arXiv:math/9911042v1 [math.OA] 8 Nov 1999 INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS RYSZARD NEST AND FLORIN RADULESCU Abstract. Let Γ be a discrete subgroup of P SL(2, R) of infinite covolume with infinite conjugacy classes. Let Ht be the Hilbert space consisting of analytic functions in L2 (D, (Im z)t−2 dzdz) and let, for t > 1, πt denote the corresponding projective unitary representation of P SL(2, R) on this Hilbert space. We denote by At the II∞ factor given by the commutant of πt (Γ) in B(Ht ). Let F denote a fundamental domain for Γ in D and assume that t > 5. ∂M = ∂D ∩ F is given the topology of disjoint union of its connected components. Suppose that f is a continuous Γ-invariant function on D whose restriction to F extends to a continuous function on F and such that f |∂M is an invertible element of C0 (∂M )˜. Let Tft = Pt Mf Pt denote the Toeplitz operator with symbol f . Then Tft is Fredholm, in the Breuer sense, with respect to the II∞ factor At and, moreover, its Breuer index is equal to the total winding number of f on ∂M . Contents 1. Introduction 2. Some results on Toeplitz operators 3. Γ-invariant Toeplitz operators 4. Γ-Fredholm operators References 1 5 8 14 16 1. Introduction In this paper we study equivariant Toeplitz operators acting on the Hilbert space Ht consisting of all square summable analytic functions in L2 (D, (Im z)t−2 dzdz). Let us first recall that the classical theory of Toeplitz operators in the unit disc yields an extension of C*-algebras 0 → K → T → C(∂D) → 0, 1 2 R. NEST AND F. RADULESCU where K denotes the algebra of compact operators on H2 and T the Toeplitz C*-algebra generated by compressions Tf to H2 of multiplication operators (by f ’s fom C(D)). In particular, for f |∂D invertible, Tf is Fredholm and the boundary map for the K-theory six term exact sequence of this extension is equivalent, via index theorem for Toeplitz operators, to the equality: Index (Tf ) = winding number of f |∂D . As it turns out, all of these facts admit suitable generalisation to the equivariant case. Let Γ be a fuchsian subgroup of P SL(2, R), which has infinite conjugacy classes and is of infinite covolume. Recall that the action of PSL(2, R) on D by fractional linear transformations lifts to projective unitary representations of PSL(2, R) on these Hilbert spaces (cf. ([21], [19])) and the commutant of πt (Γ) is a II∞ factor. We will denote by ′ At the commutant πt (Γ) ∩ B(Ht ) and by τ the normal positive nonzero trace on At . If σt denotes the 2-group cocycle corresponding to the projective unitary representation πt , then At is isomorphic ([20]) to L(Γ, σt ) ⊗ B(K), where K is an infinite dimensional separable Hilbert space and L(Γ, σt ) is the twisted group von Neumann algebra of Γ. Let F denote a fundamental domain for the action of Γ on D. We will denote by M the quotient space D/Γ and by ∂M its boundary: ∂M = (∂D ∩ F )/Γ equipped with the topology of the disjoint union of its connected components, i. e. of a countable union of disjoint circles. In particular M = F /Γ = M ∪ ∂M inherits a structure of locally compact space. C0 (∂M)˜ denotes the unitalisation of the C*-algebra of continuous, vanishing at infinity functions on ∂M. For any continuous Γ-invariant function f on D which extends to a continuous function on M we denote by Tf the Toeplitz operator on Ht with symbol f , i. e. the compression to Ht of the operator of multiplication by f on L2 (D, (Im z)t−2 dzdz). Because of the Γ invariance of the symbol such a Tf belongs to At = {πt (Γ)}′ ([20]). Suppose that t > 5 and that f |∂M is an invertible element of C0 (∂M)˜. We prove below that Tf is Fredholm (in the sense of Breuer ([6]), in At . Moreover the Breuer index is (in analogy with the classical case) equal to the winding number of f |∂M : ∂M → C \ {0}. This can be seen as an analogue of Atiyah’s index formula for coverings ([2]). The organisation of the paper is as follows. INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 3 In Section 2 we gather some more or less known results about nuclearity properties of Toeplitz operators on Ht and prove the main technical result: Let t > 5 and f, g ∈ L∞ (D) are given. Suppose that g ∈ C ∞ (D) and that inf {||z − ξ||| | z ∈ supp f and ξ ∈supp g} > ǫ for some positive number ǫ. Then both Tf Tg and Tg Tf are of trace class and Tr([Tf , Tg ]) = 0. (cf. Theorem 2 ). In Section 3 we study the L1 (τ )-properties of commutators of Toeplitz operators with Γ-invariant symbol and prove the following result. Suppose that t > 5 and f and g are Γ-invariant functions on D which are smooth on the closure of a fundamental domain for Γ. Then both [Tf , Tg ] and Tf g − Tf Tg are in M ∩ L1 (τ ) and Z 1 τ ([Tf , Tg ]) = df dg 2πi F (cf. Theorem 3 and the remarks following). Let TΓ be the C*-subalgebra of At generated by Toeplitz operators Tf with f Γ-invariant and smooth on a fundamental domain for Γ, KΓ be the C*-ideal generated by the L1 (τ )-elements in TΓ and M = D/Γ. In Section 4 we construct the extension 0 → KΓ → TΓ → C(∂M) → 0. Tf → f |∂M Let ∂ : K1 (∂M) → K0 (KΓ ) denote the boundary map in K-theory associated to this extension. We prove that, for Tf ∈ TΓ with f invertible on the boundary of M, the following equality holds: hτ , ∂[Tf ]i = winding number of f on ∂M (cf. Theorem 4). Remark 1. For notational simplicity we work throughout the paper with the case when the number of boundary components of a fundamental domain of Γ is finite. The only difference (except for typografical complications) in the general case consists of replacing the above extension of C(∂M) by KΓ by the extension: 0 → KΓ → TΓ0 → C0 (∂M) → 0 where TΓ0 stands for the (in general nonunital) C*-algebra generated by Tf with f continuous on F and in with non-zero values on finitely many components of ∂D ∩ F . 4 R. NEST AND F. RADULESCU The method of the proof are based on the equivariant Berezin’s quantization theory for such groups ([20], [17]). Let F be, as above, a fundamental domain for the action of Γ in D. Then for every Γ− equivariant, bounded function g on D, having compact support in the interior of F , the Toeplitz operator Tg ∈ At is inRL1 (At ) and has trace equal to a universal constant times the integral F g(z)(Im )−2 dzdz Moreover we will show that the commutator of two Toeplitz opertors, having symbols that are smooth and continuous on the closure of F , belongs to trace ideal of the II∞ factor. In particular if the symbol is invertible in the neighbourhood of the intersection of the boundary of D with the closure of F , the operator is Fredholm in At in Breuer’s sense ([6]). To identify the Breuer index of such a Toeplitz operator we use the Carey-Pincus theory ([8]. Let us first recall the pertinent facts. Given an operator A ∈ At such that τ [A∗ , A] < ∞, the bilinear map C[z, z] ∋ P, Q → τ [P (A∗ , A), Q(A∗ , A))] defines a cyclic one-cocycle on the algebra of polynomials (in two real variables), of the form Z (P, Q) → {P, Q}dµ(z, z), Z where Z is the of spectrum of the class of A in At /KΓ . dµ, called the principal function of A, is a finite measure on the complex plane having ′ the property that, for any connected component Z of C \ Z, dµ|Z ′ = −cπdzdz with a constant c equal to the value of τ on the index class of ∂[A − ′ λ] ∈ K0 (KΓ ) for any λ ∈ Z . Hence one has to determine the pricipal function, given by the τ -values on commutators of polynomials in Tf and its adjoint (like in the classical case in [7], [15]). To deal with the computation of those we apply the spatial theory of von Neumann algebras ([9]). One of its basic constructions gives an operator-valued