Extending obstructions to noncommutative functorial spectra

Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of nxn-matrices for n at least 3. This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce. We extend these no-go results to functors with values in (ringed) topological spaces, (ringed) toposes, schemes, and quantales. The possibility of spectra in other categories is discussed.

[1]  L. Birch,et al.  The concept of nature , 1992 .

[2]  P. Johnstone.  Sketches of an Elephant: A Topos Theory Compendium Volume 1 , 2002 .

[3]  Robin Giles,et al.  A Non-Commutative Generalization of Topology , 1971 .

[4]  A. C. Ehresmann,et al.  CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES , 2008 .

[5]  J. Rosický.  Multiplicative lattices and $C^\ast$-algebras , 1989 .

[6]  Sabine Fenstermacher,et al.  Sheaves In Geometry And Logic A First Introduction To Topos Theory , 2016 .

[7]  Miklós Rédei,et al.  Quantum Logic in Algebraic Approach , 1998 .

[8]  Andreas Döring.  Kochen–Specker Theorem for von Neumann Algebras , 2005 .

[9]  Charles A. Akemann,et al.  The general Stone-Weierstrass problem , 1969 .

[10]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[11]  Manuel L. Reyes,et al.  Obstructing extensions of the functor spec to noncommutative rings , 2011, 1101.2239.

[12]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[13]  Ladislav Beran,et al.  Orthomodular Lattices: Algebraic Approach , 1985 .

[14]  Hans Halvorson,et al.  Deep beauty : understanding the quantum world through mathematical innovation , 2011 .

[15]  G. Hicks.  An Enquiry concerning the Principles of Natural Knowledge , 1920, Nature.

[16]  Tsit Yuen Lam,et al.  Lectures on modules and rings , 1998 .

[17]  Logical Reflections on the Kochen-Specker Theorem , 1996 .

[18]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[19]  Chris Heunen,et al.  Active lattices determine AW*−algebras , 2012, 1212.5778.

[20]  Chris Heunen,et al.  Noncommutativity as a Colimit , 2010, Appl. Categorical Struct..

[21]  P. Johnstone,et al.  The point of pointless topology , 1983 .

[22]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[23]  A. Grothendieck,et al.  Théorie des Topos et Cohomologie Etale des Schémas , 1972 .

[24]  Jirí Rosický,et al.  On Quantales and Spectra of C*-Algebras , 2003, Appl. Categorical Struct..