Traces of Semigroups Associated with Interacting Particle Systems Sergio

Traces of Semigroups Associated with Interacting Particle Systems

Sergio Albeverio, Alexei Daletskii, Alexander Kalyuzhnyi

no. 264

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer- sität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems

Sergio Albeverio Inst. Ang. Math., Universit¨atBonn; BiBoS; SFB 611; IZKS; CERFIM (Locarno); Acc. Arch. (Mendrisio) Alexei Daletskii College of Science and Technology The Nottingham Trent University U.K. Alexander Kalyuzhnyi Institute of Mathematics Kiev, Ukraine

January 3, 2006 Abstract

Witten Laplacians associated with interacting particle systems on infinite coverings of compact manifolds are considered. The probablilistic representations of the corresponding heat kernels are given. Von Neumann traces of the corresponding semigroups are computed. 2000 AMS Mathematics Subject Classification. Primary: 60G55, 58A10, 58A12. Secondary: 58B99 Key words: Witten Laplacian, infinite covering, von Neumann algebra, Betti numbers, Configuration space, Gibbs measure Contents

1 Introduction 1

2 Conﬁguration spaces and measures 4

3 Random Witten Laplacian 5

4 Probabilistic representations of the heat kernels 8

5 Von Neumann algebras associated with conﬁguration spaces 15

6 Appendix: Gibbs measures on conﬁguration spaces. 21

1 Introduction

The relationship between traces of semigroups associated with elliptic operators acting in spaces of differential forms on a compact Riemannian manifold X, and geometrical and topological properties of these manifolds, is a deep and important fact of global analysis. In the case of Laplace type operators, the supertrace of the corresponding heat semigroup does not depend on time and is equal to the index of the corresponding Dirac operator (the McKean-Singer identity). On the other hand, its short time asymptotics can be expressed in terms of geometric quantities of X, which leads to the Gauss-Bonnet-Chern theorem, see e.g. [13] and [18]. Different versions of this relationship are investigated in the framework of the Atiyah-Singer In- dex Theorem and its more recent generalizations, see e.g. [9]. In the case where X is not compact but admits an infinite group G of isometries with compact quotient, the usual operator trace is substituted by the trace TrA in the commutant A of the action of G (which is a von Neu- mann algebra). This leads to the notion of L2-invariants of X (in particular, M. Atiyah’s L2-Betti numbers and the Novikov-Shubin invariants). The corresponding theory for perturbed Laplacians (Witten Laplacians associated with measures on X) leads to important results such as (asymptotic) Morse inequalities. In the case where the perturbation is G-invariant, the corre-

1 sponding semigroup Tt belongs to A, and its trace is given by the formula Z

TrA Tt = tr K(x, x; t)dx, (1) X/G where K(t; x, y) is the integral kernel of Tt and tr is the usual matrix trace. The right-hand side of the latter formula is called sometimes the Θ-function associated with H, and can be interpreted as the Laplace transform of the spectral density function of H. For a review on L2-invariants and related questions see [28], [29]. On the other hand, there is an extensive theory of random Schr¨odinger operators H = ∆ + q, i.e. Laplace operators with potentials q given by homogeneous random ﬁelds (see [30], [21]). That is, q is supposed to be G-invariant in the sense that

q(gx, ω) = q(x, Ugω), (2) where G 3 g 7→ Ug is a representation of G by measure preserving trans- formations of the probability space. For such operators, the Θ-function is defined by the formula Z Θ(t) = E tr K(t; x, x)dx, (3) X/G and can be interpreted in the following way. Let HΛ be a self-adjoint re- striction of H to a compact domain Λ. Then, under certain conditions on −tHΛ 1 −tHΛ the potential, e is a trace-class operator, limΛ↑X volΛ Tr e is a non- random quantity, and 1 Θ(t) = lim Tr e−tHΛ . (4) Λ↑X volΛ Here Tr is the usual operator trace. Thus, heuristically, Θ(t) coincides with a regularized trace of the semigroup e−tH . Sometimes it can be interpreted as a trace in a certain von Neumann algebra (for instance if the potential is a periodic or an almost periodic function). The aim of the present paper is to consider the case where H is the Witten Lapalcian associated with a random measure σ on a Riemannian manifold X, and the probability space is the space ΓX of locally finite configurations γ in X equipped with a Gibbs measure µ.

2 The measure σ has the form

−Eγ (x) σγ(dx) = e dx, (5) P where Eγ(x) = y∈γ v(ρ(x, y)), ρ is the distance on X and v is a smooth function with compact support. Measures of such type appear, via the generalized Mecke identity, in the theory of configuration spaces, and in particular in the theory of Laplace operators on differential forms over ΓX (see [4], [5], [3]). In fact, the Witten Laplacian H associated with σ is a ”part” of the Hodge-de Rham operator on ΓX associated with the Gibbs measure µ. The structure of the latter operator is very complicated in the case where µ is different from the Poisson measure. We believe that the study of spectral properties of H, which is a more realistic goal, then the study of the full Hodge-de Rham operator on ΓX , may already give interesting links between the properties of µ and geometrical and topological properties of X. Let us remark that the interest in the analysis on infinite configuration spaces has risen in recent years, because of new approaches and rich applications in statistical mechanics and quantum field theory (see [6], [7] and the review [31]). L2-Betti numbers of configuration spaces with Poisson and Lebesgue-Poisson measures, were computed in [2] and [15] respectively (see also [1], [14]). The operator H is related to the following model from statistical mechanics. Let us consider a particle with position x performing a random motion in X and interacting with a random media (gas) described by the Gibbs measure µ. The distribution of the particle is given by the random measure σγ(dx), where Eγ(x) is the energy of the interaction of the particle x and the configuration γ of gas particles. Let us remark that spectral density- and theta-functions of certain Schrödingeroperators related to particles in a random media have been discussed in [30] (see also references therein). Our situation of Witten operators acting in spaces of differential forms is significantly more complicated because of the specific structure of the corresponding potential, which is a matrix-valued function, neither positive nor bounded. As above, we suppose that there exists an infinite group G of isometries of X such that X/G is a compact Riemannian manifold. The main result of the present paper is the proof of an upper bound for the corresponding theta-function. In particular, we show that Θ(t) is

