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 p n associated with the W * -algebra generated by a semicircular system, and the C ∞ algebra S is defined as the projective limit of W p 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.
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