Abstract. For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞. IntroductionIn a couple of papers titled "Quantum Riemann Surfaces" [24] S. Klimek and A. Lesniewski have recently proved a classical limit theorem for the Poisson algebra of smooth functions on a compact Riemann surface Σ of genus g ≥ 2 (with Petersson Kähler structure) using the Toeplitz quantization procedure:{f,g} || = 0 .(1-2)Here, 1 = 1, 2, . . . are tensor powers of the quantizing Hermitian line bundle (L, h) over M , and the Toeplitz operators act on the Hilbert space of holomorphic sections of L 1/ as the holomorphic part of the operator that multiplies sections with f . As usual (1-2) gives the connection between the Poisson bracket of functions and the commutator of the associated operators and (1-1) prevents the theory from being empty. Compared to Berezin's covariant symbols [3] and to the concept of star products [2,6,9,11], where the basic idea is the deformation of the algebraic structure on C ∞ (M ) using as a formal deformation parameter, the emphasis lies here more on the approximation of C ∞ (M ) by operator algebras in norm sense.1991 Mathematics Subject Classification. 32J81, 58F06, 17B65, 47B35, 81S10.

We give a complete identi®cation of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Ka Èhler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classi®es star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szego È kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjo È strand.

Abstract. Multi-point algebras of Krichever Novikov type for higher genus Riemann surfaces are generalisations of the Virasoro algebra and its related algebras. Complete existence and uniqueness results for local 2-cocycles defining almost-graded central extensions of the functions algebra, the vector field algebra, and the differential operator algebra (of degree ≤ 1) are shown. This is applied to the higher genus, multi-point affine algebras to obtain uniqueness for almost-graded central extensions of the current algebra of a simple finite-dimensional Lie algebra. An earlier conjecture of the author concerning the central extension of the differential operator algebra induced by the semi-infinite wedge representations is proved.

For the generalized Krichever-Novikov algebras of meromorphic vector fields and their induced modules of weight A a different basis is given. With respect to this basis the module structure is generalized graded. "Local" central extensions of these algebras and their representations on the space of semi-infinite wedge product of forms of weight Aare studied. In this generalization one obtains again c = -2(6A 2 -6A+ 1) as value for the central charge.

Abstract. For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via BerezinToeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given.dedicated to the memory of Moshe Flato