3 ﬁnite for all times t. Moreover, we interpret it as a trace in a certain von Neumann algebra, and show that the corresponding analog of the McKean- Singer formula holds. Acknowledgments. We are very grateful to A. Eberle, K. D. Elworthy, Yu. Kondratiev, X.-M. Li, E. Lytvinov, S. Paycha, M. R¨ockner for interesting discussions and useful remarks. One of the authors (AD) would like to thank the Department of Stochastics , University of Bonn, for the hospitality. Financial support by SFB 611 is gratefully acknowledged.

2 Conﬁguration spaces and measures

Let X be a complete connected, oriented, C∞ Riemannian manifold of infinite volume with a lower bounded curvature. We assume that there exists an infinite discrete group of isometries of X such that X/G is a compact Riemannian manifold. The configuration space ΓX over X is defined as the set of all locally finite subsets (configurations) in X:

ΓX := {γ ⊂ X | |γ ∩ Λ| < ∞ for each compact Λ ⊂ X} . (6)

Here, |A| denotes the cardinality of a set A. We can identify any γ ∈ ΓX with the positive, integer-valued Radon measure X εx ∈ M(X), (7) x∈γ P where εx is the Dirac measure with mass at x, x∈? εx :=zero measure, and M(X) denotes the set of all positive Radon measures on the Borel σ-algebra of X. The space ΓX is endowed with the relative topology as a subset of the space M(X) with the vague topology, i.e., the weakest topology on ΓX with respect to which all maps Z X ΓX 3 γ 7→ hf, γi := f(x) γ(dx) ≡ f(x) (8) X x∈γ are continuous. Here, f ∈ C0(X) (:=the set of all continuous functions on X with compact support).

4 The action of G on X can be lifted to a diagonal action on ΓX :

ΓX 3 γ = {..., x, y, z, ...} 7→ gγ = {..., gx, gy, gz, ...} ∈ ΓX , (9) g ∈ G. Let µ be a Gibbs measure on ΓX (see Appendix). We assume that: (i) µ satisﬁes the Ruelle bound, that is, ¯ ¯ ¯ (n)¯ n kµ ≤ a (10)

(n) for some constant a, where kµ is the n-th correlation function of µ; (ii) µ is invariant w.r.t. to the G-action (9). A class of Gibbs measures which satisfy these properties is described in Appendix (see Remark 9).

3 Random Witten Laplacian

2 1 Let v ∈ C0 (R ) with supp v ⊂ [−r, r] , where r > 0 is the injectivity radius of X, and deﬁne the function V : X × X → R1 by

V (x, y) = v(ρ(x, y)), (11) x, y ∈ X, where ρ is the Riemannian distance on X. For any γ ∈ ΓX we introduce the measure

−Eγ (x) σγ(dx) = e dx (12) on X, where X Eγ(x) = V (x, y) (13) y∈γ and dx denotes the Riemannian volume on X. Let us remark that, for any ﬁxed x ∈ X, Eγ(x) < ∞ (because v has compact support). The measure σγ possesses the logarithmic derivative X λ(γ, x) = − ∇xV (x, y) ∈ TxX, (14) y∈γ

5 that is, it satisfies the integration by parts formula Z Z (∇f(x), ν(x)) σ (dx) = − f(x)[(λ(γ, x), ν(x)) TxX γ TxX X X +div ν]σγ(dx) (15) for any differentiable function f with compact support and a differentiable vector field ν on X, ∇ meaning the gradient. In what follows, we use the following notations: L2Ωp(X) - the space of p-forms on X, which are square-integrable with respect to the volume measure; 2 p Lσγ Ω (X) - the space of p-forms on X, which are square-integrable with respect to σγ; dp - the de Rham differential on p-forms on X; H(p) - the de Rham Laplacian on p-forms on X; B(H1, H2) - the space of bounded linear operators H1 → H2, H1, H2 Hilbert spaces; B(H) := B(H, H). (p) 2 p Let us consider the Witten-Bismut Laplacian Hγ in Lσγ Ω , ¡ ¢ H(p) := dp (dp)∗ + dp−1 ∗ dp−1, (16) σγ γ γ ¡ ¢ where dk ∗ : L2 Ωk → L2 Ωk−1 is the adjoint of dk : L2 Ωk−1 → L2 Ωk. γ σγ σγ σγ σγ (p) It follows from the results of [11] that Hσγ is essentially self-adjoint on the space of smooth forms with compact support. (p) On smooth forms, Hγ is given by the expression 1 ¡ ¢ H(p) = H(p) + (∇E , ∇) + ∇2E ∧p , (17) σγ 2 γ T pX γ where

k Xp z }| { ¡ 2 ¢∧p 2 ∇ Eγ = I ⊗ I ⊗ ... ⊗ ∇ Eγ ⊗... ⊗ I (18) k=1 Let

2 p 2 p U : Lσγ Ω (X) → L Ω (X) (19)

