Vol. 299, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
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

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/pjm

We have not been able to recognize your IP address 3.84.130.252 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