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.
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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. 331–363DOI 10.2977/PRIMS/1145476106