From: rblute@mathstat.uottawa.ca (Richard Blute)
To: categories@mta.ca
Subject: Paper available
Date: Fri, 27 Feb 1998 09:33:00 -0500 [thread overview]
Message-ID: <9802271433.AA22493@castor> (raw)
The following paper is available by anonymous ftp at triples.math.mcgill.ca
in the directory pub/blute as nuclear.ps.gz. It is also on Prakash Panangaden's
homepage at www-acaps.cs.mcgill.ca. Feel free to contact me if there
are any problems.
Cheers,
Rick Blute
Nuclear and Trace Ideals in Tensored *-Categories
=================================================
Samson Abramsky Richard Blute
Department of Computer Science Department of Mathematics
University of Edinburgh and Statistics
Edinburgh, Scotland University of Ottawa
Ottawa, Ontario, Canada
Prakash Panangaden
Department of Computer Science
McGill University
Montreal, Quebec, Canada
Presented to Mike Barr on the occasion of his 60th birthday.
Abstract
========
We generalize the notion of nuclear maps from functional analysis by
defining nuclear ideals in tensored *-categories. The motivation for
this study came from attempts to generalize the structure of the category
of relations to handle what might be called ``probabilistic relations''.
The compact closed structure associated with the category of relations
does not generalize directly, instead one obtains nuclear ideals.
Most tensored *-categories have a large class of morphisms
which behave as if they were part of a compact closed category, i.e. they
allow one to transfer variables between the domain and the codomain. We
introduce the notion of nuclear ideals to analyze these classes of
morphisms. In compact closed categories, we see that all morphisms
are nuclear, and in the category of Hilbert spaces, the nuclear morphisms
are the Hilbert-Schmidt maps.
We also introduce two new examples of tensored *-categories, in which
integration plays the role of composition. In the first, morphisms are a
special class of distributions, which we call tame distributions.
We also introduce a category of probabilistic relations which was
the original motivating example.
Finally, we extend the recent work of Joyal, Street and Verity
on traced monoidal categories to this setting by introducing the notion
of a trace ideal. For a given symmetric monoidal category, it is not
generally the case that arbitrary endomorphisms can be assigned a trace.
However, we can find ideals in the category on which a trace can be
defined satisfying equations analogous to those of Joyal, Street and
Verity. We establish a close correspondence between nuclear ideals and
trace ideals in a tensored *-category, suggested by the correspondence
between Hilbert-Schmidt operators and trace operators on a Hilbert space.
When we apply our notion of trace ideal to the category of Hilbert spaces,
we obtain the usual trace of an endomorphism in the trace class.
next reply other threads:[~1998-02-27 14:33 UTC|newest]
Thread overview: 7+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-02-27 14:33 Richard Blute [this message]
1998-10-30 12:24 Marta Bunge
2005-08-17 22:32 Walter Tholen
2007-09-04 17:51 Michael A. Warren
2009-01-08 22:00 Walter Tholen
2009-01-09 18:03 vs27
2009-01-10 7:08 Vaughan Pratt
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