Abstract
We introduce the notion of a von Neumann category, as a multi-object version of von Neumann algebra. A von Neumann category is a premonoidal category with compatible ∗-structure which embeds as a double commutant into a suitable premonoidal category of Hilbert spaces. The notion was inspired by algebraic quantum field theory, and we do explain that basic idea and indicate how quantum teleportation could be encoded in this setting. But we focus here on the structure of von Neumann categories. After giving the definitions and examples, we consider constructions typically associated to von Neumann algebras, and examine their extensions to the category-theoretic setting. In particular, we present a crossed product construction for ∗-premonoidal categories.
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Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. in proceedings of the 19th annual IEEE symposium on logic in computer science 2004, IEEE Comput. Soc. Press (2004)
Benton, N., Hyland, M.: Traced premonoidal categories. Theor. Informa. Appl. 37, 273–299 (2003)
Clifton, R., Halvorson, H.: Entanglement and open systems in algebraic quantum field theory, Philosophy of Science, Vol. 69, p 128 (2002)
Chuang, I., Nielsen, M.: Quantum computation and quantum information. Cambridge University Press (2000)
Coecke, B., Lal, R.: Causal categories: relativistically interacting processes. preprint (2011)
Comeau, M.: Premonoidal ∗-categories and algebraic quantum field theory, thesis, university of Ottawa (2012)
Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. math. 98, 157–218 (1989)
Ghez, P., Lima, R., Roberts, J.E.: W ∗-categories. Pac. J. Math. 120, 79–109 (1985)
Halvorson, H. In: Butterfield, J., Earman, J. (eds.) : Algebraic quantum field theory, philosophy of physics, pp 731–922 (2006). North-Holland
Jones, V., Sunder, V.: Introduction to subfactors. Cambridge University Press (1997)
Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88, 55–112 (1991)
Kadison, R., Ringrose, J.: Fundamentals of the theory of operator algebras, graduate studies in mathematics, american mathematical society (1997)
Mac Lane, S., Second Edition: Categories for the working mathematician. Springer-Verlag (2000)
Penrose, R.: Techniques of differential topology in relativity. CBMS-NSF regional conference series in applied mathematics (1972)
Penrose, R.: Applications of negative dimensional tensors, in combinatorial mathematics and its applications. In: Welsh, D. J. A. (ed.) , pp 221–244. Academic Press, New York (1971)
Power, J., Robinson, E.: Premonoidal categories and models of computation. Math. Struct. Comput. Sci. 7, 453–468 (1997)
Roberts, J. In: Kastler, D. (ed.) : Lectures on algebraic quantum field theory.In: the algebraic theory of superselection sectors, pp 1–112. World Scientic (1990)
Selinger, P.: Dagger compact closed categories and completely positive maps. Electron. Notes Theor. Comput. Sci. 170, 139–163 (2007)
Selinger, P.: Finite-dimensional Hilbert spaces are complete for dagger compact closed categories. preprint (2012)
Sunder, V.: An invitation to Von Neumann algebras. Springer-Verlag (1987)
Williams, D.: Crossed products of C ∗-algebras, american mathematical society (2007)
Williams, D. Lecture notes on C ∗-algebras (2011). http://www.math.dartmouth.edu/m123s11
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Research supported in part by NSERC.
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Blute, R., Comeau, M. Von Neumann Categories. Appl Categor Struct 23, 725–740 (2015). https://doi.org/10.1007/s10485-014-9375-6
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DOI: https://doi.org/10.1007/s10485-014-9375-6