Vol. 299, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 314: 1  2
Vol. 313: 1  2
Vol. 312: 1  2
Vol. 311: 1  2
Vol. 310: 1  2
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
KMS conditions, standard real subspaces and reflection positivity on the circle group

Karl-Hermann Neeb and Gestur Ólafsson

Vol. 299 (2019), No. 1, 117–169
Abstract

We continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions ψ on the additive group (,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V ) of bilinear forms on a real vector space V . As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of ψ to the strip

{z 0 Imz β}

with a coupling condition ψ(iβ + t) = ψ(t)¯ on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (Δ,J) (J an antilinear involution and Δ > 0 selfadjoint with JΔJ = Δ1) and an integral representation.

Our second main result is the existence of a Bil(V )-valued positive definite function f on the group τ = {id,τ} with τ(t) = t satisfying f(t,τ) = ψ(it) for 0 t β. We thus obtain a 2β-periodic unitary one-parameter group on the GNS space f for which the one-parameter group on the GNS space ψ is obtained by Osterwalder–Schrader quantization.

Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels

(λ2 d2dt2)1

on the circle β of length β.

PDF Access Denied

We have not been able to recognize your IP address 54.144.55.253 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
KMS condition, reflection positivity, standard real subspace
Mathematical Subject Classification 2010
Primary: 43A35, 43A65, 47L30
Secondary: 47L90, 81T05
Milestones
Received: 31 October 2016
Accepted: 13 September 2018
Published: 18 April 2019
Authors
Karl-Hermann Neeb
Department Mathematik
Friedrich-Alexander Universität Erlangen-Nürnberg
Erlangen
Germany
Gestur Ólafsson
Department of Mathematics
Louisiana State University
Baton Rouge, LA
United States