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Obstructing extensions of the functor Spec to noncommutative rings. (English) Zbl 1269.18001

The author proves that if \(F\) is a contravariant functor from the category of rings to the category of sets such that the restriction of \(F\) to the subcategory of commutative rings is isomorphic to \(Spec\) (the prime spectrum functor), then \(F(M_n(\mathbb{C}))=\emptyset\) for any \(n\geq 3\). An analogous result is proved in the context of \(C^*\)-algebras, by showing that for a contravariant functor \(F\) from the category of unital \(C^*\)-algebras to the category of sets whose restriction to the subcategory of commutative unital \(C^*\)-algebras is isomorphic to \(Max\) (the set of maximal ideals functor), one has \(F(M_n(\mathbb{C}))=\emptyset\) for any \(n\geq 3\).

MSC:

18A22 Special properties of functors (faithful, full, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
16D90 Module categories in associative algebras
46L05 General theory of \(C^*\)-algebras
46M15 Categories, functors in functional analysis
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References:

[1] M. Artin and W. Schelter, Integral ring homomorphisms, Advances in Mathematics 39 (1981), 289–329. · Zbl 0461.16014 · doi:10.1016/0001-8708(81)90005-0
[2] B. van den Berg and C. Heunen, Noncommutativity as a colimit, Applied Categorical Structures (2011), available at http://dx.doi.org/10.1007/s10485-011-9246-3 .
[3] B. van den Berg and C. Heunen, No-go theorems for functorial localic spectra of noncom-mutative rings, arXiv:1101.5924, Proceedings of the 8th Workshop on Quantum Physics and Logic, to appear.
[4] P. M. Cohn, The affine scheme of a general ring, in Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Lecture Notes in Mathematics, Vol. 753, Springer, Berlin, 1979, pp. 197–211.
[5] K. R. Davidson, C*-Algebras by Example, Fields Institute Monographs, Vol. 6, American Mathematical Society, Providence, RI, 1996.
[6] B. Fuglede, A commutativity theorem for normal operators, Proceedings of the National Academy of Sciences of the United States of America 36 (1950), 35–40. · Zbl 0035.35804 · doi:10.1073/pnas.36.1.35
[7] O. Goldman, Rings and modules of quotients, Journal of Algebra 13 (1969), 10–47. · Zbl 0201.04002 · doi:10.1016/0021-8693(69)90004-0
[8] C. Heunen, N. P. Landsman and B. Spitters, A topos for algebraic quantum theory, Communications in Mathematical Physics 291 (2009), 63–110. · Zbl 1209.81147 · doi:10.1007/s00220-009-0865-6
[9] N. Jacobson, Schur’s theorems on commutative matrices, Bulletin of the American Mathematical Society 50 (1944), 431–436. · Zbl 0063.03016 · doi:10.1090/S0002-9904-1944-08169-X
[10] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, Graduate Studies in Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1997, Reprint of the 1983 original. · Zbl 0888.46039
[11] S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17 (1967), 59–87. · Zbl 0156.23302
[12] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189, Springer-Verlag, New York, 1999. · Zbl 0911.16001
[13] E. S. Letzter, On continuous and adjoint morphisms between non-commutative prime spectra, Proceedings of the Edinburgh Mathematical Society. Series II 49 (2006), 367–381. · Zbl 1126.16001 · doi:10.1017/S0013091504000628
[14] S. Mac Lane, Categories for the Working Mathematician, second edn., Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998. · Zbl 0906.18001
[15] C. Procesi, Non commutative Jacobson-rings, Annali della Scuola Normale Superiore di Pisa (3) 21 (1967), 281–290.
[16] A. L. Rosenberg, The left spectrum, the Levitzki radical, and noncommutative schemes, Proceedings of the National Academy of Sciences of the United States of America 87 (1990), 8583–8586. · Zbl 0767.14001 · doi:10.1073/pnas.87.21.8583
[17] A. L. Rosenberg, Noncommutative local algebra, Geometric and Functional Analysis 4 (1994), 545–585. · Zbl 0830.14002 · doi:10.1007/BF01896408
[18] I. Schur, Zur Theorie vertauschbaren Matrizen, Journal für die Reine und Angewandte Mathematik 130 (1905), 66–76.
[19] S. P. Smith, Subspaces of non-commutative spaces, Transactions of the American Mathematical Society 354 (2002), 2131–2171. · Zbl 0998.14003 · doi:10.1090/S0002-9947-02-02963-X
[20] F.M. J. Van Oystaeyen and A. H. M. J. Verschoren, Non-commutative Algebraic Geometry: An Introduction, Lecture Notes in Mathematics, Vol. 887, Springer-Verlag, Berlin, 1981. · Zbl 0477.16001
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