6 − 1 E (x) be the unitary isomorphism deﬁned by the multiplication by e 2 γ . Then the operator

(p) (p) −1 Hγ := UHσγ U (20) in L2Ωp(X) has the form

(p) (p) (p) Hγ = H + Wγ , (21) where

(p) 2 ¡ 2 ¢∧p Wγ = k∇Eγk + ∆Eγ + ∇ Eγ (22)

[19]. Let us remark that Wγ is G-invariant in the sense that

(p) −1 (p) Wγ (x) = (dg) Wgγ (gx) (23)

¡ ∧p ∧p¢ for any g ∈ G, γ ∈ ΓX and x ∈ X. Here dg ∈ B (TxX) , (TgxX) is the corresponding group translation in the ﬁbers of the tensor bundle (TX)∧p. (p) −tHγ 2 p Let us consider the corresponding heat semigroup Tt = e in L Ω (X) and let

(p) ¡ ∧p ∧p¢ Kγ (x, y; t) ∈ B (TxX) , (TyX) (24) be its integral kernel. We introduce the function Z (p) (p) θ (x, t) = tr Kγ (x, x; t) µ(dγ), (25) ΓX where tr is the usual matrix trace. Our ﬁrst aim is to prove the following result.

Theorem 1 For any t > 0 and p = 0, 1, ..., dim X

sup θ(p)(x, t) < ∞. (26) x∈X

The proof of the theorem will be given in the next section. Moreover, we will give an estimate of θ(p) in terms of the heat kernel on X, potential V and correlation functions of µ.

7 (p) The G-ivariance (23) of the potential Wγ implies the G-ivariance of the (p) (p) kernel Kγ (x, x; t) and, consequently, of the function θ (x, t) (the latter follows from the G-ivariance of µ). Thus, θ(p)(·, t) deﬁnes a function θe(p)(·, t) on X/G, and we can deﬁne the theta-fuction Z Θ(p)(t) = θe(p)(x, t) dx. (27) X/G

The next statement follows immediatly from the theorem above and compactness of X/G.

Corollary 1 For any t > 0 and p = 0, 1, ..., dim X

Θ(p)(t) < ∞. (28)

4 Probabilistic representations of the heat kernels

In this section we give a probabilistic representation of the semigroup e−tH(p) and prove with its aid Theorem 1. (p) According to the Weitzenb¨ock formula, the Witten Laplacian Hγ has the form

(p) (p) (p) (p) Hγ = ∆ + R + Wγ , (29)

(p) 2 p (p) ∧p where ∆ is the Bochner Laplacian in L Ω (X), and R (x) ∈ B((TxX) ) is the corresponding Weitzenb¨ock term (see e.g. [19], [13]). Let ξx(s), s ∈ [0, t] be the Brownian motion on X in the time interval [0, t] starting at x, and denote by B(x, r) the ball of radius r with center at x. We need the following general result.

Theorem 2 Let f : X × X → R1 be a bounded function such that, for some r ∈ R and any x ∈ X, suppf(x, ·) ⊂ B(x, r). Deﬁne X fγ(x) := hf(x, ·), γi = f(x, y). (30) y∈γ

8 Then, for all t ∈ R1, the following estimate holds: Z ·Z ¸ t £ ¤ sup W etfγ (ξx(s)) ds µ(dγ) < ∞. (31) x∈X ΓX 0

Here W is the expectation w.r.t. the Brownian motion ξx.

Proof. For any measurable function g on X with compact support, the Laplace transform of µ has the form Z ethg,γi µ(dγ) = 1 + Γ X Z X∞ 1 ¡ ¢ ¡ ¢ tg(y1) tg(yn) (n) + e − 1 ... e − 1 kµ (y1, ..., yn)dy1...dyn, (32) n! n n=1 X

(n) which follows from formula (107) in Appendix. Here kµ is the n-th correlation function of µ. According to the Ruelle bound (10) the following estimate holds: Z ·Z ¸ ¡ ¢ ethg,γi µ(dγ) ≤ exp a etg(y) − 1 dy . (33) ΓX X The right-hand side of the latter is ﬁnite because g has compact support. Thus Z F(z, t) := etfγ (z) µ(dγ) ΓX µZ ¶ ¡ ¢ ≤ exp a etf(z,y) − 1 dy < ∞. (34) X Moreover, for any z ∈ X, Z Z ¡ ¢ ¡ ¢ etf(z,y) − 1 dy = etf(z,y) − 1 dy X B(z,r) ¯ ¯ ≤ vol B(z, r) max ¯etf(z,y) − 1¯ =: C(t) < ∞. (35)

This implies the estimates

F(z, t) ≤ eaC(t) (36)

9 for any z ∈ X, and Z tfγ (ξx(s)) aC(t) W e µ(dγ) = W F(ξx (s), t) ≤ e . (37) ΓX Then, by Fubini’s theorem, Z ·Z ¸ Z ·Z ¸ t £ ¤ t £ ¤ W etfγ (ξx(s)) ds µ(dγ) = W etfγ (ξx(s)) µ(dγ) ds ΓX 0 0 ΓX ≤ t exp aC(t). (38) The latter estimate is uniform in x, which implies (31). ¤ The following corollary is immediate.

