# Noncommutative Sobolev Spaces, $C_{∞}$ Algebras and Schwartz Distributions Associated with Semicircular Systems

### Masaru Mizuo

Tohoku University, Sendai, Japan

## 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 $W_{n}$ associated with the $W_{∗}$-algebra generated by a semicircular system, and the $C_{∞}$ algebra $S$ is deﬁned as the projective limit of $W_{n}$. The Schwartz distribution space is then deﬁned as the dual space of $S$ and the Fourier representation theorem is obtained for Schwartz distributions. We furthermore discuss vector ﬁelds on the $C_{∞}$ algebra $S$. Appendix treats the $K$-theory of the noncommutative cubic space.

## Cite this article

Masaru Mizuo, Noncommutative Sobolev Spaces, $C_{∞}$ Algebras and Schwartz Distributions Associated with Semicircular Systems. Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, pp. 331–363

DOI 10.2977/PRIMS/1145476106