (I.3) for every positive t is called decomposable. It is also easy to show that if E is isomorphic to the product system Eα of some E0-semigroup α, then E is decomposable iff α is (Proposition 12.2). 4 WILLIAM ARVESON If we think of the multiplication in a product system E as representing the tensor product operation, then the vectors in D(t) are “elementary tensors”. In the heuristic picture in which one thinks of E(t) as a continuous tensor product O E(t) = Hs,Hs = H 0<s<t of copies of a Hilbert space H, the decomposable vectors have a corresponding heuristic representation (I.4) x = ⊗ xs 0<s<t where xs is a nonzero vector in H for every s ∈ (0, t). Since finite tensor products of Hilbert spaces of the form H ⊗ H ⊗ · · · ⊗ H | {z } n times are always spanned by vectors of the form x = x1 ⊗ x2 ⊗ · · · ⊗ xn with xi ∈ H, decomposability appears as a natural property of product systems. However, notice that Theorem 11.1 below implies that only the simplest product systems are decomposable. Product systems occur naturally in probability theory. For example, any ran- dom distribution over the interval (0, ∞) having stationary independent increments gives rise to a product system (see [2, pp 14–16] for an example based on Poisson processes). Some of these examples (such those arising from Poisson processes) do not appear to have “enough” units to be divisible in the sense of [2]. Nevertheless, they do. They do because in constructing these examples, one essentially constructs the sets D(t), t > 0, and then defines the product system E by setting E(t) = spanD(t), t > 0. Such a product system is clearly decomposable and thus, by Theorem 11.1 below, is simply one of the standard ones. Our approach to decomposable product systems is based on the concept of a metric path space. Roughly speaking, a metric path space is a structure which behaves as if it were the “logarithm” of a product system. Path spaces are very general objects, and we give a variety of examples. The most natural constructions of product systems all begin with a metric path space of some kind...an example of this being the construction starting with a Poisson process as above. We give an essentially complete classification of metric path spaces in part I. In part II, we show that every decomposable product system is associated with an essentially unique metric path space, and in part III these results are applied to the classification of product systems and E0-semigroups.