Quantum Riemann surfaces I. The unit disc - Lesniewskilesniewski.us/papers/published/QuantumRiemannSurfacesI.pdf · Quantum Riemann Surfaces I. The Unit Disc* Slawomir Klimek and

Commun. Math. Phys. 146, 103-122 (1992) CommunicaUons in Mathematical

�9 Springer-Verlag 1992

Quantum Riemann Surfaces I. The Unit Disc*

Slawomir Klimek and Andrzej Lesniewski Harvard University, Cambridge, MA 02138, USA

Received April 10, 1991; in revised form September 20, 1991

Abstract. We construct a non-commutative C*-algebra Cj,(tT) which is a quantum deformation of the algebra of continuous functions on the closed unit disc tT. C,(tT) is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions on U.

1. Introduction

Alain Connes has shown [8] that a substantial part of differential geometry can be extended to a non-commutative setup in which a non-commutative .-algebra replaces an algebra of functions on a manifold. Clearly, not every non-commutative algebra has an interesting geometry and, while a satisfactory concept of a "non- commutative differentiable manifold" has not been formulated yet, it is desirable to study examples of such structures.

Recently, a growing number of examples of "non-commutative differentiable manifolds" has been studied, see e.g. [16, 18, 22]. These examples form, in a sense, a testing ground for probing general concepts of non-commutative differential geometry and so give us a better insight into the properties of a non-commutative manifold. Also, some important applications of these ideas have been found in other areas of mathematics and physics [3, 7, 9].

In this paper, we begin a program of developing a theory of non-commutative Riemann surfaces. We believe that non-commutative Riemann surfaces should play a distinguished role in non-commutative differential geometry, very much like ordinary Riemann surfaces in the commutative case. Also, there are some speculations [ 13] that quantum Riemann surfaces might be helpful in studying certain integrable systems arising in string theory.

Conceptually, the simplest method of constructing non-commutative manifolds is the framework of deformation quantization, see e.g. [2, 4, 19, 22]. This is the starting point of our approach.

* Supported in part by the National Science Foundation under grant DMS/PHY 88-16214

104 S. Klimek and A. Lesniewski

Let M be a Riemann surface and let co be a symplectic form on M. Then, the *-algebra Ca(M) of smooth functions on M comes equipped with a Poisson bracket:

{f~ O}:= c0- t(df, d9), f , oeC~~ (I.1)

We wish to construct a family of IE*-algebras st . , 0 < / t < 1, called quantum Riemann surfaces, together with "quantization maps" T("): C~176 ~ s l . such that:

(i) T (u) is linear and .-preserving and (ii) lim i rr(,) (~) (u) ,-.o i#" ; ' T~ ] - T{s'~ = 0. Further-

more, we require that the biholomorphisms of M act on d , and that d , is, roughly speaking, the fixed point algebra of the quantum universal covering of M.

In this paper we construct a quantum deformation of the unit disc (-- universal covering space of hyperbolic Riemann surfaces). Our construction draws some ideas from Berezin's paper [4] on quantization by covariant symbols. In a subsequent paper [11] we will describe quantum deformations of hyperbolic Riemann surfaces. In another development [12] we study a two-parameter deformation of the unit disc which is closely related to the quantum group SUq(1, 1). We should also mention that quantum Riemann surfaces of genus 0 and 1 have already been studied in the literature, see [16,20] (quantum sphere) and [8, 18] (quantum torus). Also, quantizations of the unit disk using various symbolic calculi have been studied (see e.g. [14, 21] and references therein).

Recall that the group of biholomorphisms SU(1, 1)/TZ 2 of U = {zOl;: [z[ < 1} consists of fractional transformations

z ~ ( a z +b)(-bz +~) -~, [ a [ 2 - [ b [ 2 = 1 . (I.2)

The (essentially unique) SU(1, 1)/TZ2-invariant symplectic form on U is

i co =~(1 --Izl2)-Zdz ^ dY., (I.3)

with the corresponding Poisson bracket

{f, g} = i(1 - [ z 1 2 ) 2 ( 6 3 f ~ 0 - ~fOo). (I.4)

In Sect. II we give the definition of the quantum unit disc C,(U) in terms of generators and construct an SU(1, 1)/~ 2 action on C,(U). In Sects. III and IV we study the representation theory of C~((7) in terms of the algebra of Toeplitz operators on a suitable Hilbert space. This leads us to structural theorems for the quantum unit disc which are closely related to the Brown-Douglas-Fi l lmore theory [5]. In Sect. V we show that the SU(1, 1)/7Z 2 action can be implemented on this space by a unitary projective representation of SU(1, 1)/Z2. Finally, in Sect. VI we study asymptotic expansions of products of Toeplitz operators and prove that, indeed, T r satisfies requirement (ii) formulated above.

II. Quantum Unit Disc

For 0 < # < 1 we let P~(U) denote the unita1112-algebra generated by two elements z and ~ with the following relation:

[z, ~] = I~(l - z~)(I - ~z). (II.1)

Quantum Riemann Surfaces 105

We call P,(/]) the algebra of polynomials on the closed quantum unit disc The goal of this section is to study some general properties of representations of Pu(CT). We show that Pu(U) has a nontrivial universal enveloping (E*-algebra. We call this C*-algebra the algebra of continuous functions on the closed quantum unit disc.

Let Jg be a Hilbert space and let n: P,({7) --+ s be a representation of Pu(U) by bounded linear operators on J/g satisfying the condition n(z)* = re(S). By abuse of terminology, we will refer to such representations as ,-representations (strictly speaking, Pu(U) is not a ,-algebra). Setting re(z) = d ~ re(J) = e-t0, 0 < 0 < 2re, we see that nontrivial *-representations of Pu({7) exist. By re(Pu(U))- we denote the ~*-algebra of operators on atg defined as the norm closure of re(Pu(U)).

We will frequently use the following notation:

x := zi, y := ~z. (II.2)

Applying re and taking the adjoint of (II.1) we find that

[n(x), re(y)] = 0, (II.3)

i.e., re(x) and n(y) generate an abelian subalgebra of re(Pu(U)).

Theorem ILl. The operators re(z) and re(i) satisfy

[I re(z)II = II re(J)II = 1. (II.4)

Proof. Let (E*(x, y) denote the abelian unital sub-~*-algebra of ~(~r ~ generated by n(x) and re(y). As a consequence of a Gelfand-Naimark theorem, ~*(x, y) is isometrically isomorphic with C(X), the ~*-algebra of continuous functions on the spectrum X of ~*(x, y). Let x(a) and y(a), asX, denote the images of re(x) and re(y) under this isomorphism. Clearly, x(a), y(a) >= O.

