Abstract
We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramsky S., Brandenburger A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011)
Abramsky S., Heunen C.: H*-algebras and nonunital Frobenius algebras: first steps in infinite dimensional categorical quantum mechanics. AMS Proc. Symp. Appl. Math. Clifford Lect. 71, 1–4 (2012)
Abramsky S., Heunen C.: Logic and algebraic structures in quantum computing and information.Chapter Operational Theories and Categorical Quantum Mechanics. Cambridge University Press, Cambridge (2015)
Aguiar M.: A note on strongly separable algebras. Bol. Acad. Nac. Ciencias 65, 51–60 (2000)
Albert A.A., Muckenhoupt B.: On matrices of trace zero. Mich. Math. J. 4(1), 1–3 (1957)
Antonevich A., Krupnik N.: On trivial and non-trivial n-homogeneous C*-algebras. Integr. Equ. Oper. Theory 38, 172–189 (2000)
Auslander M., Goldman O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)
Blackadar B.: Operator Algebras: Theory of C*-Algebras and Von Neumann Algebras. Springer, Berlin (2006)
Blanchard E., Kirchberg E.: Global Glimm halving for C*-bundles. J. Oper. Theory 52, 385–420 (2004)
Blecher D.P., LeMerdy C.: Operator Algebras and Their Modules, an Operator Space Approach. Oxford University Press, Oxford (2004)
Blute R., Comeau M.: Von Neumann categories. Appl. Categ. Struct. 23(5), 725–740 (2015)
Bos R.: Continuous representations of groupoids. Houst. J. Math. 37(3), 807–844 (2011)
Buss A., Zhu C., Meyer R.: A higher category approach to twisted actions on C*-algebras. Proc. Edinb. Math. Soc. 56, 387–426 (2013)
Coecke, B., Heunen, C., Kissinger, A.: Categories of quantum and classical channels. Quantum Information Processing (2014)
Coecke B., Lal R.: Causal categories: relativistically interacting processes. Found. Phys. 43(4), 458–501 (2012)
Dauns J., Hofmann K.H.: Representation of Rings by Sections. American Mathematical Society, Providence (1968)
DeMeyer F., Ingraham E.: Separable algebras over commutative rings Number 181 in Lecture Notes in Mathematics. Springer, Berlin (1971)
Dixmier, J.: C*-algebras. North Holland (1981)
Dixmier J., Douady A.: Champs continues d’espaces Hilbertiens et de C*-algèbres. Bull. Soc. Math. Fr. 91, 227–284 (1963)
Dupré M.J.: Classifying Hilbert bundles. J. Funct. Anal. 15(3), 244–278 (1974)
Enrique Moliner, P., Heunen, C., Tull, S.: Space in monoidal categories. Quantum Phsyics and Logic (2017). arXiv:1704.08086
Fell J.M.G.: An extension of Mackey’s method to Banach *-algebraic bundles Number 90 in Memoirs. American Mathematical Society, Providence (1969)
Fell J.M.G., Doran R.S.: Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles Number 125 126 in Pure and Applied Mathematics. Academic, Berlin (1988)
Frank M.: Hilbert C*-modules over monotone complete C*-algebras. Math. Nachr. 175, 61–83 (1995)
Frank,M., Paulsen,V.: Injective and projective Hilbert C*-modules, and C*-algebras of compact operators (2006). arXiv:math.OA/0611348
Furber R.W.J., Jacobs B.P.F.: From Kleisli categories to commutative C*-algebras: probabilistic Gelfand duality. Log. Methods Comput. Sci. 11(2), 5 (2015)
Ghez P., Lima R., Roberts J.E.: W*-categories. Pac. J. Math. 120, 79–109 (1985)
Gogioso, S., Genovese, F.: Infinite-dimensional categorical quantum mechanics. In: Quantum Physics and Logic (2016)
Hattori A.: On strongly separable algebras. Osaka J. Math. 2, 369–372 (1965)
Heunen, C.: An embedding theorem for Hilbert categories. Theory Appl. Categ. 22(13), 321–344 (2009)
Heunen, C., Jacobs, B.: Quantum logic in dagger kernel categories 27(2), 177–212 (2010)
Heunen C., Karvonen M.: Monads on dagger categories. Theory Appl. Categ. 31(35), 1016–1043 (2016)
Heunen, C., Kissinger, A.: Can quantum theory be characterized in terms of information-theoretic constraints? (2016). arXiv:1604.05948
Heunen, C., Kissinger, A., Selinger, P.: Completely positive projections and biproducts. In: Quantum Physics and Logic X, Number 171 in Electronic Proceedings in Theoretical Computer Science, pp. 71–83 (2014)
Heunen, C., Tull, S.: Categories of relations asmodels of quantumtheory. In: Quantum Physics and Logic XII, Number 195 in Electronic Proceedings in Theoretical Computer Science, pp. 247–261 (2015)
Heunen C., Vicary J.: Categories for Quantum Theory: An Introduction. Oxford University Press, Oxford (2017)
Heunen, C., Vicary, J., Wester, L.: Mixed quantum states in higher categories. In: Quantum Physics and Logic, volume 172 of Electronic Proceedings in Theoretical Computer Science, pp. 304–315 (2014)
Ivankov, P.: Quantization of noncompact coverings (2017). arXiv:1702.07918
Kelly G.M., Laplaza M.L.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980)
Lance E.C.: Hilbert C*-modules, volume 210 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1995) A toolkit for operator algebraists
Paschke W.L.: Inner product modules over B*-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)
Paulsen V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
Pavlov A.A., Troitskii E.V.: Quantization of branched coverings. Russ. J. Math. Phys. 18(3), 338–352 (2011)
Pluta R.: Ranges of Bimodule Projections and Conditional Expectations. Cambridge Scholars, Cambridge (2013)
Raeburn I., Williams D.P.: Morita Equivalence and Continuous-Trace C*-Algebras, volume 60 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1998)
Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Selinger, P.: A survey of graphical languages for monoidal categories. In: New Structures for Physics, Lecture Notes in Physics, pp. 289–355. Springer, Berlin (2009)
Størmer E.: Positive Linear Maps of Operator Algebras. Springer, Berlin (2013)
Takahashi A.: Hilbert modules and their representation. Rev. Columbiana Mat. 13, 1–38 (1979)
Taylor M.E.: Measure Theory and Integration. American Mathematical Society, Providence (2006)
Tomiyama J.: Topological representation of C*-algebras. Tohoku Math. J. 14(2), 187–204 (1962)
Tomiyama J., Takesaki M.: Applications of fibre bundles to the certain class of C*-algebras. Tohoku Math. J. 13(3), 498–522 (1961)
Tomiyama Y.: On the projection of norm one in W*-algebras. Proc. Jpn. Acad. 33(10), 608–612 (1957)
Vicary J.: Categorical formulation of quantum algebras. Commun. Math. Phys. 304(3), 765–796 (2011)
Vicary, J.: Higher quantum theory (2012). arXiv:1207.4563
Wegge-Olsen N.E.: K-Theory and C*-Algebras. Oxford University Press, Oxford (1993)
Yamagami S.: Frobenius duality in C*-tensor categories. J. Oper. Theory 52, 3–20 (2004)
Zito P.A.: 2-C*-categories with non-simple units. Adv. Math. 210(1), 122–164 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Heunen, C., Reyes, M.L. Frobenius Structures Over Hilbert C*-Modules. Commun. Math. Phys. 361, 787–824 (2018). https://doi.org/10.1007/s00220-018-3166-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3166-0