weight E : B(Ht ) → At such that, for a trace-class operator A0 in the domain of E, τ ◦ E(A0 ) = T r(A0 ). This allowes one to replace the computation of values of τ on Γ-invariant operators by computation of value of the classical trace on certain trace-class operators on Ht . In fact we prove in Section 3 that commutators of Γequivariant Toeplitz operators are of the form E(A0 ) for A0 given by a suitable (trace class) commutator of polynomials in Toeplitz operators and hence the computation reduce to the classical case. But in this case the principal function for a pair of Toeplitz operators that commute modulo the trace ideal in B(Ht ) is well understood - it INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 5 is basically given by the fact that the index of Tz is equal to one ([1]) and gives explicit formulas that lead to the results stated above. 2. Some results on Toeplitz operators Let D denote the unit circle in the complex plane and set dz̄dz dµt (z) = (1 − |z|2 )t (1) (1 − |z|2 )2 We set Ht = {f ∈ L2 (D, dµt ) | f holomorphic on D} (2) As is well known, Ht is a closed subspace of L2 (D, dµt) and the orthogonal projection Pt : L2 (D, dµt ) → Ht is called the Toeplitz projection. Given a function f ∈ L∞ (D, dµt ) we denote by Mf the operator of multiplication by f on L2 (D, dµt) and set • the Toeplitz operator associated to f : Tf = Pt Mf Pt • the Henkel operator associated to f : Hf = (1 − Pt )Mf¯Pt Note, for future computations, that Tf is an integral operator on L2 (D, dµt ) with integral kernel t − 1 f (ξ) Kf (z, ξ) = (3) ¯t 2πi (1 − z ξ) and Hf∗¯Hg = Tf g − Tf Tg . (4) We will denote by δ the absolute value of the cosine of hyperbolic distance on D, i. e. δ(a, b) = (5) (1 − ||a||2 )(1 − ||b||2 ) . |1 − ab̄|2 As is well known, ||Tf || = ||f ||∞ (6) and (7) ||f ||2S2 = ||Hf ||22 + ||Hf¯||22 t−1 2 ) =( 2πi Z |f (a) − f (b)|2 δ t (a, b)dµ0 (a, b). D×D In particular, since both Hz and Hz̄ are of finite rank, the function f (a, b) = (a − b) is square integrable with respect to the measure 6 R. NEST AND F. RADULESCU δ t (a, b)dµ0 (a, b) and hence all functions which are Lipschitz with exponent one on D have finite S2 -norm. We let PSL(2,R) act on D by fractional linear transformations and denote by πt the induced projective unitary representation on L2 (D, dµt) (and Ht ). Both dµ0 and δ(a, b) are PSL(2,R)-invariant, which gives a useful formula Z Z 4π t δ (a, b)dµ0 (a) = δ t (a, 0)dµ0(a) = (8) . t−1 D D The following result is probably well known to specialists, however, since we do not have a ready reference, so we will include the proof below. Theorem 1. Let f and g belong to C ∞ (D). Then Tf Tg −Tf g is a trace class operator and, moreover, Z t−1 2 T r([Tf , Tg ]) = ( (f (a)g(b) − f (b)g(a))δ t (a, b)dµ0 (a, b) ) 2πi D×D Z Z 1 1 df dg = f dg. = 2πi D 2πi ∂D Proof. By the smoothness assumption, both f and g belong to the S2 class and hence Tf Tg −Tf g is a trace class operator. For the computation of the trace we can just as well assume that both f and g are realvalued. To begin with, for a real-valued function f , (7) gives Z 1 t−1 2 2 (f (a) − f (b))2 δ t (a, b)dµ0 ) T r(Tf 2 − Tf ) = ( 2 2πi D×D and hence, by an application of the polarisation identity, Z t−1 2 T r(Tf g − Tf Tg ) = ( f (a)(g(a) − g(b))δ t (a, b)dµ0 ) 2πi D×D which implies immediately the first equality. To get the second equality recall that, for a pair of (non-commutative) polynomials P (z̄, z) and Q(z̄, z), the Carey-Pincus formula holds: Z Z 1 1 ∗ ∗ dP dQ = P dQ, T r([P (Tz , Tz ), Q(Tz , Tz )]) = 2πi D 2πi ∂D see [7]. Since P (Tz∗, Tz ) = TP mod L1 (T r), this implies the second equality for f and g polynomial. Approximating arbitrary pair of smooth functions uniformly with their first derivatives on D completes the proof of the second equality. The following is