1 Corollary 2 For a.a. γ ∈ ΓX and any x ∈ X, t ∈ R Z t £ ¤ W etfγ (ξx(s)) ds < ∞. (39) 0 Let us consider the diﬀerential equation D ω(s) = −R(p)(ξ (s)) − W (p)(ξ (s))ω(s), (40) ds x γ x ∧p ∧p D ω(0) ∈ (TxX) , in the tensor bundle (TX) , where ds is the covariant derivative along the trajectories of ξx(s) (see [18]). Let f(x, y), x, y ∈ X satisfy the conditions of Theorem 2. We assume that for any γ ∈ ΓX and x ∈ X the following estimate hold:

(p) 2 −(Wγ (x)h, h) ≤ fγ(x) khk , (41) ∧p h ∈ (TxX) . For instance, we can set ° ° °¡ 2 ¢∧p° f(x, y) := −∆xV (x, y) + ° ∇ V (x, y) ° . (42) x ∧p B(TxX )

Theorem 3 A solution ω(t) = ωγ(t) of equation (40) exists for all time t and a.a γ ∈ ΓX and satisﬁes the estimate Z t £ ¤ tcp 1 tfγ (ξx(s)) W kω(t)k ∧p ≤ kω(0)k ∧p e W e ds, T X (TxX) ( ξx(t) ) t 0 (43) ° ° (p) 1 where cp = − infx∈X °R (x)° , t ∈ R .

10 Proof. We use arguments similar to [27, Th. 5.1] (in fact, our situation is simpler). We have D ³ ´ kω(t)k2 = −2 Wf(p)(ξ (t)) ω(t), ω(t) , (44) dt γ x

(p) (p) (p) where Wfγ = R + Wγ , or ³ ´ f(p) D Wγ (ξx (t)) ω(t), ω(t) kω(t)k2 = −2 kω(t)k2 , (45) dt kω(t)k2 which implies  ³ ´  Z f(p) t Wγ (ξx (s)) ω(t), ω(s) 2 2   kω(t)k = kω(0)k exp − 2 2 ds 0 kω(s)k

Z t 2 tcp ≤ kω(0)k e exp 2fγ(ξx (s)) ds. (46) 0 Then Z t tcp W kω(t)k ≤ kω(0)k e W exp fγ(ξx (s)) ds Z 0 1 t £ ¤ ≤ kω(0)k etcp W etfγ (ξx(s)) ds (47) t 0 by Jensen’s inequality, which together with (39) implies the result. ¤

Thus, for a.a. γ ∈ ΓX , the equation (40) deﬁnes an evolution operator family ¡ ¢ U (p) (s) ∈ B (T X)∧p, (T X)∧p (48) ξx,γ x ξx(s) by the formula

U (p) (s)ω(0) = ω(s). (49) ξx,γ It satisﬁes the estimate ° ° Z t ° (p) ° 1 £ ¤ W U (t) ≤ W etfγ (ξx(s)) ds. (50) ° ξx,γ ° t 0

11 Remark 1 The mapping ΓX 3 γ 7→ f(γ) := W kωγ(t)k ∧p is measur- (Tξx(t)X) able. Indeed, let Λ be a compact in X. Then, for γ = {x1, ..., xm} ⊂ Λ, the solution of (40) depends continuously (in the mean sense) on x1, ..., xm ∈ Λ. The latter follows from the general theory of stochastic diﬀerential equations depending on parameters. Thus the cylinder function fΛ(γ) := f(Λ ∩ γ) is continuous on ΓX [6]. It can be shown by the argumets similar to those used in the proof of the theorem above that fΛ(γ) → f(γ) as Λ ↑ X, which implies the measurability of f. Thus the left-hand side of (50) is measurable as a function of γ.

Remark 2 In the case where V ≡ 0 the operator U (p)(s) coincides with the ξx parallel translation along ξx, see [18].

Remark 3 Let ξx,y(s), s ∈ [0, t], be the Brownian bridge from x to y. Then ¡ ¢ U (p) (t) ∈ B (T X)∧p , (T X)∧p (51) ξx,y,γ x y and

° ° Z t ° (p) ° 1 £ ¤ W U (t) ≤ etcp W etfγ (ξx,y(s)) ds. (52) ° ξx,y,γ ° t 0

(p) Let us consider the semigroup e−tHγ in L2Ωp(X). Let K(x, y; t) be the heat kernel on X. It is known [17] that K is a strictly positive C∞ function on [0, ∞) × X × X.

(p) −tHγ Theorem 4 For any t ∈ R and γ ∈ ΓX the semigroup e has an integral kernel

(p) ¡ ∧p ∧p¢ Kγ (x, y; t) ∈ B (TxX) , (TyX) , (53) which satisﬁes the relation

³ ´∗ K(p)(x, y; t) = K(x, y; t)W U (p) (t) , (54) γ ξx,y,γ x, y ∈ X.

12 (p) ∞ Proof. Let us recall that Hγ is essentially self-adjoint on C0 (X). Thus, ∞ for ω ∈ C0 (X), we have the following probabilistic representation of the (p) semigroup e−tHγ : D E D E (p) (p) e−tHγ ω(x), ν = W ω(ξ (t)),U (t)ν (55) x ξx,γ

∧p [18], where ν ∈ (TxX) . This can be rewritten as D E Z D E (p) (p) e−tHγ ω(x), ν = K(x, y; t)W ω(ξ (t)),U (t)ν dy x,y ξx,y,γ ZX D E = K(x, y; t)W ω(y),U (p) (t)ν dy (56) ξx,y,γ ¿XZ À ³ ´∗ = K(x, y; t)W U (p) (t) ω(y)dy, ν , ξx,y,γ X which implies (54). ¤ Proof of Theorem 1. Formulae (52) and (54) imply that the following estimate holds for any γ ∈ ΓX and all x ∈ X, t ∈ R : µ ¶ ° ° dim X ° (p) ° tr K(p)(x, x; t) ≤ K(x, x; t)W °U (t)° (57) γ p ξx,x µ ¶ Z dim X 1 t £ ¤ ≤ K(x, x; t)etcp W etfγ (ξx(s)) ds. p t 0 Note that k(x) := K(x, x; t) is G-invariant and C∞, which together with compactness of X/G implies that it is bounded. Formula (31) immediatly implies that the θ-function Z (p) (p) θ (x, t) = tr Kγ (x, x; t) µ(dγ) (58) ΓX is bounded in x ∈ X for any t > 0 and p = 0, 1, ..., dim X . ¤

Remark 4 More precisely, the θ-function satisﬁes the estimate µ ¶ dim X θ(p)(x, t) ≤ etcp K(x, x; t) F(x, t), (59) p where F is the Laplace transform of the measure µ deﬁned by (34).

13 Remark 5 We can also give a lower bound of θ(p)(x, t). Indeed, let g : X × X → R1 be such that for X gγ(x) := hgx, γi = g(x, y), (60) y∈γ any γ ∈ ΓX and x ∈ X, the estimate

(p) 2 −(Wγ (x)h, h) ≥ gγ(x) khk (61)

∧p hold for all h ∈ (TxX) . It follows then from (46) that

° ° R ° (p) ° tb t g (ξ (s))ds W U (t) ≥ e p We 0 γ x,y (62) ° ξx,y ° for all t ∈ R1, and, consequently, µ ¶ Z R (p) d tb(p) t g (ξ (s))ds θ (x, t) ≥ e K(x, x; t) W e 0 γ x,y µ(dγ), (63) p ΓX ° ° ° (p) ° where bp = − supx∈X R (x) . For instance, we can set ° ° 2 °¡ 2 ¢∧p° gx(y) := − k∇xV (x, y)k − ∆xV (x, y) − ° ∇ V (x, y) ° . x ∧p B(TxX ) (64)

Remark 6 If X is a symmetric space, that is, there exists a group of isometries acting on X transitively, then both K(x, x; t) and F(x, t) do not depend of x, and the estimate (59) gets the form µ ¶ dim X θ(p)(x, t) ≤ etcp k(t), (65) p where k(t) := K(x, x; t)F(x, t).

Example 1 Euclidean space. Let X = Rd, G = Zd. Then R(p)(x) = 0 and −d/2 Kt(x, x) = (4πt) . Formula (65) can be rewritten in the form µ ¶ d θ(p)(x, t) ≤ (4πt)−d/2 F(0, t) (66) p

14 Then Z Θ(p)(t) = θ(p)(x, t)dx Td µ ¶ d ≤ (4πt)−d/2 F(0, t). (67) p

Here Td is the d-dimensional torus.

Example 2 Hyperbolic space. Let X = Hd. Then R(p)(x) = −p(d − p) and we have µ ¶ Z Z d 1 t £ ¤ θ(p)(x, t) ≤ etp(d−p)K(x, x; t) W etfγ (ξx(s)) ds µ(dγ), p t ΓX 0 (68) where K is the heat kernel on Hd. It is known that the group SL(d, R) acts transitively on Hd by isometries. Thus, according to Remark 6, the latter estimate obtains the form µ ¶ d θ(p)(x, t) ≤ etp(d−p)k(t). (69) p

5 Von Neumann algebras associated with con- ﬁguration spaces

In this section, we construct a W ∗ (von Neumann) algebra containing the semigroup

(p) (p) −tHγ Tγ,t := e (70) and interpret the theta-function Θ(p)(t) as its trace. We refer to [34] for general notions of the theory of von Neumann algebras. 2 p (p) Let Ug be the action of G in L Ω (X). It follows from (23) that Hγ satisﬁes the commutation relation

(p) −1 (p) UgHγ Ug = Hgγ (71)

15 for any g ∈ G and γ ∈ ΓX , where the action of G on ΓX is deﬁned by formula (p) (9). Obviously, the semigroup Tγ,t and the orthogonal projection

(p) 2 p (p) Pγ : L Ω (X) → Ker Hγ (72) satisfy similar relations:

(p) −1 (p) (p) −1 (p) UgTγ,t Ug = Tgγ,t,UgPγ Ug = Pgγ . (73)

(p) (p) (p) Remark 7 If Hγ commuted with Ug, g ∈ G, then both Tγ,t and Pγ would p 0 belong to the commutant U := {Ug}g∈G, and we would have the equality Z (p) −tHγ (p) TrU e = tr Kγ (t; x, x)dx. (74) X/G This holds however only for γ such that gγ = γ for all g ∈ G, which form a µ-zero set.

Let us consider the space

2 p 2 p LµΩ := L (ΓX × X → T X, dµ ⊗ dx) 2 2 p = L (ΓX , dµ) ⊗ L Ω (X). (75) The diagonal action

ΓX × X 3 (γ, x) 7→ g(γ, x) := (gγ, gx). (76) of the group G on the space ΓX × X generates the action G 3 g 7→ Ug 2 p p 0 2 p in the space of forms LµΩ . We denote by A := {Ug}g∈G ⊂ B(LµΩ ) the commutant of Ug. Next, we introduce the algebra Cp of µ-essentially bounded maps

2 p A :ΓX → B(L Ω (X)) (77) such that

A(gγ) = UgA(γ)Ug−1 (78)

p for any g ∈ G and γ ∈ ΓX . C can be naturally identiﬁed with a subalgebra of Ap. Moreover

p p ∞ 2 p C = A ∩ Lµ (ΓX → B(L Ω (X))) (79)

16 and thus is a W ∗-algebra. p Let A ∈ C and, for any γ ∈ ΓX , denote by aγ(x, y) the integral kernel of A(γ). Deﬁne the functional Z Z

TrA = tr aγ(x, x)µ(dγ)dx. (80) X/G ΓX Theorem 5 Tr is a faithful normal semiﬁnite trace on the W ∗-algebra Cp.

Proof. 1) Let us prove that Tr is cyclic, i.e. for any A, B ∈ Cp such that TrAB is ﬁnite we have

TrAB = TrBA. (81)

Assume without loss of generality that A and B are symmetric. Then + their integral kernels aγ and bγ satisfy the relation aγ(x, y) = aγ(y, x) and + + bγ(x, y) = bγ(y, x) respectively, where n : TyX → TxX is the adjoint of an operator n : TxX → TyX with respect to the Riemannian structure of X. Then Z Z µZ ¶

TrAB = tr aγ(x, y)bγ(y, x)dy µ(dγ)dx ZX/G ZΓX Z X ¡ + +¢ = tr bγ(x, y) aγ(y, x) dyµ(dγ)dx (82) ZX/G ZΓX ZX

= tr (bγ(x, y)aγ(y, x)) dyµ(dγ)dx X/G ΓX X = TrBA.

∗ 2) Let us show that Tr is faithful. AssumeR that Tr(A A) = 0. The integral ∗ + kernel cγ of A A has the form cγ(x, z) = X aγ(y, x) aγ(y, z)dy. We have Z Z Z ∗ + Tr(A A) = tr aγ(y, x) aγ(y, x)dy dx µ(dγ) ZX/G ZΓx Z X ¡ + ¢ = tr aγ(y, x) aγ(y, x) dy dx µ(dγ) = 0, (83) X/G Γx X which implies that

2 ¡ + ¢ |aγ(y, x)| := tr aγ(y, x) aγ(y, x) = 0 (84)

17 and aγ(y, x) = 0 for almost all γ ∈ ΓX , y ∈ X/G and x ∈ X. Since the measures µ(dγ) and dy are G-invariant, using the equality (78) we get Z Z Z 2 0 = |aγ(y, x)| dy µ(dγ) dx X/G Γx X X Z Z Z 2 = |aγ(y, x)| dy µ(dγ) dx g∈G X/G Γx X X Z Z Z 2 = |ag−1γ(y, x)| dy µ(dγ) dx (85) g∈G X/G Γx X X Z Z Z 2 = |aγ(gy, gx)| dy µ(dγ) dx g∈G X/G Γx X X Z Z Z 2 = |aγ(y, gx)| dy µ(dγ) dx X/G Γx X Zg∈GZ Z 2 = |aγ(y, x)| dy µ(dγ) dx, X ΓX X which implies that aγ(y, x) = 0 for almost all γ ∈ ΓX and x, y ∈ X. Thus we have A = 0. 3) Let us show that Tr is normal. We define the operator Z P : A(γ) 7→ A(γ)µ(dγ), (86) ΓX A ∈ Cp. Because of G-invariance of µ we have P(A) ∈ U p. It is known that p U possesses a faithful normal semifinite trace TrU defined by the formula Z

T rU B = tr b(x, x)dx, (87) X/G where b is the integral kernel of B ∈ U (see [8], [20]). Thus, for A ∈ Cp, we have obviously

Tr A = TrU P(A). (88)

The normality of Tr follows now from the normality of TrU and the Lemma below. ¤

18 Lemma 1 The mapping P is normal.

Proof. Let us ﬁrst show that P is a Schwarz mapping, i.e. that

P(A)∗P(A) ≤ P(A∗A) (89) for all A ∈ Cp. We remark that the following esimate holds: Z

kid ⊗ P(A)kCp = k A(γ)µ(dγ)k ≤ ess supkA(γ)k = kAkCp . ΓX (90)

Thus the mapping id ⊗ P is a projection of norm one [35] from W ∗-algebra Cp onto its W ∗-subalgebra 1 ⊗ U p consisting of constant maps

p ΓX 3 γ 7→ 1 ⊗ B ∈ 1 ⊗ U (91)

