O HE EXISECE OF E -SEMIGROUPS WILLIAM ARVESO Abstract. Product systems are the classifyig structures for semigroups of edomorphisms of B(H), i that two E -semigroups are cocycle cojugate iff their product systems are isomorphic. hus it is importat to kow that every abstract product system is associated with a E -semigrouop. his was first proved more tha fiftee years ago by rather idirect methods. Recetly, Skeide has give a more direct proof. I this ote we give yet aother proof by a very simple costructio. 1. Itroductio, formulatio of results Product systems are the structures that classify E -semigroups up to cocycle cojugacy, i that two E -semigroups are cocycle cojugate iff their cocrete product systems are isomorphic [Arv89]. hus it is importat to kow that every abstract product system is associated with a E -semigroup. here were two proofs of that fact [Arv9], [Lie3] (also see [Arv3]), both of which ivolved substatial aalysis. I a recet paper, Michael Skeide [Ske6] gave a more direct proof. I this ote we preset a ew ad simpler method for costructig a E -semigroup from a product system. Our termiology follows the moograph [Arv3]. Let E {E(t) : t > } be a product system ad choose a uit vector e E(1). e will be fixed throughout. We cosider the Fréchet space of all Borel - measurable sectios t (, ) f(t) E(t) that are locally square itegrable (1.1) f(λ) 2 dλ <, >. Defiitio 1.1. A locally L 2 sectio f is said to be stable if there is a λ > such that f(λ +1)f(λ) e, λ λ. ote that a stable sectio f satisfies f(λ + ) f(λ) e for all 1 wheever λ is sufficietly large. he set of all stable sectios is a vector space S, ad for ay two sectios f,g S, f(λ + ),g(λ + ) becomes idepedet of whe λ is sufficietly large. hus we ca defie a positive semidefiite ier product o S as follows (1.2) f,g lim +1 lim 2 Mathematics Subject Classificatio. 46L55, 46L9. 1 1 f(λ + ),g(λ + ) dλ.
2 WILLIAM ARVESO Let be the subspace of S cosistig of all sectios f that vaish evetually, i that for some λ > oe has f(λ) for all λ λ. Oe fids that f,f ifff. Hece, defies a ier product o the quotiet S/, ad its completio becomes a Hilbert space H with respect to the ier product (1.2). Obviously, H is separable. here is a atural represetatio of E o H. Fix v E(t), t>. For every stable sectio f S, let φ (v)f be the sectio { v f(λ t), λ > t, (φ (v)f)(λ), <λ t. Clearly φ (v)s S. Moreover, φ (v) maps ull sectios ito ull sectios, hece it iduces a liear operator φ(v) o S/. he mappig (t, v), ξ E S/ φ(v)ξ H is obviously Borel-measurable, ad it is easy to check that φ(v)ξ 2 v 2 ξ 2 (see Sectio 2 for details). hus we obtai a represetatio φ of E o the completio H of S/ by closig the desely defied operators φ(v)(f + )φ (v)f +, v E(t), t>, f S. heorem 1.2. Let α {α t : t } be the associated E-semigroup o B(H) (1.3) α t (X) φ(e (t))xφ(e (t)), X B(H), t >, 1 where e 1 (t),e 2 (t),... is a orthoormal basis for E(t) for every t>. he for every t oe has α t (1) 1. 2. Proof of heorem 1.2 he followig observatio implies that we could just as well have defied the ier product of (1.2) by +1 f,g lim. Lemma 2.1. For ay two stable sectios f,g, there is a λ > such that f,g for all real umbers λ. +1 Proof. For iteger values of k, the itegral k+1 k becomes idepedet of k whe k is large. hus, for sufficietly large ad the iteger satisfyig < + 1, it is eough to show that (2.1) +1 +1.
EXISECE OF E -SEMIGROUPS 3 he itegral o the left decomposes ito a sum + +1. For λ, f(λ),g(λ) E(λ) f(λ) e, g(λ) e E(λ+1) f(λ+1),g(λ+1) E(λ+1), hece It follows that +1 ad (2.1) is proved. f(λ +1),g(λ +1) dλ +1 ( +1 +1 + +1 +1 +1 ). o show that φ is a represetatio, we must show that for every t>, every v, w E(t), ad every f,g Soe has φ (v)f,φ (w)g v, w f,g. Ideed, for sufficietly large we ca write φ (v)f,φ (w)g +1 +1 v, w v, w φ (v)f(λ),φ (w)g(λ) dλ v f(λ t),w g(λ t) dλ +1 t+1 t f(λ t),g(λ t) dλ v, w f,g, where the fial equality uses Lemma 2.1. It remais to show that φ is a essetial represetatio, ad for that, we must calculate the adjoits of operators i φ(e). he followig otatio from [Arv3] will be coveiet. Remark 2.2. Fix s> ad a elemet v E(s); for every t> we cosider the left multiplicatio operator l v : x E(t) v x E(s + t). his operator has a adjoit l v : E(s + t) E(s), which we write more simply as v η l vη, η E(s + t). Equivaletly, for s<t, v E(s), y E(t), we write v y for l vy E(s). ote that v y is udefied for v E(s) ad y E(t) whe t s. Give elemets u E(r), v E(s), w E(t), the associative law (2.2) u (v w) (u v) w makes sese whe r s (t > ca be arbitrary), provided that it is suitably iterpreted whe r s. Ideed, it is true verbatim whe r<sad t>, while if s r ad t>, the it takes the form (2.3) u (v w) v, u E(s) w, u, v E(s), w E(t).