From Eq. (II.1) we have

y(a)[laX(a) + 1 -- #] = (# + 1)x(a) -- #. (11.5)

Since I~x(a) + 1 - # > 0, we have for all a s S ,

(~ + 1 ) x ( , r ) -

y(a) = #x(a) + 1 - / ~ " (II.6)

The function 0 < t --+ [(# + 1)t - I~]/[#t + 1 - #] is continuous and monotonically increasing, and thus as a consequence of (II.6),

(kt + 1)II re(x)II -- II re(Y)II : ~ II re(x)II + 1 -- #" (II .7)

But from (II.2), II re(x)[I = II re(y)II = II re(z)If 2. Substituting t h i s i n t o (11.7) and solving the resulting algebraic equation yields II re(z)II = 1, as claimed. []

Lemma 11.2.

(i) Ker(I - n(x)) = Ker(I - n(y)),

(ii) R a n ( / - n(x)) = R a n ( / - n(y)).

Proof. (i) This is an immediate consequence of (II.1) and (11.3).

(II.8)

(II.9)


(ii) If~k = (I - rt(x))q~, then from (II.1) and (II.3), (I + #(I - rc(y)))~b = (I - rt(y))~o. But II # ( I - rt(.v))I[ < 1, and thus r = ( I - rt(y))(I + r t (y) ) ) - l~pr rt(y)), prov- ing that R a n ( / - re(x)) c R a n ( / - rt(y)). By a similar argument, R a n ( / - rt(y)) c R a n ( / - rt(x)). []

Lemma II.3. (i) Ker(I - rt(x)) is invariant under rt(P~(U_))-,

(ii) R a n ( / - rt(x)) is invariant under rt(P~(U)).

Proof. (i) Indeed, if ~eKer ( I - rt(x)), then by (II.8), (I - rt(x))rr(z)~ = rt(z)(I - zr(y))~k = 0, and (I - rt(y))rt($)~, = rt(~)(I - rt(x))~ = 0, i.e., rt(z)ff, rt(~)ffeKer(I - rt(x)). Since Ker(I - rt(x)) is closed, the claim follows.

(ii) If ~k = (I - rt(x))~p~Ran(I - rt(x)), then by (II.9), rt(~)ff = (I - rt(y))rt(~)~pr R a n ( / - rt(y)) = R a n ( / - rt(x)). Furthermore, ~b = (I - ~t(y))~0' and thus rt(z)~, = (I - rrt(x))rt(z)~o' e R a n ( l - rt(x)). []

We now write

= Ker( l - rt(x)) ~ ~ , (II.10)

where ~ is the or thogonal complement of Ker ( I - r t (x ) ) . As a consequence of Lemma 11.3 (i) and the fact that ~P , ( [7 ) ) - is a .-algebra, both direct summands in (II.10) are invariant under r t (P , (U) ) - .

Let now 7ESU(1 , 1). We will write the inverse of V as

where

lal e - I b l 2 = 1. (II.12)

Lemma 11.4. I f TESU(1 , 1), then (brt(z) + a ) - i Ert(Pu(t.7))-.

Proof. Consider the series

a -1 ~ ( - b/a)"rt(z)". (11.13) n>O

As a consequence of (II.12), Ib/al < 1. Since by Theorem II.1, I1 re(z)II = 1, the series (11.13) converges to an element of .W(~). A routine calculation show that its limit is, in fact, (brt(z) + a) - x. []

We now set for 7eSU(1, 1),

p~(n(z)):= (tirt(z) + b)(brt(z) + a ) - t,

p~(/~(e)):= (brt(e) + 4)-l(art(z-) + b). (II.14)

In the theorem below we assert that p~ defines an action of SU(1, 1) on rt(Pu(U))-.

Theorem II.$. Le t y~SU(1, 1). Then:

(i) pr(rt(z))* = pr(rt(~)), (II. 15)

(ii [pr(rt(z)), pr@(~))] = #( I - pr(rt(z))pr(rt(z--)))(l - pr(rt(z-))p~(rt(z))). (11.16)

Proof . (i) This is clear.


(ii) We verify (I1.16) on K e r ( / - re(x)) and ~ separately. We set z ' := pr(rc(z)) and 2 ' := pr(r~(5)). If coeKer(I - re(x)), then a s traightforward calculation shows that

z ' 5 ' 0 = ~ ' z ' 0 = 0 ,

which proves that (II.16) holds on Ker ( I - re(x)). T o prove that (II.16) holds on ~ we observe that the inverses ( I - r e ( x ) ) - 1

and ( I - r e ( y ) ) -1 exist as densely defined unbounded operators. Fur thermore , as a consequence of Lemma II.3 (ii), Dom(( I - re(x))- l) = Dom(( I - re(y))- i) = Ran(I - re(x)).

Lemma II.6. R e l a t i o n (11.1) r e s t r i c t ed to ~,@ is equ i va l en t to the f o l l o w i n g e q u a t i o n on Ran(I - re(x)):

( I - r e ( x ) ) - 1 _ ( I - r e ( y ) ) - 1 = #. (II.17)

P r o o f . Multiplying (II. 17) by I - re(x) on one side and by I - n(y) on the other yields (II.1). Conversely, writing (II.1) as

((I - 7z(y)) -- (I -- x(x)))tp = # ( I - rc(x)) ( I - rc(y))q), q ) e ~ ,

and using (11.3) yields (11.17). []

We can now conclude the proof of the theorem. As a consequence of (II.12) and Lemma 11.4,

I - z '5' = (bn(z ) + a ) - 1(1 - rc(x))(brc(5) + •)- 1, (11.18)

and

I - 5'z' = (bre(5) + ci)- l(I - r~(y))(brc(z) + a)- 1. (II.19)

We claim that I - z'5' and I - 5'z' are invertible on Ran(I - re(x)). Indeed, we set for ~b ~Ran(I - re(x))

(I - z ' 5 ' ) - i~k = (brc(~) + gO(I - re (x ) ) - l(brc(z) + a)~k, (II.20)

and

(I - 5' z ' ) - t ~ = (brc(z) + a) ( I - re(y)) - l(brr(5) + gt)~k. (II.21)

As a consequence of Lemma II.3 (ii) these are well defined. A simple calculation shows that (II.20) and (II.21) are indeed the inverses of I - z '5 ' and I - 5'z', respectively.

As a consequence of the above considerat ions we have, as opera tor equat ions on Ran(I - re(x)):

(I - z ' z t ) - 1 - ( I - 5 t z ' ) - 1 = ( b 7 ~ ( 5 ) + a)(I - re(x))- i(b~(z) + a)

- (brc(z) + a) ( I - re(y) ) - l(brc(~) + a)

= I b l 2 { ~ ( y ) ( I - re(y)) -1 - 7r(x)(I - - r~(x))- 1}

+ l a12{(I - re(x))-1 _ (I - re(y))- i}

= - [b [2{ Ire(z), 7r(~)] (I - re(y)) -1 + / ~ ( x ) ) + ktla] 2

= N l a l 2 - Ibl2)=/~.

This shows that z' and s satisfy (II. 17) and the proof of the theorem is complete. []


We now let for aeP, (U) ,

It a II := sup II ~(a)II, (11.22) / t

where the supremum is taken over all possible .-representations of Pu(U). As a consequence of Theorem II.1, I[ a II < go forall aeP , ( t2 ) . Let N : = {aeP,(U_): II a II = 0} be the null-ideal in Pu(U). We define C,(U) to be the completion of Pu(U)/N in the norm induced by I1"11 and call it the algebra of continuous functions on the closed quantum unit disc. By definition, C,(t~) is a unital 112*-algebra with involution defined by z*:= i. The following theorem is a consequence of the above considerations.

Theorem 11.7. The ll2*-algebra Cu(U ) is generated by two elements z and ~ satisfying (11.1) and such that II z ]l = 1. The mapping

F --* p~(z):= (gtz + b)(bz + a)- 1

defines an action of SU(1, 1) on Cu(U).

IlL Representation Theory

In this section we classify all irreducible *-representations of the algebra of polynomials on the quantum unit disc Pu([7). As in Sect. II, let n: P,(tT) ~ ~ ( ~ ) be a representation of P,(U) on a Hilbert space ~,vg.

Proposition III.1. Let 2, := np(1 + n#)- t , n = 0, 1, 2 . . . . . Then

(i) Spec(g(x)) C {/~n}n> 1 U {1}, (III.1)

(ii) Spec(n(y)) = {2, }, => o u { l }. (III.2)

Proof. By Theorem II.1, II re(x)I] = I] n(y)II = 1. Therefore, Spec(n(x)), Spec(rc(y)) = [0, 1]. Furthermore, Spec(n(x))\{0} = Spec(n(y))\{0}. Let I , , n = 0, 1,2 . . . . be a sequence of open subsets of [0,1] defined by I , := (,~,, 2, +1). We claim that 1. c~ Spec(n(x)) = ~ and I , c~ Spec(n(y)) = ~ , for n > 0. Indeed, as a consequence of (II.1),

~(y) = {(# + 1)~(x) - #} {#zt(x) + 1 - #} - 1, (III.3)

and thus r~(x) ~ 2i. Therefore, Io c~ Spec(zc(x)) = ~ . Consequently, Io c~ Spec(rc(y)) = ~Z~. We now proceed inductively. Assume that Io , I 1 . . . . . I , do not intersect the spectra of re(x) and re(y). If 2e i ,+1 is in the spectrum of rc(x),then by (III.3) 2 = {(# + 1)2 - #} {/~2 + 1 - #}- 1 is in the spectrum of re(y). But 2eI , , which is a contradiction. Therefore, I,+1 does not intersect the spectrum of re(x) and, consequently, it does not intersect the spectrum of re(y). []

Theorem III.2. An irreducible *-representation (2/g, zt) o f p , ( u ) is unitarily equivalent to one of the two following representations:

(~) a one-dimensional representation r~(z) = e -i~ r~(~) = e i~ where 0 __< 0 < 2rt; (~) an infinite dimensional representation defined as follows: let ~f" = 12(TZ + ) (where

2g+ is the set o f nonnegative integers) and let {f,},>=o be an orthonormal basis for ~ .


Then (n + 1)# "~1/2

u ( 2 ) f . = l + ( n + l ) # ) f"+~' n > 0 , (III.4)

0 n = O , ~ ( z ) f . = j. n# "~1/: r (III.5)

[ 1 + , ~ S , . - 1 , ,__>1.

Proof. We assume that ( ~ , ~) is an irreducible *-representation of Pu(U!. We write as in (II.10) and observe that either Ker ( I - ~(x)) = 0 or ~ = 0. If Jg = 0, then

~(z)n(z)* =~z(z)*~(z)=I, i.e., re(z) is unitary. As a consequence, Jr is one- dimensional and ~(z)=e -~~ Let K e r ( l - ~ ( x ) ) = 0 and let f t , , n = 1,2 . . . . be the eigenspaces of n(x) corresponding to the eigenvalue 2.. By the spectral theorem,

= ( ~ ft, . Observe that, as a consequence of (II.1), ~(~) maps ~r into ~q,+l, and n>l

~(z) maps ~q,+ 1 into ft, . Indeed, if f e f f , , then

(1 + # - #~(~))~(z~)~(~)f = #~(2)f - # ~ ( ~ ) ~ ( z ~ ) f

+ ~(2)~(z~)f = (# - #2. + 2.)~(~)f.

Consequently,

~(z2)lr(2)f = (# - #2n + 2.)(1 + # - #n(2z))- 1~(2)f = (# - #2. + 2.)n(2)(I + # - #~z(z2))- i f

= (# - #2. + 2.)(1 + # - #2 . ) - l~z(2)f

='~ .+ 1~(2)L

i.e,, ~ (2 ) fe f f .+ 1. In the same way we show that ~(z): fr 1 ~ fr Now, let i. denote the restriction of ~(2) to ft . and let r. denote the restriction of 7c(z) to ft. . Then i.r.+ 1 = 2.1 and r.+~i. =2.1. This shows that the subspaces ft . and fr are isomorphic, n = 1, 2 , . . , , We claim that ffl is one-dimensional. Indeed, if el, e2efr are nonzero and mutually orthogonal, then the subspaces generated by {Tr(2")el }.,> 1 and {~(2")e2 }m~l are nonzero, invariant under ~(P~(t~)) and mutual ly orthogonal . As a consequence, all fr are one-dimensional. Let foe f r with lifo I] = 1, and let f . , n > 1, be defined by

f . : = H ~(2")fo II -1 ~(2")fo efr +1. (III.6)

Then { f . }. > o forms an o r thonormal basis for ocg. A simple computa t ion shows that

~(z)f = {x~Ofo n = 0 -1, n > l . [ ]

We now describe a representat ion of Pu(U) on a Hilbert space of holomorphic functions on the unit disc U c IIL This space was previously studied by Berezin [4] in the context of quant izat ion of covariant symbols.


We set

1 r := - + 1, (III.7)

#

and observe that r > 2. Let Wv be the Hilbert space of holomorphic functions on U equipped with the Petersson scalar product

We set

(m.8)

d#r(0 = 2 (1 - 1(]2) r- 2d( ^ d(. (III.9)

It is well known that the functions

1 F(r + n) ~ 1 / 2 (III.10) q~n(~) = ( ~ n!F(r -- 1)J

form an orthonormal basis for 3ff u. This leads us to the following expression for the corresponding Bergman kernel:

K((, t/) = n - l ( r - 1)(1 - (q)-*. (III.11)

Recall that K((, t/) is the integral kernel of the orthogonal projection P of L2(U, d#,) onto 3ff v. For each continuous function f on tT we consider an operator Tr on oW v defined by

Tf q~ (0 := (PM f ~p)(() = S K((, q)f(tl)q~ (t/)d#,(r (III. 12) U

Here M ; denotes the multiplication operator on L2(U, d#,). Observe that Tr ~ ~ ( ~ v ) . An operator of the form Tf is called a Toeplitz operator with symbol f (see e.g. [10]). Note that Ty = T~. Let ~--u(U) denote the ~*-algebra generated by the Toeplitz operators.

We define the following Toeplitz operators:

(n(~)tp)(0:= S K(~, n)r/(p(t/)d#,(q) = ~ ( ( ) , V

(n(z)tp)(0:= ~ K(~, q)O~o(~/)d#,(q) = (1r(~)*~o)(0. (III. 13) U

Lemma 111.3. We have

(n(z)q~)(0 = ( - r ~ t/r-1 qr (III. 14) 0

Proof. We set ~b(0:-- 0r(z)q~)(0. Then using (III.11),

~(() = n - ~(r - 1)~ q(1 - ( q ) - r U

Quantum Riemann Surfaces l 11

1 d - - ~ (1 - ~q)K(~,//)~o(~)d#r(//)

r - ld~v

1 d - r : 1 [qr - ~ ( ~ ( ~ ) ) ] "

As a consequence, we obtain a differential equation on q~,

~0' + rO = tp'.

Solving this equation yields

(III.15)

~,(() = ( - ' ~ ~ ' - 1 e,(~)d~, 0

as claimed. []

T h e o r e m III.4. (~r n) defines a *-representation of P,(U). This representation is unitarily equivalent with representation (fl) of Theorem 111.2.

Proof. From (III.10) and (III.13), 1/2

( n + r ) q:)n + 1(~),

which proves (III.4). Likewise, from (III.14),

(~(z)~o.)(~) = ~-" I ~ - ~e'.(~)d~ = ~o._~ (~), o - l + r

for n > 1 and (n(z)cpo)(~) = 0. This proves (III.5). []

In other words, n is a homomorphism of Pu(U) into ~ ( U ) . Our next result states that, in fact, Cu(U ) is isomorphic with Y-u(U). Therefore, the concrete C*-algebra _Y-u(U) may serve as a universal representation of the abstract C*- algebra C u( U ).

T h e o r e m ILLS. n: Pu( / ] )~ 3-.(t]) induces an isomorphism of the ff~*-algebras Cu(O ) and ~--.(U).

We will prove this theorem in Sect. IV.

IV. T o e p l i t z O p e r a t o r s

Our goal in this section is to prove Theorem III.5. The technique to achieve this is a detailed analysis of the structure of the ll:*-algebra ~--,(U). Our analysis follows the standard methods (in particular those of [10, 1 and 6]). For the reader's con- venience we include the details of the proofs referring to the literature at appropriate places. Throughout this section, f denotes a continuous function on U and II f II denotes its sup-norm.

Lemma IV.1. (i) II TI II ~ FI f II 0o, (ii) Ty = 0, /f and only if f = O.


Proof. Part (i) is clear. To prove part (ii) we observe that T I = 0 if and only if (era, Tiq~,) = 0, m, n = 0, 1, 2 . . . . . where q~. is defined by (III.10). Since

(~", Ty(") = ~ ~m~"f(Od#,(~), U

and since the monomials (~(-" generate a dense subspace in LE(u, d#r ) this is possible if and only i f f = O . []

Let :,T'(ovfv) denote the ~2*-algebra of compact operators on ~ v .

Lemma IV.2. T rac~(gfv) i f and only i f f ~ ov = O.

Proof. We follow [1] and [6]. Suppose that f ' rov = 0. There is a sequence {f,) of continuous functions such that suppf , c U and II f - f . II ~--,0. Let {(Pm} be a sequence in o~ v such that (Pro --* 0 weakly (recall that a weakly convergent sequence is bounded). We claim that T:q)m--*O in norm which means that T: is compact. Indeed, since (pro(()--. 0 pointwise, it follows that, for a fixed n, q~m(()~ 0 uniformly on supp f , . Therefore,

II T f ~ [I < II Tyjp,, II + II Tf-T,(Pm [I

< C ( sup I(P,,(~)I l[f,[l + [ I f - f , iloo[IfP,.[[~,

/ %

\ ~ s u p p f n /

f ) 1/2 where C = ~ ~ d#r(0 ~ < oo. Choosing n sufficiently large and using the fact that

. J

t1 r [I < K, we see that the right-hand side of this inequality is arbitrarily small. To prove the converse we consider the sequence of elements of ~ v

(0,(0:= K((, tl.)/K(tl., tl,) ~/2,

where q,~ U, q, ~ t/sdU. Then II (o, l] = 1 and (0.(0 ~ 0 uniformly outside a neigh- borhood V of q. Passing to a subsequence, if necessary, we may assume that r ~ 0 weakly. Since

It (Tf - f(~l))q~, II 2 __< j- If(l) - f(t/)l 21 r U

= If(() - fifO] + Cn(( [1 f __ f(q)11 2

we see that [] (T I - f ( t / ) ) r [[ ~ 0 . As a consequence,

If(t/)l = II f(q)r II < II rfq),l[ + I I ( ry - f(tt))r II---,0,

since Tf is compact. Therefore, f(t/) = 0. []

Lemma IV.3. T y T o - TIg, [ T f , To]s~(Wv).

