Next Article in Journal
Wave-Like Exact Models with Symmetry of Spatial Homogeneity in the Quadratic Theory of Gravity with a Scalar Field
Next Article in Special Issue
Towards a Measurement Theory for Off-Shell Quantum Fields
Previous Article in Journal
Clues about the Nature of Multiquark States
Previous Article in Special Issue
Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs
Article

Category Algebras and States on Categories

Nagahama Institute of Bio-Science and Technology, 1266 Tamura, Nagahama, Shiga 526-0829, Japan
Academic Editors: Motoichi Ohtsu and Alexey Kanel-Belov
Symmetry 2021, 13(7), 1172; https://doi.org/10.3390/sym13071172
Received: 28 May 2021 / Revised: 23 June 2021 / Accepted: 26 June 2021 / Published: 29 June 2021
(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of -categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups. View Full-Text
Keywords: category; algebra; state; category algebra; state on category; noncommutative probability; quantum probability; GNS representation category; algebra; state; category algebra; state on category; noncommutative probability; quantum probability; GNS representation
MDPI and ACS Style

Saigo, H. Category Algebras and States on Categories. Symmetry 2021, 13, 1172. https://doi.org/10.3390/sym13071172

AMA Style

Saigo H. Category Algebras and States on Categories. Symmetry. 2021; 13(7):1172. https://doi.org/10.3390/sym13071172

Chicago/Turabian Style

Saigo, Hayato. 2021. "Category Algebras and States on Categories" Symmetry 13, no. 7: 1172. https://doi.org/10.3390/sym13071172

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop