Dagger Categories and Formal Distributions

A nuclear ideal is an ideal contained in an ambient monoidal dagger category which has all of the structure of a compact closed category, except that it lacks identities. Intuitively, the identities are too “singular” to live in the ideal. Typical examples include the ideal of Hilbert-Schmidt maps contained in the category of Hilbert spaces, or the ideal of test functions contained in the category DRel of tame distributions on Euclidean space.

[1]  A. Paul,et al.  Pacific Journal of Mathematics , 1999 .

[2]  John E. Roberts,et al.  A new duality theory for compact groups , 1989 .

[3]  D. A. Higgs,et al.  Nuclearity in the category of complete semilattices , 1989 .

[4]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[5]  Radha Jagadeesan,et al.  New foundations for the geometry of interaction , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[6]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[7]  P. Ghez,et al.  $W^\ast$-categories. , 1985 .

[8]  V. Kac.  Vertex algebras for beginners , 1997 .

[9]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[10]  Dusko Pavlovic,et al.  Quantum measurements without sums , 2007 .

[11]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[12]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[13]  Bob Coecke,et al.  POVMs and Naimark's Theorem Without Sums , 2006, QPL.

[14]  Peter Selinger,et al.  Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract) , 2007, QPL.

[15]  A. Grothendieck,et al.  Produits Tensoriels Topologiques Et Espaces Nucleaires , 1966 .

[16]  P. Panangaden,et al.  Nuclear and trace ideals in tensored-categories , 1998, math/9805102.

[17]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, LICS 2004.

[18]  I. Stamatescu,et al.  Decoherence and the Appearance of a Classical World in Quantum Theory , 1996 .

[19]  M. A. Al-Gwaiz.  Theory of distributions , 1992 .

[20]  Richard Blute,et al.  Feedback for linearly distributive categories: traces and fixpoints , 2000 .

[21]  Samson Abramsky,et al.  Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories , 2005, CALCO.

[22]  F. Trèves.  Topological vector spaces, distributions and kernels , 1967 .

[23]  R. Borcherds.  VERTEX ALGEBRAS FOR BEGINNERS (University Lecture Series 10) , 1998 .

[24]  S. Majid.  Foundations of Quantum Group Theory , 1995 .

[25]  L. Schwartz.  Théorie des distributions , 1966 .

[26]  Erkki J. Brändas,et al.  Decoherence and the Appearance of a Classical World in Quantum Theory : E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch and I.-O Stamatescu, Springer-Verlag, New York, 2003 , 2004 .

[27]  R. Borcherds.  Vertex algebras , 1997, q-alg/9706008.

[28]  Samson Abramsky,et al.  Abstract Physical Traces , 2009, ArXiv.

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[30]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .