Quantum Riemann surfaces I. The unit disc

Abstract

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

Noncommutative Riemann Surfaces by Embeddings in $${\mathbb{R}^{3}}$$

“…Encouraged by the explicit construction and by the fact that for the two-torus our results almost coincide with the older results of [BHSS91], we are quite optimistic that for the case of genus g ≥ 2 this embedding approach may give more explicit constructions than the existence proof in [KL92] and [BMS94].…”

“…The result was dubbed "Fuzzy Sphere" in [Mad92]. The papers [KL92] prove that the (complexified) Poisson algebra of functions on any Riemann surface arises as a N → ∞ limit of gl(N , C) -which had been conjectured in [BHSS91]. This result was extended to any quantizable compact Kähler manifold in [BMS94], the technical tool being geometric and Berezin-Toeplitz quantization.…”

Quantum Hall Effect on the Hyperbolic Plane

“…The group G also has an induced σ representation W on this space, and we shall suppose that there is an irreducible subrepresentation on a subspace which is projected out by P . (This is certainly true in the case considered in [Klim+Les1].) The algebra P M (C c (G/K))P then gives the non-commutative analogue of C c (G/K).…”

Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

“…The formula (73) also implies that TzTz = Tz2 (even though T z T z = T z 2 ) and, in particular, that T * z = Tz on H s h,hol ; thus we see that the validity of (21) and (22) is indeed restricted to subspaces of L 2 in general.…”

“…Finally, since D is a bounded domain and thus polynomials are automatically bounded on D, the validity of (17) [22]; this time, with the total set V = C ∞ (D). (Again, they only proved (6) for N ≤ 1, but their argument easily extends to arbitrary N .…”