Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras

2012/12/11 by Andreas Doering, Doering, Andreas · 1 citation
Materials Science · Mathematics · #18F20 #81R60 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Lanthanide and Transition Metal Complexes #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Primary 46L05 #Quantum Physics (quant-ph) #Secondary 81R15

paper · pdf · doi:10.48550/arxiv.1212.2613

openalex publication_date 2012/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To each unital C*-algebra A we associate a presheaf Σ^A, called the spectral presheaf of A, which can be regarded as a generalised Gelfand spectrum. We develop a categorical notion of local duality and show that there is a contravariant functor from the category of unital C*-algebras to a suitable category of presheaves containing the spectral presheaves. We clarify how much algebraic information about a C*-algebra is contained in its spectral presheaf. A nonabelian unital C*-algebra A that is neither isomorphic to C² nor to B(C²) is determined by its spectral presheaf up to quasi-Jordan isomorphisms. For a particular class of unital C*-algebras, including all von Neumann algebras with no type I₂ summand, the spectral presheaf determines the Jordan structure up to isomorphisms.

Citations

• Self-adjoint Operators as Functions II: Quantum Probability

Cited by

• Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory

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