the main technical result of this section INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 7 Theorem 2. Let t > 5 and f, g ∈ L∞ (D) are given. Suppose moreover, that g ∈ C ∞ (D) and that inf {||z − ξ||| | z ∈ supp f and ξ ∈supp g} > ǫ for some positive number ǫ. Then both Tf Tg and Tg Tf are of trace class and Tr([Tf , Tg ]) = 0. Proof. We use ∂ to denote the unbounded operator Ht ∋ h → ∂z h ∈ Ht . defined on the subspace of holomorphic functions h such that their first derivative is smooth up to the boundary of the disc and in Ht , and by ∂ −1 the unique extension to a bounded operator on Ht of (9) zn → 1 n+1 z . n+1 It is easy to see that ∂ −1 is Hilbert-Schmidt, in fact, since ||z n ||2 ∼ O(1) as n → ∞, the characteristic values of ∂ −1 are of the order O(n−1 ). Moreover Id −∂∂ −1 is of finite rank. Since we can write Tf Tg |C[z] = Tf Tg ∂∂ −1 + Tf Tg (1 − ∂∂ −1 ), to prove that Tf Tg is trace class it is sufficient to show that the densely defined operator Tf Tg ∂ has a (unique) extension to a Hilbert-Schmidt operator on Ht . Suppose first that h and ∂h both belong to Ht and are smooth up to the boundary of D. Given an a ∈ / supp(g), we have R g(b) dµt (b) (1−a ∂ h(b) (Tg ∂h)(a) = t−1 2πi D b̄)t b R 2 )t−2 g(b)(1−|b| = − t−1 dhdb̄ 2πi D  (1−ab̄)t  R 2 )t−2 g(b)(1−|b| hdb̄, ∂ = t−1 b t 2πi D (1−ab̄) where we used Stokes theorem and the fact that the integrand is smooth and vanishes at the boundary of D. But this implies that the densely defined operator Tf Tg ∂ is in fact given by an integral operator with kernel Z (1 − |a|2 )t−2 (1 − |b|2 )t−3 F (a, b) K(z, ξ) = const dλ(a, b) ¯ t (1 − āz)t (1 − ξb) (1 − b̄a)t D×D where dλ is the Lebesque measure on D × D and F is an L∞ -function which vanishes on a neighbourhood of the diagonal in D × D given by {(a, b) | |a − b| > ǫ}. In particular, sup | a,b F (a, b) |<∞ (1 − b̄a)t 8 R. NEST AND F. RADULESCU and, by Cauchy-Schwartz inequality, Z (1 − |a|2 )t−2 (1 − |b|2 )t−3 2 2 2 |. dλ(a, b)| |K(z, ξ)| ≤ const(V ol (D, dλ) ¯ t (1 − āz)t (1 − ξb) D×D To estimate the L2 -norm of K(z, ξ) we can first integrate over z and ξ which gives, in view of (8), the estimate Z 2 ||K||2 ≤ const (1 − |a|2 )t−4 (1 − |b|2 )t−6 dλ(a, b) D×D which is finite for t > 5. To finish the proof Tg Tf = (Tf¯Tḡ )∗ and hence is also trace class by applying the above argument to Tf¯Tḡ . As a direct consequence we get T r[Tf , Tg ] = 0. 3. Γ-invariant Toeplitz operators Let Γ be a countable, icc and discrete subgroup of PSL(2,R). ′′ The von Neumann algebra (πt (Γ) ) is a II1 factor with unique normal ′ normalized trace τ given by ′ τ (πt (γ)) = 0 ′ for γ 6= e. Its commutant At = (πt (Γ)) is a factor of type II. We will assume from now on that Γ has infinite covolume in D, i. e. M = D/Γ is an open Riemannian surface which can (and will) be thought of as an open subset of an ambient closed Riemannian surface N. We assume moreover that M has finitely many boundary components (the boundary in N), hence ∂M = ∪i Ci , a finite union of disjoint smooth simple closed contractible curves in N. In this case M is a II∞ factor with a unique (up to the normalisation) positive normal trace τ . By general theory (see [14], [9],[12]) there exists a unique, normal, semifinite operator-valued weight E : B(Ht ) → M such that, for A ∈ L1 (T r) in the domain of E, τ ◦ E(A) = T r(A). E is uniquely determined by the equality of normal linear functionals m → τ (E(A)m) = T r(Am) for A ∈ L1 (T r) and m ∈ At . Below we list some of the properties of E used later. INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 9 A vector ξ ∈ Ht is called Γ-bounded if the densely defined map Rξ X cγ πt (γ −1 )ξ ∈ Ht l2 (Γ) ∋ {cγ }γ∈Γ → γ∈Γ is bounded. Let pξ denote the orthogonal projection onto the one dimensional subspace spanned by vector ξ. Then it is easy to see