4 WILLIAM ARVESO Lemma 2.3. Choose v E(t). For every stable sectio f S, there is a ull sectio g such that (φ (v) f)(λ) v f(λ + t)+g(λ), λ >. Proof. A straightforward calculatio of the adjoit of φ (v) :S Swith respect to the semidefiite ier product (1.2). Lemma 2.4. Let <s<t, let v 1,v 2,... be a orthorormal basis for E(s) ad let ξ E(t). he (2.4) vξ 2 ξ 2. 1 Proof. For 1, ξ E(t) v (vξ) E(t) defies a sequece of mutually orthogoal projectios i B(E(t)). We claim that these projectios sum to the idetity. Ideed, sice E(t) is the closed liear spa of the set of products E(s)E(t s), it suffices to show that for every vector i E(t) of the form ξ η ζ with η E(s), ζ E(t s), we have v (vξ) ξ. For that, we ca use (2.2) ad (2.3) to write hece v (v ξ) v (v (η ζ)) v ((v η) ζ) η, v v ζ, v (vξ) ( η, v v ) ζ η ζ ξ, 1 1 as asserted. (2.4) follows after takig the ier product with ξ. Proof of heorem 1.2. Sice the projectios α t (1) decrease with t, it suffices to show that α 1 (1) 1; ad for that, it suffices to show that for ξ H of the form ξ f + where f is a stable sectio, oe has (2.5) α 1 (1)ξ,ξ φ (v ) f 2 f 2 ξ 2, 1 v 1,v 2,... deotig a orthoormal basis for E(1). Fix such a basis (v ) for E(1) ad a stable sectio f. Choose λ > 1 so that f(λ +1)f(λ) e for λ>λ.forλ>λ we have λ +1> 1, so Lemma 2.4 implies vf(λ +1) 2 f(λ +1) 2 f(λ) e 2 f(λ) 2. 1 It follows that for every iteger >λ, vf(λ +1) 2 dλ 1 1 vf(λ +1) 2 dλ f(λ) 2 dλ f + 2 H.
EXISECE OF E -SEMIGROUPS 5 Lemma 2.3 implies that whe is sufficietly large, the left side is (φ (v ) f)(λ) 2 dλ φ (v )f 2, 1 ad (2.5) follows. Remark 2.5 (otriviality of H). Let L 2 ((, 1]; E) be the subspace of L 2 (E) cosistig of all sectios that vaish almost everywhere outside the uit iterval. Every f L 2 ((, 1]; E) correspods to a stable sectio f Sby extedig it from (, 1] to (, ) by periodicity f(λ) f(λ ) e, < λ +1, 1, 2,..., ad for every 1, 2,... we have +1 f(λ) 2 dλ +1 1 f(λ ) e 2 dλ 1 f(λ) 2 dλ. Hece the map f f + embeds L 2 ((, 1]; E) isometrically as a subspace of H; i particular, H is ot the trivial Hilbert space {}. Remark 2.6 (Purity). A E -semigroup α {α t : t } is said to be pure if the decreasig vo euma algebras α t (B(H)) have trivial itersectio C 1. he questio of whether every E -semigroup is a cocycle perturbatio of a pure oe has bee resistat [Arv3]. Equivaletly, is every product system associated with a pure E -semigroup? While the aswer is yes for product systems of type I ad II, ad it is yes for the type III examples costructed by Powers (see [Pow87] or Chapter 13 of [Arv3]), it is ukow i geeral. It is perhaps worth poitig out that we have show that the examples of heorem 1.2 are ot pure; hece the above costructio appears to be iadequate for approachig that issue. Sice the proof establishes a egative result that is peripheral to the directio of this ote, we omit it. Refereces [Arv89] W. Arveso. Cotiuous aalogues of Fock space. Memoirs Amer. Math. Soc., 8(3), 1989. [Arv9] W. Arveso. Cotiuous aalogues of Fock space IV : essetial states. Acta Math., 164:265 3, 199. [Arv3] W. Arveso. ocommutative Dyamics ad E-semigroups. Moographs i Mathematics. Spriger-Verlag, ew York, 23. [Lie3] V. Liebscher. O the geerators of quatum dyamical semigroups. 23. arxiv:pr/36365. [Pow87] R.. Powers. a o-spatial cotiuous semigroup of -edomorphisms of B(H). Publ. RIMS (Kyoto Uiversity), 23(6):154 169, 1987. [Ske6] M. Skeide. A simple proof of the fudametal theorem about Arveso systems. If. Dim. Aal. Quatum Prob. (to appear), 26. arxiv:math.oa/6214v1. Departmet of Mathematics, Uiversity of Califoria, Berkeley, CA 9472 E-mail address: arveso@mail.math.berkeley.edu