Noncommutative Sobolev Spaces, C^∞ Algebras and Schwartz Distributions Associated with Semicircular Systems

Abstract

When the C^*-algebra and the W^*-algebra generated by a semicircular system are viewed from the viewpoints of noncommutative topology and noncommutative probability theory, we may consider the C^*-algebra as a certain kind of a “noncommutative cubic space” and the W^*-algebra as a “noncommutative cubic measure space.” In this paper we introduce the Sobolev spaces W_n^p associated with the W^*-algebra generated by a semicircular system, and the C^∞ algebra \mathcal{S} is defined as the projective limit of W_n^p. The Schwartz distribution space is then defined as the dual space of \mathcal{S} and the Fourier representation theorem is obtained for Schwartz distributions. We furthermore discuss vector fields on the C^∞ algebra \mathcal{S}. Appendix treats the K-theory of the noncommutative cubic space.