2 (here 1 is the identity operator in Hilbert space L (ΓX , dµ)), which implies the estimate (89), see [35, Th.1]. It is known that any Schwarz mapping between W ∗-algebras is continouos in the σ-weak topology (that is, normal) if it is continouos in strong topology. Moreover, a stronger statement is true: for Schwarz mappings continuity in weak topology is equivalent to the continuity in σ∗-strong topology, see [33]. Thus we need only to prove that P is strongly continuous, which follows from the estimate Z ¯Z ¯ ¯ ¯2 2 ¯ ¯ kP(A)fkL2Ωp(X) = ¯ Aγf(x)µ(dγ)¯ dx ZX Z ΓX 2 ≤ |Aγf(x)| µ(dγ)dx (92) ZX ZΓX ¯ ¯2 ¯ e ¯ = ¯Aγf(γ, x)¯ µ(dγ)dx X Γ ° °X ° °2 = °Afe° , 2 p LµΩ

2 p e 2 p e where f ∈ L Ω (X) and f ∈ LµΩ , f(γ, x) = f(x). ¤

19 (p) (p) (p) (p) Let us now consider the maps Tt : γ 7→ Tγ,t and P : γ 7→Pγ . Com- (p) p mutation relations (73) imply that Tt, P ∈ C .

Theorem 6 1) For all time t and any p = 0, ..., dim X

(p) (p) TrTt = Θ (t) < ∞. (93)

2) For any p = 0, ..., dim X

TrP(p) < ∞. (94)

3) The following McKean-Singer formula holds for all time t:

dimXX dimXX p (p) p (p) (−1) TrTt = (−1) TrP . (95) p=0 p=0

Proof. 1) Formula (93) follows immediately from (80) and Corollary 1. 2) We have obviously

(p) (p) (p) (p) Tγ,t (I − Pγ ) = Tγ,t − Pγ , (96) or

(p) (p) (p) (p) Tt (I − P ) = Tt − P . (97)

Thus

(p) (p) (p) (p) TrP = TrTt − TrTt (I − P ) < ∞. (98)

3) Formula (95) follows from the McKean-Singer formula in von Neumann L p algebras (see [12, (5.1.10)]), applied to the algebra C = p C and operators P p ∗ P p ∗ L 2 p D := d ,D = (d )γ in p LµΩ . ¤

Remark 8 The right-hand side of formula (95) can be understood as a regularized index of the Dirac operator D + D∗.

20 6 Appendix: Gibbs measures on conﬁgura- tion spaces.

Here we briefly discuss the definition and some properties of Gibbs measures on ΓX , associated with pair potentials. For a detailed exposition see e.g. [6] A pair potential is a measurable symmetric function φ: X × X → R ∪ {+∞} We will also suppose that φ(x, y) ∈ R for x 6= y. For a compact φ Λ ⊂ X, a conditional energy EΛ :ΓX → R ∪ {+∞} is defined by ( P P φ(x, y), if |φ(x, y)| < ∞, φ {x,y}⊂γ, {x,y}∩Λ6=? {x,y}⊂γ, {x,y}∩Λ6=? EΛ(γ) := +∞, otherwise (99)

Given Λ , we deﬁne for γ ∈ Γ and ∆ ∈ Bor(ΓX ), the corresponding Borel σ-algebra of ΓX . z,φ z,φ −1 ΠΛ (γ, ∆):=1{Zz,φ<∞}(γ)[ZΛ (γ)] (100) ZΛ h i 0 φ 0 0 × 1∆(γΛc + γΛ) exp −EΛ(γΛc + γΛ) πz(dγ ), Γ where Z h i z,φ φ 0 0 ZΛ (γ):= exp −EΛ(γΛc + γΛ) πz(dγ ). (101) Γ

A probability measure µ on (ΓX , Bor(ΓX )) is called a grand canonical Gibbs measure with interaction potential φ if it satisﬁes the Dobrushin– Lanford–Ruelle equation z,φ µΠΛ = µ for all compact Λ ⊂ X. (102) Let G(z, φ) denote the set of all such probability measures µ. It can be shown [22] that the unique grand canonical Gibbs measure corresponding to the free case, φ = 0, is the Poisson measure πz. We suppose that the interaction potential φ satisﬁes the following conditions:

(S) (Stability) There exists B ≥ 0 such that, for any Λ ∈ Oc(X) and for all γ ∈ ΓΛ, X φ EΛ(γ):= φ(x, y) ≥ −B|γ|. (103) {x,y}⊂γ

21 (I) (Integrability) We have Z C:= ess sup |e−φ(x,y) − 1| dy < ∞. (104) x∈X X (F) (Finite range) There exists R > 0 such that φ(x, y) = 0 if ρ(x, y) ≥ R. (105)

Theorem 7 [24, 25, 26] 1) Let (S), (I), and (F) hold, and let z > 0 be such that 1 z < (e2BC)−1, (106) 2e where B and C are as in (S) and (I), respectively. Then, there exists a Gibbs measure µ ∈ G(z, φ) such that for any n ∈ N and any measurable symmetric function f (n) :(X)n → [0, ∞], we have Z X (n) f (x1, . . . , xn) µ(dγ) (107) Γ {x ,...,x }⊂γ 1 n Z 1 (n) (n) = f (x1, . . . , xn)kµ (x1, . . . , xn) dx1 ··· dxn, n! (Rd)n

