Extending obstructions to noncommutative functorial spectra. (English) Zbl 1308.46078

The dual equivalence between the category of commutative \(C^{\ast}\)-algebras and that of compact Hausdorff topological spaces was established by I. Gelfand and M. Neumark [Contemp. Math. 167, 3–20 (1994; Zbl 0824.46060)] more than half a century ago, which was the starting point of Alain Connes’ noncommutative geometry and inspired Alexander Grothendieck to construct his famous scheme theory in the middle of the 20th century. Although many noncommutative spectra have been proposed, M. L. Reyes [Isr. J. Math. 192, Part B, 667–698 (2012; Zbl 1269.18001)] has observed that none of them behaves functorially, establishing that they could not without taking pain of trivialization on the prototypical noncommutative rings \(M_{n}(\mathbb C)\). His elaborate proof is based upon S. Kochen and E. P. Specker’s famous no-go theorem [J. Math. Mech. 17, 59–87 (1967; Zbl 0156.23302)]. The main result of this article is that this daunting obstruction cannot be circumvented simply by allowing locales, toposes, ringed spaces or even quantales as pointless extensions of the classical notion of space. The result is technically based upon the authors’ works [Appl. Categ. Struct. 20, No. 4, 393–414 (2012; Zbl 1261.46051); ibid. 21, No. 1, 103–104 (2013; Zbl 1328.46043)].
The second author and M. L. Reyes [J. Math. Anal. Appl. 416, No. 1, 289–313 (2014; Zbl 1310.46053)] have proposed a completely new notion of spectrum for arbitrary \(AW^{\ast}\)-algebras. Classically, C. A. Akemann [J. Funct. Anal. 4, 277–294 (1969; Zbl 0177.17603); Pac. J. Math. 39, 1–11 (1971; Zbl 0203.44502)] and R. Giles and H. Kummer [Indiana Univ. Math. J. 21, 91–102 (1971; Zbl 0205.26801)] have proposed a noncommutative notion of spectrum, but the desired correspondence is merely functorial partially. The so-called Bohrification, giving rise to a functor from the category of \(C^{\ast}\)-algebras to that of localic topoi, loses some essential information, enabling one only to reconstruct the partial \(C^{\ast}\)-algebra structure of the original \(C^{\ast}\)-algebra. J. Rosický [Cah. Topologie Géom. Différ. Catég. 30, No. 2, 95–110 (1989; Zbl 0676.46047)] constructed a functor from the category of \(C^{\ast}\)-algebras to that of quantum frames, taking into account only the Jordanian structure of the original \(C^{\ast}\)-algebra. There is no nondegenerate functor between the category of quantum frames and that of quantales.


46M15 Categories, functors in functional analysis
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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