Noncommutative Sobolev Spaces, <em>C</em>^∞ 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 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.