Abstract
We study the functor ℓ2 from the category of partial injections to the category of Hilbert spaces. The former category is finitely accessible, and in both categories homsets are algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces.
An erratum for this chapter can be found at http://dx.doi.org/10.1007/978-3-642-38164-5_26
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramsky, S.: Retracing some paths in process algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 1–17. Springer, Heidelberg (1996)
Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored *-categories. Journal of Pure and Applied Algebra 143, 3–47 (1999)
Abramsky, S., Heunen, C.: H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. In: Abramsky, S., Mislove, M. (eds.) Clifford Lectures. Proceedings of Symposia in Applied Mathematics, vol. 71, pp. 1–24. American Mathematical Society (2012)
Abramsky, S., Jung, A.: Domain theory. In: Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford University Press (1994)
Adámek, J., Rosicky, J.: Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press (1994)
Barr, M.: Algebraically compact functors. Journal of Pure and Applied Algebra 82, 211–231 (1992)
Bousfield, A.K.: Constructions of factorization systems in categories. Journal of Pure and Applied Algebra 9(2-3), 207–220 (1977)
Robin, J., Cockett, B., Lack, S.: Restriction categories I: categories of partial maps. Theoretical Computer Science 270(1-2), 223–259 (2002)
Danos, V., Regnier, L.: Proof-nets and the Hilbert space. In: Advances in Linear Logic, pp. 307–328. Cambridge University Press (1995)
Freyd, P., Kelly, M.: Categories of continuous functors I. Journal of Pure and Applied Algebra 2 (1972)
Grandis, M., Tholen, W.: Natural weak factorization systems. Archivum Mathematicum 42, 397–408 (2006)
Haghverdi, E.: A categorical approach to linear logic, geometry of proofs and full completeness. PhD thesis, University of Ottawa (2000)
Haghverdi, E., Scott, P.: A categorical model for the geometry of interaction. Theoretical Computer Science 350, 252–274 (2006)
Halmos, P.: A Hilbert space problem book, 2nd edn. Springer (1982)
Heunen, C.: An embedding theorem for Hilbert categories. Theory and Applications of Categories 22(13), 321–344 (2009)
Heunen, C., Jacobs, B.: Quantum logic in dagger kernel categories. Order 27(2), 177–212 (2010)
Hines, P.: The algebra of self-similarity and its applications. PhD thesis, University of Wales (1997)
Hines, P.: Quantum circuit oracles for abstract machine computations. Theoretical Computer Science 411(11-13), 1501–1520 (2010)
Hines, P., Braunstein, S.L.: The structure of partial isometries. In: Semantic Techniques in Quantum Computation, pp. 361–389. Cambridge University Press (2009)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras. Academic Press (1983)
Korostenski, M., Tholen, W.: Factorization systems as Eilenberg-Moore algebras. Journal of Pure and Applied Algebra 85(1), 57–72 (1993)
Lawson, M.V.: Inverse semigroups: the theory of partial symmetries. World Scientific (1998)
Lane, S.M.: Categories for the Working Mathematician, 2nd edn. Springer (1971)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Functional Analysis, vol. I. Academic Press (1972)
Rosicky, J., Tholen, W.: Lax factorization algebras. Journal of Pure and Applied Algebra 175, 355–382 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Heunen, C. (2013). On the Functor ℓ2. In: Coecke, B., Ong, L., Panangaden, P. (eds) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science, vol 7860. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38164-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-38164-5_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38163-8
Online ISBN: 978-3-642-38164-5
eBook Packages: Computer ScienceComputer Science (R0)