Noncommutative Sobolev Spaces, $C^{\infty }$ Algebras and Schwartz Distributions Associated with Semicircular Systems

Abstract

When the $C^{\ast }$-algebra and the $W^{\ast }$-algebra generated by a semicircular system are viewed from the viewpoints of noncommutative topology and noncommutative probability theory, we may consider the $C^{\ast }$-algebra as a certain kind of a “noncommutative cubic space” and the $W^{\ast }$-algebra as a “noncommutative cubic measure space.” In this paper we introduce the Sobolev spaces $W_n^p$ associated with the $W^{\ast }$-algebra generated by a semicircular system, and the $C^{\infty }$ 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 $C^{\infty }$ algebra $\mathcal{S}$. Appendix treats the $K$-theory of the noncommutative cubic space.