Spectral-Geometric Deformations of Function Algebras on Manifolds

Abstract: We propose a novel method for deforming the algebra of smooth functions on a compact Riemannian manifold based on spectral data rather than group actions or Poisson structures. The deformation is defined through spectral coefficients of the Laplacian and a weight function satisfying a fusion-type cocycle condition, producing a noncommutative product that depends intrinsically on Laplacian spectrum of the manifold. We develop the analytic and algebraic foundations of this construction, establishing associativity, continuity, and the existence of a group structure on admissible weights. We show that the construction is functorial with respect to spectrum-preserving maps of manifolds. This spectral-geometric deformation quantization approach provides a fully intrinsic analytic model of noncommutative geometry based solely on spectral data of the manifold.