# Noncommutative Sobolev Spaces, <em>C</em><sup>∞</sup> 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 *Wnp* associated with the _W_∗-algebra generated by a semicircular system, and the _C_∞ algebra *S* is deﬁned as the projective limit of *Wnp*. 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, <em>C</em><sup>∞</sup> 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