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

  • Masaru Mizuo

    Tohoku University, Sendai, Japan

Abstract

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

Cite this article

Masaru Mizuo, Noncommutative Sobolev Spaces, 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