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 \mathcal{S} is defined as the projective limit of
Wnp. 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∞ algebra \mathcal{S}. Appendix treats the
K-theory of the noncommutative cubic space.
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