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.
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Copenhagen
E-mail address: rnest@math.ku.dk
University of Iowa, Math. Dept.,Iowa City Iowa 52242, USA