(n) d n where kµ is a non-negative measurable symmetric function on (R ) , called the n-th correlation function of the measure µ, and this function satisfies the following estimate d n (n) n ∀(x1, . . . , xn) ∈ (R ) : kµ (x1, . . . , xn) ≤ a , (108) where a > 0 is independent of n.(the Ruelle bound). 2) Let φ be a non-negative potential which fulfills (I) and (F). Then, for each z > 0, there exists a Gibbs measure µ ∈ G(z, φ) such that the correlation (n) functions kµ of the measure µ satisfy the Ruelle bound (108). Remark 9 Let us assume that the potential φ(x, y) has the form φ(x, y) = Φ(ρ(x, y)) (109) and Φ : R → R such that supp Φ ⊂ [−r, r], where r > 0 is the injectivty radius of X. Then the conditions (S), (I) and (F) are fulfilled. Thus, under the condition (106), the corresponding measure µ exists and satisfies conditions (i), (ii) of Section 2.

22 In the case where X = Rd, the existence of Gibbs measures satisfying the Ruelle bound is known for arbitrary z > 0 under additional conditions of super stability and lower regularity (Ruelle measures [32]). We present two classical examples of potentials φ(x, y) = Φ(x−y) satisfying these conditions. Example 1. (Lennard–Jones type potentials)Φ ∈ C2(Rd \{0}), Φ ≥ 0 d −α c on R , Φ(x) = c|x| for x ∈ B(r1), Φ(x) = 0 for x ∈ B(r2) , where c > 0, α > 0, 0 < r1 < r2 < ∞. Example 2. (Lennard–Jones 6–12 potentials) d = 3, Φ(x) = c(|x|−12 − |x|−6), c > 0.

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Verzeichnis der erschienenen Preprints ab No. 235

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236. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Inverse Spectral Problems for Coupled Oscillating Systems: Reconstruction by Three Spectra

237. Albeverio, Sergio; Cebulla, Christof: Müntz Formula and Zero Free Regions for the Riemann Zeta Function

238. Marinelli, Carlo: The Stochastic Goodwill Problem; erscheint in: European Journal of Operational Research

239. Albeverio, Sergio; Lütkebohmert, Eva: The Price of a European Call Option in a Black- Scholes-Merton Model is given by an Explicit Summable Asymptotic Series

240. Albeverio, Sergio; Lütkebohmert, Eva: Asian Option Pricing in a Lévy Black-Scholes Setting

241. Albeverio, Sergio; Yoshida, Minoru W.: Multiple Stochastic Integrals Construction of non-Gaussian Reflection Positive Generalized Random Fields

242. Albeverio, Sergio; Bernabei, M. Simonetta; Röckner, Michael; Yoshida, Minoru W.: Homogenization of Diffusions on the Lattice Z d with Periodic Drift Coefficients; Application to Logarithmic Sobolev Inequality

243. Albeverio, Sergio; Konstantinov, Alexei Yu.: On the Absolutely Continuous Spectrum of Block Operator Matrices

244. Albeverio, Sergio; Liang, Song: A Remark on the Nonequivalence of the Time-Zero \phi_3^4-Measure with the Free Field Measure

245. Albeverio, Sergio; Liang, Song; Zegarlinski, Boguslav: A Remark on the Integration by Parts Formula for the \phi_3^4-Quantum Field Model

246. Grün, Günther; Mecke, Klaus; Rauscher, Markus: Thin-Film Flow Influenced by Thermal Noise

247. Albeverio, Sergio; Liang, Song: A Note on the Renormalized Square of Free Quantum Fields in Space-Time Dimension d > 4

248. Griebel, Michael: Sparse Grids and Related Approximation Schemes for Higher Dimensional Problems; erscheint in: Proc. Foundations of Computational Mathematics 2005

249. Albeverio, Sergio; Jushenko, Ekaterina; Proskurin, Daniil; Samoilenko, Yurii: *-Wildness of Some Classes of C*-Algebras

250. Albeverio, Sergio; Ostrovskyi, Vasyl; Samoilenko, Yurii: On *-Representations of a Certain Class of Algebras Related to Extended Dynkin Graphs

251. Albeverio, Sergio; Ostrovskyi, Vasyl; Samoilenko, Yurii: On Functions on Graphs and Representations of a Certain Class of *-Algebras

252. Holtz, Markus; Kunoth, Angela: B-Spline-Based Monotone Multigrid Methods

253. Albeverio, Sergio; Kuzhel, Sergej; Nizhnik, Leonid: Singularly Perturbed Self-Adjoint Operators in Scales of Hilbert Spaces

254. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: On Spectra of Non-Self-Adjoint Sturm-Liouville Operators

255. Albeverio, Sergio; Nizhnik, Leonid: Schrödinger Operators with Nonlocal Point Interactions

256. Albeverio, Sergio; Alimov, Shavkat: On One Time-Optimal Control Problem Associated with the Heat Exchange Process; erscheint in: Applied Mathematics and Optimization

257. Albeverio, Sergio; Pustylnikov, Lev D.: Some Properties of Dirichlet L-Functions Associated with their Nontrivial Zeros II

258. Abels, Helmut; Kassmann, Moritz: An Analytic Approach to Purely Nonlocal Bellman Equations Arising in Models of Stochastic Control

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