Proof. We follow [1]. Obviously, the second claim is a consequence of the first, so we need prove that T I T o - Tlo~f{( .gf v). Clearly T c o - T I T o = PMy(1 - P)M o. Let

.1. 2 Jgv denote the orthogonal complement of ougv in L (U,d#,) and let H I : = (1 - P)M yP: ~ v --* ~r ~v. Then

T,o - T , T , = H~H, . (IV.I)


Furthermore,

H:o = So:H o + H f T o, (IV.2)

where • • S:: oug v ~ 3/g v is defined by S: := (1 - P)M:. Let now ~ : = { feC(0 ) : H : is compact}. As a consequence of(IV.2), ~ is a closed subalgebra of C(U). Also, H1 = 0, H e = 0 and by (IV.l),

H ~ H~ = Tr - T~ T~ = [n(z), n(z)] ~YY(iY:v).

So 1, ~, ~-~), and thus by the Stone-Weierstrass theorem ~ = C(8). []

Let J c Y-u(8) be the commutator ideal of J , ( 8 ) , i.e., the smallest norm-closed, two sided ideal in : - , ( 8 ) containing all the commutators.

Lemma IV.4. J = ~(JFv) .

Proof. We follow [1]. To show that YF(~v) = ~--u(J/Fv) we use Theorem 5.39 in [10]._Firstly, o~ff(3egv)c~ 3-u(~v) ~ ~ , as [T~, T~] ~ ( J g v ) . Secondly, we claim that ~--,(U) is irreducible. Indeed, let Q be an orthogonal projection in o~fv which commutes with 3-,(U). Then (Q~0)(~)- QTod~I = r for all polynomials e s W v . From Q2= Q we have Q1 = 1 a n d ' t h e claim follows. Therefore, by the quoted theorem, Yf(Jefv) ~ J , ( 8 ) . As a consequence of Lemma IV.3, J c YY(o~v). Since : f (o~v) is simple [10], ,Uf(Hv) = J. []

Theorem IV.5. (i) There is a short exact sequence of C*-algebra (the Brown-Douglas- Fillmore sequence)

0 ~ Yd(2/t~v) ~ ~--.(8)~ C(8 U ) ~ O. (IV.3)

(ii) Every element Ae5-u(U) can be written as

A = T: + K, (IV.4)

where K a.Yl ( 2,~ v). Furthermore,

I[ T: + K I1 _-__ sup If (0l . (IV.5)

Proof (i) We claim that

as C*-algebras. Indeed, a ( f ) := Ty (mod o~(()r ). By Lemma IV.3, a is a *-homomorphism whose range is dense. By Lemma IV.2, Ker(a) -- C(~ { f ~ C ( O ) : f r0v = 0}. So a induces an injective *-homomorphism

a: c ( 8 ) / 0 ~ ~_ c ( ~ u ) ~ 9-.(8)/~(~& By Proposition 4.67 in [10], a is an isometry, so Ran(a) = Ran(a) = .Y,(O)/YY(~ugv) , and a is an isomorphism of C*-algebras.

(ii) As a consequence of (i), every element in Y-,(0) is of the form (IV.4). Since j : : -u (8 ) -*C(OU) , j (T :+K) :=fTaU is a *-monomorphism of C*-algebras, it follows from Proposition 4.67 in [10] that sup If(01 _= II T: + K ]J. []

~eSU Now we are in a position to prove Theorem III.5.

:-,(O)/J~ff (~f v) ~- C(~U), (IV.6)

let a:C(8)~.Y-u(U)/.gY(jCYv) be the map defined by


Proof of Theorem 111.5. Let

a= ~ ~kzzk~ z .... zk"~I"EPu(U), (IV.7) kl, . . . ,kn ll,.. . ,ln

where ~u~l12. By (II.22) and Theorems III.2 and II.4,

, t a l l = m a x { 7r(~k,Zk~'~'"zk"~ '") ,Is up ~ k l f f l + ' " + k " ~ ~+'''+l" }.

By (IV.5), the maximum is achieved at r ~ ( 2 ~kt zk~ ~' .... zk"~ "~ . Therefore li a II \ k,l /

[] 7r(a)[]. The claim follows. []

As a consequence of Theorems III.5 and IV.5 we obtain:

Corollary IV.6. The (E*-aloebra Cu(U ) is a GCR algebra.

Let now ~--12(7Z+) and let {f,},=>o be an orthonormal basis for ~ . The operator S defined by Sf,:= f,+ 1, n > O, is called the unilateral shift [10]. Let C*(S) be the unital ~E*-algebra generated by S.

Theorem IV.7. C~(U) is isomorphic with (E*(S).

Proof. Let Em,n~.q~(o~) be defined by Em,nfv:=(~nvfrn. Then S = Y' E,+I, , and n>_O

~r(~) = ~ {(n + 1)#/(1 + (n + 1)/~)}1/2E,+ 1,,. It is clear that S - rc(~)so~f(~). Conse- n_>-O

quently, IE*(S)~ r~(Cu(U))- Cu(U). On the other hand, since S + KeC*(S) (see [10]), it follows that r~(~)OE*(S), i.e., ~r(Cu([7)) c IE*(S). []

V. The Action of SU(1, 1)

In Sect. II we constructed a group action SU(1, 1 )~ Aut(Cu(U)). Under the identi- fication Cu(U ) ~ 3-u(U), this action can be implemented by a projective unitary representation of SU(1, 1) on out'v.

For 2OI;*:= IE\{0} we set log2:= logl2[ + iarg2, where - ~ < arg2 < ~. If y~SU(1, 1), we let a~ and b~ denote the matrix elements of y-1 as in (II.11). The function [14]

1 ~/{1og(~211 ~- 1( + ti~2 ) _ log(~lr2 ( + d~1~2) + log(b~,( + tit1)}, (v.1)

where ~1, ]12 ~ S U(1, 1), (e U, is independent of ( (to see this, differentiate with respect to 0. We denote the value of (V.1) by 2(71, ]12). Furthermore, setting ( = 0 it is easy to see that 2(]11, Y2)~{ - 1, 0, 1}. In fact, the function 2: SU(1, 1) x SU(1, 1) ~2g defines a cohomology class in H2(SU(1, 1),2g), i.e.,

2(Y2, ]13) - - 2(]11112' ]13) + 2(]11, ]12113) - - 2(]11, ]12) = 0 (v.2) (to see this, use the fact that (V.1) is independent of 0- We set

a(yl, y2): = exp {27rir2(71,72)}. (V.3)

Quantum Riemann Surfaces t t 5

For yeSU(1, 1) and q~e~u we define

(U,q~)(~):= (b,~ + a~)-'q~(7- ~ ~), (V.4)

where (b~ + d~)-~:= exp{ - r log(br~ + d~)}. Then Ur is a unitary operator on i f v, and

U~,~ = a(7 ~, 72) U . U~. (V.5)

Therefore, as a consequence of (V.2), 7 ~ Ur is a projective unitary representation of SU(1, 1).

Theorem V.1. The SU(1, 1) action 7 ~ Pr on Cu(O) is impldmented by 7 ~ Ur.

Proof. We need verify that

p~(i) = UrSU~-,. (V.6)

Indeed, from (III.1 I),

- , - , a ~ + b ~ r r ( u ~ e u ~ _ ,~o)(O = (t,~ + ~ ) -~ (eu~- ,q~)(~- ~ ~) = ( ~ + a) _ ~ , ~ ~- ~o)(~- ~ ~) b(+~

as claimed. []

= (b~ + a)-r a~ + b ( _ b7 - 1 ~ + a)-"~0(~) = a~ + b b( +a b( +~

- - e ( ~ ) = p~(e)~0(0,

VI. Asymptotics of Products of Toeplitz Operators

In this section we show that, in a suitable sense, C~(U) is a quantum deformation

of the unit disc. To be more specific, we prove that for smooth f and g, 1 [Ts, To] approaches its classical value, as # ~ 0. #

Let C~~ denote the Fr6chet space of smooth functions on U whose derivatives extend to continuous functions on U. For f , geC~176 we define

{f, 9} (~):= i(1 -- I~ [z) 2 [Of(~)Sg(~) - ~f(~)Og(~)], (VI. 1)

the Poisson bracket o f f and g corresponding to the Poincar6 symplectic form on

U, c~:= 2(1 -[([2)-Zd~ ^ d~. Obviously, {f,g} eC*(tT). The goal of this section is I

to prove the following theorems.

Theorem VI.1. Let f EC(U). Then

I[ Till < [Ifll~ < [r Tf[[ + o(1), (VI.2)

as r --* oo. In particular, lim II Zf II = tl f II ~o. r~oo

Theorem VI.2. Let f , #~C~(U) and let r be sufficiently large. Then:

(i) There exists a constant C (depending on f and g) such that

II r(TTT9 - Tfo) + T(1 -[~[')'Of~g [I ~ C r - 1/2. (VI.3)

116

(ii)

S. Klimek and A. Lesniewski

There is a constant C (depending on f and g) such that

~[T:, To] -- T~f,o } < (VI.4) Cr-1/2.

Remark. It should be remembered that the operators and the operator norms in (VI.2-4) depend on r even though, for the sake of notational simplicity, this is not explicit in our formulas. Estimate (VI.2) says that lim 1I Ts [1 = il f 11, while estimate

(IV.4) says that lira u~o [T I, To] - T~y,g I = 0. This shows that Cu(U) is a quantum

deformation of the unit disc in a sense close to Rieffel's "strict deformation quantization" [17].

Proof of Theorem VI.1. The first inequality in (VI.2) is clear. To prove the second one, we define 7~SU(1, 1), (~U, by

Observe that 7~(0) = ~, and

l y~(q)- ~,r 1 < I~1 = l i l n I

In particular, if Ir/] < 6 < 1/2, then

[7~(r/) - 7~(0) 1 < 26. (VI.6)

Lemma VI.3. I f f ~C(U), then

supply f ( 0 - (r - 1) v ~ f(Y;(t/))d#r(q) = o(1), (VI.7)

as r ~ ov.

We shall prove this lemma after completing the main line of the argument. Using the reproducing property of the Bergman kernel, we write

f(() = ( r - 1)!7~ f(rl)d#,(rl)+ ( f ( 0 - ( r - 1)!7~' f(~l)d#r(rl))

<%. <%> + ( - <,- ,> .s :,.>...,.> ). where 7~f(~/):= f(7~(t/)), and where q~o is given by (III.10). As a consequence of Lemma VI.3,

sup If(~)f < sup II Tzs II + o(1). ~U ~U �9

But, T~f = U~cTsUy~ 1, with Ur~ unitary, and so Ilf[I co < II Tfll + o(1), as claimed.


Proof of Lemma VI.3. Since f is continuous on a compact set, it is uniformly continuous. Given e > 0, we take 6 < 1/2 so that I f 0 1 ) - f(~/')l < e/2, whenever [q - q'[ < 26. Then,

f(~) - (r - 1) ~ f(y~(r/))d#,(~/) = (r - 1) ~ (f(7~(O)) - f(~(t/)))d#~(q) U U

= I ' " + I 1~]<6 6<1~1<1

Now, using (VI.6),

Furthermore,

II~l =< sup If(yr - f(yr < e/2. I,iI < a

[I2[ ~ 2[[f[[ ~ 5 d/407) = 2 [[ f [I oo( 1 - 5 2 ) r - l , a < l q l < l

which is less than e/2, for r sufficiently large.

Proof of Theorem V1.2. Obviously, (ii) is a consequence of (i) and (VIA). To prove (i) we write

(Tf Tg~o, ~) = I K(~, tl)f(Og(~l)r (VI.8) U2

where go, @e ~ u . In the ~/-integral we substitute q = ?~0, where ?r 1) is defined by (VI.5). Then

(Tf Ego, r = n - l ( r -- 1) I f(Og(7~o)(a~:* (o)(O)~b(O(1 - ]~I 2)-€ U 2

(vi.9)

This form of (VI.8) is suitable for asymptotic analysis. From Taylor's theorem,

g(~o) - g ( O + (1 - I Gl2)ag(OO + (1 - I Gl~)ag(O0 + (1 - I r - ~~g(O +1( 1 -IGl2)~Ug(O_] 0 / + (1 - I G I 2 ) [ - {~g(0 +�89 --l~12)~2g(0]0 2

+ (1 -- [~12)zO~g(O00+ G(O, O, (VI. 10)

where G(O, 0 is the second order remainder. Let us assume for a moment that the following fact is true.

Lemma IV.4. There is a constant C (depending on f and g) such that

I= - l(r -- 1) ff f(OO(O, O(U; , ~o)(0)$(0(! -- I~l 2)-'/2d#,(Od#,(O)l U 2

~ Cr -312 ][ ~0 [[ U ~/[I. ( V I . 1 1 )

Substituting (VI.IO) into (VI.9) and using the fact that O-+(U(~o)(O) is a holomorphic function we find:

(T• r = j" f(Og(O~(O~'(Od~(O + zt-*(r - 1) U

5 f(O3g(O c;v~8~ (Uv; - ' ~0)(0)~0 (0(1 - Ir ) -r/2 + 1[ O[2d#r(r @

U 2

1 1 8 S. Klimek and A. Lesniewski

+ n - l(r -- 1) ~ f ( 0 ( 1 -- l~12)2tggg(~)40(~)qs(~)lO[2d#,(Od#,(O) U s

0 2 + re- ~(r - 1) J" f (~ ) [ -- ~gg(~) + �89 -- I~ I2)g2g(~)] ~ (U i , 40)(0)

U 2

�9 0 (0(1 - I (I 2)-*/2 +11 o 14d#.(Od#,(O) + R, (VI. 12) where R denotes the remainder term which obeys the estimate (VI. 11). Observe that the first term on the right-hand side is equal to (Tio40, ~). Furthermore , using

n - l(r - 1) j" lO12d#,(O) = r - 1, U

and

0 ~12)-,+ 1 f~ 2 r l2 (1-1~12)-*/2~(Uu,40)(0)=(1-1 1(1 "-I- ~ -0 ) - {(1 -1~1 ) (U?,40)(0)}, (VI.13)

we write the second and third terms in (VI.12) as

_1 S f(Ogg(~)~s(~)(1 -- I~12) -* + 2 ~ r v ~ {(1 - I~ 12) '40(0}d#, (0

+ 1_ ~ f(~)(1 - l~12)20gg(~)40(()~,(~)d#,(~). r v

Integrating by parts we see that this sum is equal to

1 1 S(1 - I ~12)2~f(ff)~g(040(~)%O(0d##(~ ) = - -(Tr _ ,r ~O). r F

We claim that the fourth term in (VI.12) can be estimated by Cr -2 II 40 II II ~s II, with C independent of r. We proceed as follows. Using (VI. 1 3) twice, we find that

z -,lZ 62 2 ,+1 -~9 ( 1 - 1 ~ 1 ) ~(uu,40)(O)=-(1-1~ '1 ) - ~{(1-1(1~) '40(0}

+ (, - I~l~)-*+ 1 ~ {(, - l~l~)~ r(, - 1~12)'40(0]}.

We substitute this into (VI.12) and use the fact that

2 x - l(r - 1) e~lO[gd#r(O) = r(r + 1)" (VI.14)

Integrating by parts we write the integral as

with F(~) cont inuous r - 2 sup [F(0L [[ 40 i[ l] • II-

Ceu

Lemma VI.4. []

1 S F(040(~)~k(0d#,(~), (VI. 15)

r(r + 1) v

and bounded on U. Finally, we estimate (VI.15) by This concludes the proof of (VI.3) up to the p roof of


Proof of Lemma VIA. We claim that G(O, 0 can be written as

G(O'O= E gJ( O'()Oj'~3-j' (VI.16) O=<j-<3

where gj, O <__j __< 3, are smooth functions on U x U with the following properties:

(i) Igj(0, ()[ < 0(1)(1 -1(15)(1 -101) -6, (VI.17)

I 63 igj(O, ~)} (ii) ~{(1 -[[[2)-lgj(O. O} + ~_{(1 -[(12)- __< 0(1)(1 -[0[) -8, (VI.18)

0 0 8 0 = -- [(12)(1 - (iii) ~ 9 j ( , 0 + ~ g j ( , ( ) <O(1)(1 101) -8. (VI.19)

Indeed, we write

G(0,() = ~i(1--S)2~:c:O(SO)) dS

= 2 o-_<i=<3 \ j / ! (1 - s)Z(?~oJ~o-iytg(sO)dsOJO 3 -J,

and define

1~3)!/\x. - ( ) OsoSso ?~g(sO)ds. (VI.20) at(0,0:= (1 j - 3 - j ,

To verify that g j(0,() obeys (VI.17)-(VI.19) we carry out the differentiations in (VI.20) according to the chain rule. Using the fact that for k > 1

~k ~0 k ?~(0) = ( - I) k - ik!(l - I (12)~- I(l + ~0) -k- i, (Vl.21)

as well as the inequality

I I+sCO[- l<(1 - s lOI ) - l< (1 -101 ) -1, O < s < l , (VI.22)

we easily obtain the bound

,oj(O. (), ~ 0(1){ ~ ,, c?~9 [, oo }(1 --, (,2)(1 -- ,0,)- 6. l < a + f l < 3

which proves (VI.17). Inequalities (VI.18) and (VI.19) follow from an analogous argument for the derivatives of (VI.20).

Let us now estimate the contribution that each term in (VI.16) gives to (VI.11). Let j = 0. Using

1 t? if(1 -1012) r-z - (1 -1012) "-1, (VI.23)

r - lc~O

and integrating by parts with respect to 0 we find that

n - l(r - 1) I f(()go(O, OO3(U [, (P)(O)O(O(1 - ](12)-':adl~,(Odkt,(O) U 2


2 0 = _ ~ -1 ~ f(r - IOl ) ~ go(O, r ~,o)(o)0(0(1 - I~1 z) -"/2dl#(()d#,(O)

U 2

-1 ~)"/(()(1 -- I 012)0o(0, r _~o ( u , ? ' ~o)(0)0(r - Ir z ) -'/2d#r(r 7~

(VI.24)

The first term on the right-hand side can be estimated by

sup I/(r sup (1 -1r 1(1 -IOI)Sg~,,go(O,r 2 g - 1

r ~,o ol7

"[O[4(1-[OI)-14dt2r(O)) ~[(1-lr162 ) ]l (0 [] II ~t l]

<_ Cr- 3/2 II ~o II II ~ [I,

where we have used that

j" 1014(1 - I OD- 14d#,(0) < O(1)r- 3, (VI.25) U

and r~ -~ ~ (1 - l([2)2-rd/#(r = 1. (VI.26)

v

To estimate the second term in (VI.24) we first use (VI.13) and integrate by parts with respect to r The result is

~z -1 ~ 0f(r - ]O[2)go(O, 0ffz(U [ , q~)(0)~b(0(1 - [~ 12)-r/2 + ~(1 + C0)- ld#,(r uE

~z -1 ~ f(~)(1 - 1 0 1 2 ) ~ { ( a -1~12)-Xgo(O,O} + U 2 cg

�9 f f z ( u ? , ~0)(0)~(r - Ir -~/2 + 2(1 + CO)- ld~(()d~r(O),

Using (VI.22) (with s = i) and properties (i)-(iii) above, we estimate each of these terms by

{ ! ~/2C )1/2 Cll~olt IlOll 1014(1-[Ol)-16d#~(O)tf t ! ( 1 - 1(12)-r+2dp~(r j k

< Cr- 3/2 II • II I[ 0 II.

This completes the proof fo r j = 0. Let j = 1. Using (VI.23) and integrating by parts with respect to 0 we find that

- l(r -- 1) ~ f(~)9~ (0, r ~o)(0)~b(()(a - Ir 2)- r/zdl~r(~)d#r(O ) U 2

0 = - n - ~ ~ / ( 0 ( 1 - I012) ~ 01(, r ~0)(0)4,(0(1 - I~1 ~)-'/2d#,.(Od#,.(O)

-- ~z- 1 ~ f(~)(1 - 1012)9 a (0, r 2 ~0 (U~-' (p) 0)~(r -- Ir z)-'/zd#,(~)d#,(o) U 2


- - r~- 1 ~f(~)(1 -- 1012)o 1 (0, ~)ff(u; , ~o)(0)r - I ffl 2)-'/2dm(0d~,(0). U 2

The first two terms on the r ight-hand side of this equat ion can be estimated in the same way as the terms in (VI.24) in the case o f j = 0. To estimate the third term we use (VI.23) (with r replaced by r + 1), integrate by parts with respect to 0 (by doing so we produce the missing factor of O( r - 1)), and use the Schwarz inequality as in the analysis o f j = 0. The resulting bound is Cr- 3/211 tp II II ~ II.

Let j = 2. We use

1 t3 0(1 -1012) ̀ -2 - r - 10ff (1 -1012) ' -1 ' (VI.27)

and integrate by parts with respect to 0 to obtain

- ~(r -- 1) j" f(~)g2(O, ()02ff(U r (p)(O)O(~)(1 -- I~ 12)-'/2d#,(()d#,(0) U 2

= - rc -1 ~ f(~)(1 - I012 ) ov~g2 (0, ~)lO12(u?, ~0)(0)~(~)(1 - I~12 ) -"/2d~,(~)d~,(O) U 2

_ ~ - 1 ~ f ( ~ ) ( 1 - 1012)g2(0, ~)O(U? 1~0) (0 )~(~) (1 - 1Wl2)-'/2d#r(~)d~,(O). U z

Both terms on the r ight-hand side of this equat ion have a familiar structure and, by the methods explained above, they can be estimated by Cr- 3/2 II ~0 II II ~ II.

Finally, let j = 3. Using (VI.27) and integrating by parts with respect to 0 we obtain

g - l ( r - - 1) ~ f(~)ga(O, ~)03(U (1 ~0)(0)~(()(1 - I~ 12)-'/2d#r(~)d#,(0) U 2

rc-x ~ f(()(1 - 10 [2) tTv~-O3 (0, ()02(U (~ ~0)(0)~(~)(1 - l(]2)-'/2d#,(()d#,(0). U 2

Again, as in the case o f j = 0, this can be estimated by Cr- 3/2 [I (P 11 I1 $ I[. []

In conclusion, we notice that formula (VI.9) provides a systematic basis for generating an asymptot ic expansion of Ty T o in powers of r - 1. The methods we have used to estimate the second order remainder in this expansion can presumably be used to estimate the remainder of arbi t rary order.

Acknowledgements. We would like to express our gratitude to Arthur Jaffe, Palle Jorgensen, Marc Rieffel and especially Raul Curto for very helpful discussions.

R e f e r e n c e s

1. Axler, S., Conway, J. B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. 34, 466-483 (1982)

2. Bayen, R., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 100, 61-151 (1978)

3. Bellisard, J.: K-theory of ~*-algebras in solid state physics. Lecture Notes in Physics vol 257, pp. 99-156. Berlin, Heidelberg, New York: Springer 1986

4. Berezin, F. A.: General concept of quantization. Commun. Math. Phys. 40, 153-174 (1975)


5. Brown, L. G., Douglas, R. G., Fillmore, P. A.: Extensions of ~*-algebras and K-homology. Ann. Math. 105, 265-324 (1977)

6. Coburn, L. A.: Singular integral operators and Toeplitz operators on odd spheres. Indiana Univ. Math. J. 23, 433-439 (1973)

7. Connes, A.: A survey of foliations and operator algebras. Proc. Symp. Pure Math. 38, 521- 628 (1982)

8. Connes, A.: Non-commutative differential geometry. Publ. Math. IHES 62, 94-144 (1986) 9. Connes, A., Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups.

Topology 29, 345-388 (1990) 10. Douglas, R. G.: Banach algebra techniques in operator theory. New York, San Francisco,

London: Academic Press 1972 11. Klimek, S., Lesniewski, A.: to appear 12. Klimek, S., Lesniewski, A.: to appear 13. Moore, G.: Geometry of the string equations. Commun. Math. Phys. 133, 261-304 (1990) 14. Moreno, C., Ortega-Navarro, P.: Deformations of the algebra of functions on Hermitian

symmetric spaces resulting from quantization. Ann. Inst. H. Poincar6 38, 215-241 (1983) 15. Petersson, H.: Zur Analytischen Theorie der Grenzkreisgruppen, Teil I. Math. Z. 115, 23-67

(1938) 16. Podles, P.: Quantum spheres. Lett. Math. Phys. 14, 193-202 (1987) 17. Rieffel, M.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122,

531-562 (1989) 18. Rieffel, M.: Non-commutative Tori - a case study of non-commutative differentiable mani-

folds. Contemp. Math. 105, 191-211 (1990) 19. Rieffel, M.: Deformation quantization and operator algebras. Proc. Symp. Pure Math. 51,

411-423 (1990) 20. Sheu, A. J.-L., Lu, J.-H., Weinstein, A.: Quantization of the Poisson SU(2) and its Poisson

Homogeneous space the 2-Sphere. Commun. Math. Phys. 135, 217-232 (1991) 21. Unterberger, A. J.: A quantization of the Cartan domain BD 1 (q = 2) and operators on the

light cone. J. Funct. Anal. 72, 279-319 (1987) 22. Woronowicz, S. L.: Twisted SU(2) group: an example of non-commutative differential calculus.

Publ. RIMS 23, 117-181 (1987)

Communicated by A. Jaffe

Documents

Quantum Riemann surfaces I. The unit disc - Lesniewskilesniewski.us/papers/published/QuantumRiemannSurfacesI.pdf · Quantum Riemann Surfaces I. The Unit Disc* Slawomir Klimek and