2016/05/23 by Shane Farnsworth · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Holomorphic and Operator Theory #Homotopy and Cohomology in Algebraic Topology #Product (mathematics) #Spectral properties #Tensor (intrinsic definition) #Tensor product #Tensor product of Hilbert spaces #hep-th #math-ph #math.MP
paper · pdf · doi:10.1063/1.4975410
published as J. Math. Phys. 58, 023507, 2017 · 15 pages, no figures
arxiv created 2016/05/23 · openalex created_date 2016/06/24 · openalex publication_date 2017/02/01 · arxiv updated 2020/11/23 · openalex updated_date 2026/08/05
Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.