that X Rξ Rξ∗ = πt (γ)Pξ πt (γ −1 ) γ∈Γ P i. e. ξ is Γ-bounded precisely in the case when the sum γ∈Γ πt (γ)pξ πt (γ −1 ) converges in the strong operator topology to a bounded oprator on Ht , in fact equal to E(pξ ) and in this case τ (E(pξ )) = T r(pξ ) = 1. Let us introduce the following. Definition 1. A bounded operator A is called Γ-bounded if the sums X πt (γ)Aπt (γ −1 ) γ∈Γ converge in the strong operator topology. Let A be a positive trace class operator of the form X Ax = λi pξi i where {ξi } is an orthonormal system in Ht . A is in the domain of E if it is Γ-bounded and in this case X E(A) = πt (γ)Aπt (γ −1 ) and τ (E(A)) = T rA. γ∈Γ Proposition 1. Let f0 be an L∞ function D satisfying the conditions: • the euclidean distance from the essential support of f0 to ∂D is strictly positive; P • the sum γ∈Γ f0 ◦ γ is locally finite. P If, moreover, t > 2, the associated Γ-invariant L∞ -function f = γ∈Γ f0 ◦ γ on D satisfies Tf ∈ At ∩ L1 (τ ) Proof. Since f is Γ-invariant and in L∞ (D), Tf ∈ At . for the rest of the claim it is sufficient to look at f0 positive. But then Tf0 is a positive operator with smooth kernel, hence, by Lidskii theorem, it is of trace class. By the second assumption it is Γ-bounded and hence, according to the remarks above, it is in the domain of E and E(Tf0 ) = TP γ f0 ◦γ = Tf . 10 R. NEST AND F. RADULESCU In particular τ (Tf ) = τ (E(Tf0 )) = T r(Tf0 ) < ∞ as claimed. Let us introduce some notation connected with fundamental domains for the action of Γ on D. Suppose we choose points Pi on ∂M, one on each connected component Ci . We’ll call this a cut of M. To each such cut we can associate a fundamental domain F such that the chosen points are in bijective correspondence with end-points of the intervals F ∩ ∂D From now on F will (unless explicitly stated to the contrary) denote a generic fundamental domain for Γ on D. Our goal is to compute the τ -trace of commutators of the form [Tf , Tg ], where f and g are sufficiently general Γ-invariant functions on D. To see what is the problem, suppose first that f0 , g0 ∈ C ∞ (F ) P satisfy suppf0 ⊂ F int and suppg0 ⊂ F int . Let f = γ∈Γ f0 ◦ γ and P g = γ∈Γ g0 ◦ γ be the corresponding Γ-invariant function on D. Looking at kernels, we obtain that ([20], [17]) Z 1 t−1 2 2 τ (Tf 2 − Tf ) = ( (f (a) − f (b))2 δ t (a, b)dµ0 ) 2 2πi D×F and hence, by an application of the polarisation identity, Z t−1 2 f (a)(g(a) − g(b))δ t (a, b)dµ0 . ) τ (Tf g − Tf Tg ) = ( 2πi D×F Note that the right hand side is bounded by Z 1 t−1 2 |a − b|2 δ t (a, b)dµ0 , ( ) 2 2πi D×F which is convergent by [1]. Hence Z t−1 2 (f (a)g(b) − f (b)g(a))δ t (a, b)dµ0 , τ ([Tf , Tg ]) = ( ) 2πi D×F Consequently, since the formula in Theorem 1 extends by continuity for functions f, g that are smooth and Γ-invariant (by replacing D × D by D × F ), it follows that, for such f and g, Z Z Z t−1 2 1 1 t ( (f (a)g(b)−f (b)g(a))δ (a, b)dµ0 = d(f0 )dg = df dg. ) 2πi 2πi D 2πi F D×F But it is not obvious from the outset neither that τ ([Tf , Tg ]) is in the domain of τ nor that its trace is approximated by the trace of commutators of Toeplitz operators associated to functions of the form INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS P f = γ∈Γ f0 ◦ γ and g = from the boundary! P γ∈Γ 11 g0 ◦ γ with f0 and g0 supported away Theorem 3. Let f and g be two smooth functions on M (i.e. continuous with all their derivatives up to the boundary of M). We will use the same notation to denote their representatives as Γ-invariant functions on D. Suppose that t > 5. Then [Tf , Tg ] is in M ∩ L1 (τ ) and Z 1 τ ([Tf , Tg ]) = df dg. 2πi F Proof. Let us begin with the following observations. 