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Abstract
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We continue our investigations of the representation theoretic
side of reflection positivity by studying positive definite functions
on the
additive group
satisfying a suitably defined KMS condition. These functions take values in the space
of bilinear forms on
a real vector space .
As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic
continuation of
to the strip
with a coupling condition
on the boundary. Our first main result consists of a characterization of these functions in terms of
modular objects
( an antilinear
involution and
selfadjoint with
)
and an integral representation.
Our second main result is the existence of a
-valued positive
definite function
on the group
with
satisfying
for
. We thus obtain
a
-periodic
unitary one-parameter group on the GNS space
for which the one-parameter group on the GNS space
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
on the circle
of length
.
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Keywords
KMS condition, reflection positivity, standard real
subspace
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Mathematical Subject Classification 2010
Primary: 43A35, 43A65, 47L30
Secondary: 47L90, 81T05
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Milestones
Received: 31 October 2016
Accepted: 13 September 2018
Published: 18 April 2019
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