1. Suppose that h0 , . . . hn is a finite family of functions on D satisfying the conditions of the proposition 1 and we set A = Th0 . . . Thn . Since ! Y |A∗ |2 ≤ ||hi ||∞ T|h0 |2 , i6=0 ∗ 2 the averages γ πt (γ)|A | πt (γ −1 ) converge in the strong operator topology to E(|A∗ |2 ). Moreover, for any normal linear functional ψ on B(Ht ), X ψ(πt (γ)|A∗ |2 πt (γ −1 )) = ψ(E(|A∗ |2 )). P γ To see the equality it is sufficient to consider positive ψ, but then all that is involved is an exchange of the order of summation for a double series consisting of positive terms. P 2. There exists a smooth partition of unity of D of the form i φi where φi are smooth, positive functions such that, for each i, the family of functions {φi ◦ γ}γ∈Γ is localy finite. To see this it is sufficient to notice that, for any disc Dǫ = {z ∈ D||z| ≤ ǫ} with ǫ < 1, the number of γ ∈ Γ such that γ(Dǫ )∩Dǫ 6= ∅ is finite. This follows from the fact that Γ is discrete as a subgroup of PSL(2,R. This implies that the set of normal linear functionals m → T r(BmA∗ ) with A and B as above is total in B(Ht )∗ . The proof of the theorem will be done in two steps. First part. Suppose that we are given a fundamental domain F for Γ such that X X f= f0 ◦ γ, g = g0 ◦ γ γ γ 12 R. NEST AND F. RADULESCU where f0 and g0 are both smooth on D and their supports have positive Euclidean distance to the complement of F in D. We will compute T r(A∗ [Tf , Tg ]A) where A = Th0 . . . Thn . The operator under trace has a smooth kernel and the integral of its restriction to the diagonal has (up to a constant) the form Z Z Z Z A(a, d)(f (b)g(c) − g(b)f (c)) dµt (d) dµt (c) dµt (b) dµt (a) ¯ t (1 − āb)t (1 − b̄c)t (1 − c̄d)t (1 − da) D D D D where A(a, d) is a smooth kernel with support of strictly positive euclidean distance from ∂D × D ∪ D × ∂D. Hence the function A(a, d)(f (b)g(c) − g(b)f (c)) F (a, b, c, d) = (1−|b|2 )t/2 (1−|c|2 )t/2 ¯ t (1 − āb)t (1 − b̄c)t (1 − c̄d)t (1 − da) is uniformly bounded on D4 (the only singularity in the denominator appears for b = c and it is controlled by the fact that |δ| ≤ 1) and our integral can be written as the integral of L∞ -function F (a, b, c, d) with respect to the finite measure dω = dµt ⊗ dµt/2 ⊗ dµt/2 ⊗ dµt . Hence Z F dω = XZ γ∈Γ i. e. T r(A∗ [Tf , Tg ]A) = X F dω, D×γ(F )×D×D T r(A∗ (Tf0 ◦γ Tg − Tg0 ◦γ Tf )A). γ Since Tf0 Tg − Tg0 Tf is of trace class, we can exchange the summation over γ ∈ Γ with the trace and get the identity ! X T r(A∗ [Tf , Tg ]A) = T r πt (γ)AA∗ πt (γ)−1 (Tf0 Tg − Tg0 Tf ) . γ But this shows that τ (E(|A∗ |2 )[Tf , Tg ]) = τ (E(|A∗ |2 )E(Tf0 Tg − Tg0 Tf )). Since for any m ∈ At τ (E(AA∗ )m) = T r((AA∗ )m) and the set of such linear functionals is separating for B(Ht ), we get E(Tf0 Tg − Tg0 Tf ) = E(Tf Tg0 − Tg Tf0 ) = [Tf , Tg ]. But, since Tf Tg0 − Tg Tf0 = [Tf0 , Tg0 ] + Tf −f0 Tg0 − Tg−g0 Tf0 INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 13 is of trace class, [Tf , Tg ] is in M ∩ L1 (τ ) and 1 τ ([Tf , Tg ]) = T r([Tf0 , Tg ] − [Tg0 , Tf ]) 2 By the theorem 3 Z Z 1 1 d(f0 )dg = df dg. τ ([Tf , Tg ]) = 2πi D 2πi F Second part. By the proposition 1 we can assume that both f and g are, as functions on M , supported on a neighbourhood of the boundary of M diffeomorphic to (∪i Ci )×] − ǫ, 0]. Now, using partition of unity, we can split both f and g into finite sums X X f= fk , g = gs s k i so that for any pair of indices (k, s) there are open intervals Ik,s of nonzero length on each of the boundary components Ci such that both fk i and gs vanish on (∪i Ik,s ×]) − ǫ, 0] - possibly with a smaller, but still positive value of ǫ. But then, choosing a cut of M given by a choice i of points Pi in the interior of Ik,s will provide us with a fundamental domain Fk,s such that the conditions of the first part of this proof hold for (fs , gk , Fk,s) and hence [Tfk , Tgs ] ∈ L1 (τ ) and Z 1 τ ([Tfk , Tgs ]) = df dg 2πi Fk,s To complete the proof note that the expression df dg for Γ-invariant R functions is Γ-invariant, hence the integral F df dg is independent on the choice of the fundamental domain and the result follows. Corollary 1. Suppose that f and g are smooth functions on M . Then Z τ ([Tf , Tg ]) = f dg. ∂M Proof. This follows immediately from R the fact that under the natural diffeomorphism F \ ∂F the integral df dg becomes identified with F R df dg and the Stokes theorem. M Remarks 1. Virtually the same proof shows that, for f and g smooth on M, the operator Tf Tg − Tf g is in M ∩ L1 (τ ) and Z t−1 2 f (a)(g(b) − g(a))δ t (a, b)dµ0 (a, b). ) τ (Tf Tg − Tf g ) = ( 2πi D×F 14 R. NEST AND F. RADULESCU 2. All of the results above can be easily extended to the case when f and g are in Li nf ty(M) and Lipschitz with exponent one in a tubular neighbourhood of ∂M in M . 4. Γ-Fredholm operators Let TΓ denote the C*-subalgebra of At generated by Toeplitz operators Tf with Toepltz symbol f ∈ C(M ) and denote by KΓ the C*-ideal generated by elements in L1 (τ ) ∩ TΓ . An element A of At is called Γ-Fredholm if it has an inverse, say R, modulo KΓ and, in this case, the commutator [A, R] has well-defined trace Γ-index of A = τ ([A, R]) which depends only on the class of A in K1 (TΓ /KΓ). Remark 2. The number” Γ-index of A” is also known as Brauer index of A. According to the proposition 1, there exists a surjective continuous map (10) q : C(∂M) → TΓ /KΓ sending function f |∂M to Tf mod KΓ - this map is well defined since ||Tf || = ||f ||∞. Theorem 4. Let Γ be a countable, discrete, icc subgroup of PSL(2,R) such that D/Γ has infinite covolume and M = D/Γ is an open Riemannian surface with finitely many boundary components. Assume that t > 5 and that the trace τ on At is normalized by its value on a Toeplitz operators Tf with symbols f ∈ Cc∞ (M) by Z t−1 τ (Tf ) = f (z)dµ0 (z). 2πi F The following holds. 1. For any function f ∈ C(M ) such that f |∂M is invertible, the operator Tf is Γ-Fredholm and its Γ-index is equal to the sum of the winding numbers of restriction of f to the boundary of M. 2. The map q : C(∂M) → TΓ /KΓ is injective and yields a nontrivial extension 0 → KΓ → TΓ → C(∂M) → 0. Proof. Step 1. Suppose that f ∈ C ∞ (M ) be invertible on the boundary of M. Then, for any function g smooth in the closure of M, and such that INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 15 supp(1 − f g) ⊂ M the theorem 3 gives 1 − Tf Tg ∈ KΓ and hence Tf is Γ-Fredholm. Let now P and Q be two non-commutative polynomials in (z̄, z). Then, again by the theorem 3 and its corollary, Z 1 ∗ ∗ τ ([P (Tf , Tf ), Q(Tf , Tf )]) = τ ([TP (f¯,f ) , TQ(f¯,f ) ]) = P (f¯, f )dQ(f¯, f ). 2πi ∂M On the other hand, by Carey-Pincus formula for traces of commutators (see [7]), Z ∗ ∗ τ ([P (Tf , Tf ), Q(Tf , Tf )]) = {P, Q}dν |z|<||Tf || where dν is a finite measure supported on the convex hull of the essential spectrum of Tf mod(KΓ ) and, since Tf is Γ-Fredholm, there exists an open ball Bǫ around the origin such that dν|Bǫ = cdλ, where 2πic = Γ-index of Tf . If we set dν = dν|Bǫ + dν1 , the two formulas above give Z Z Z 1 1 ¯ ¯ P (f , f )dQ(f , f ) = Γ-index of Tf dP dQ+ {P, Q}dν1 . 2πi ∂M 2πi |z|≤ǫ ǫ≤|z|≤||Tf || Applying Stokes theorem, we get the equality Z Z Z 1 1 ¯ ¯ P (f , f )dQ(f , f ) = (Γ-index of Tf ) P dQ+ {P, Q}dν1. 2πi ∂M 2πi |z|=ǫ ǫ≤|z|≤||Tf || If we now set P (z̄, z) = z and approximate z1 uniformly on the annulus ǫ ≤ |z| ≤ ||Tf || by polynomials Q, since both sides are continuous in the uniform topology on C(ǫ ≤ |z| ≤ ||Tf ||) we get, in the limit, Z Z 1 1 −1 f df = (Γ-index of Tf ) z −1 dz = (Γ-index of Tf ). 2πi ∂M 2πi |z|=ǫ Step 2. We will now prove injectivity of q. To this end it is enough to show that, for any open interval I ⊂ ∂M, we can find a function in C(M) which is zero when restricted to ∂M \I but for which the corresponding toeplitz operator Tf is not in KΓ . But given such an interval, we can easily find a smooth function f such that f |∂M is supported within I and such that the winding number of 1 + f on ∂M is nonzero. But then, by step 1 above, the Γ-index of 1 + Tf is nonzero and hence Tf is not an element of KΓ , which proves the second part of the theorem. Now, let f be continuous and with invertible restriction to the boundary of M. By part two of the theorem, this implies that the image of Tf in TΓ /KΓ is f |∂M and hence invertible, i. e. Tf is Γ-Fredholm, and the formula for its Γ-index follows from the fact that it is a functional 16 R. NEST AND F. RADULESCU on K1 (TΓ /KΓ ) and hence homotopy invariant of the class of f |∂M in K1 (C(∂M)). The normalisation statement follows immediately from the proposition 1. References [1] Arazy, J., Fisher, S., Peetre, J, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), no. 6, 989–1053. [2] Atiyah, M. F. Elliptic operators, discrete groups and von Neumann algebras. Colloque ”Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), pp. 43–72. Asterisque, No. 32-33, Soc. Math. France, Paris, 1976. [3] M.F. Atiyah, W. Schmidt, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math., 42, (1977), 1-62. [4] F. A. Berezin, General concept of quantization, Comm. Math. Phys., 40 (1975), 153-174. [5] P. Bressler, R.Nest and B.Tsygan, Riemann-Roch theorems via deformation quantization, I.Math. Res. Letters 20, (1997), page 1033. [6] Breuer, M. Fredholm theories in von Neumann algebras. II, Math. Ann. 180 1969 313–325. [7] Carey, R. W.; Pincus, J. Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977), no. 3-4, 153–218. [8] Carey, R. W.; Pincus, J., An invariant for certain operator algebras, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 1952–1956. [9] A. Connes, Non-commutative Differential Geometry, Publ. Math., Inst. Hautes Etud. Sci., 62, (1986), 94-144. [10] Connes, A., On the spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980), no. 2, 153–164. [11] A. Connes, M. Flato, D. Sternheimer, Closed star products and Cyclic Cohomology, Letters in Math. Physics, 24, (1992), 1-12. [12] M. Enock, R. Nest, Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras, J. Funct. Anal. 137 (1996), no. 2, 466–543. [13] F. Goodman, P. de la Harpe, V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Springer Verlag, New York, Berlin, Heidelberg, 1989. [14] U. Haagerup, Operator-valued weights in von Neumann algebras. I., J. Funct. Anal. 32 (1979), no. 2, 175–206. [15] W. J. Helton, R. Howe, Traces of commutators of integral operators. Acta Math. 135 (1975), no. 3-4, 271–305. [16] F. J. Murray, J. von Neumann, On ring of Operators,IV, Annals of Mathematics, 44, (1943), 716-808. [17] R. Nest, T. Natsume, Topological approach to quantum surfaces, Comm. Math. Phys. 202 (1999), no. 1, 65–87. [18] R. Nest, B. Tsygan, Algebraic index theorem for families, Adv. Math. 113 (1995), no. 2, 151–205. [19] Puknszky, L., On the Plancherel theorem of the 2 × 2 real unimodular group, Bull. Amer. Math. Soc. 69 1963 504–512. [20] F. Rădulescu, The Γ-equivariant form of the Berezin quantization of the upper half plane, Mem. Amer. Math. Soc. 133 (1998), no. 630, INDEX OF Γ-EQUIVARIANT TOEPLITZ OPERATORS 17 [21] P. Sally, Analytic continuation of the irreducible unitary representations of the universal covering group of SL(2, R), Memoirs of the American Mathematical Society, No. 69 American Mathematical Society, Providence, R. I. 196 [22] D. Voiculescu, Circular and semicircular systems and free product factors. In Operator Algebras, Unitary Representations, Enveloping algebras and Invariant Theory. Prog. Math. Boston, Birkhauser, 92, (1990), 45-60. Copenhagen E-mail address: rnest@math.ku.dk University of Iowa, Math. Dept.,Iowa City Iowa 52242, USA