Disintegration of positive isometric group representations on
$$\varvec{\mathrm {L}^{p}}$$
L
p
-spaces

Disintegration of positive isometric group representations on
$$\varvec{\mathrm {L}^{p}}$$...
Jeu, Marcel; Rozendaal, Jan
2017-05-06 00:00:00
Let G be a Polish locally compact group acting on a Polish space
$${{X}}$$
X
with a G-invariant probability measure
$$\mu $$
μ
. We factorize the integral with respect to
$$\mu $$
μ
in terms of the integrals with respect to the ergodic measures on X, and show that
$$\mathrm {L}^{p}({{X}},\mu )$$
L
p
(
X
,
μ
)
(
$$1\le p<\infty $$
1
≤
p
<
∞
) is G-equivariantly isometrically lattice isomorphic to an
$${\mathrm {L}^p}$$
L
p
-direct integral of the spaces
$$\mathrm {L}^{p}({{X}},\lambda )$$
L
p
(
X
,
λ
)
, where
$$\lambda $$
λ
ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of
$$\mathrm {L}^{p}({{X}},\mu )$$
L
p
(
X
,
μ
)
as an
$${\mathrm {L}^p}$$
L
p
-direct integral of order indecomposable representations. If
$$({{X}}^\prime ,\mu ^\prime )$$
(
X
′
,
μ
′
)
is a probability space, and, for some
$$1\le q<\infty $$
1
≤
q
<
∞
, G acts in a strongly continuous manner on
$$\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )$$
L
q
(
X
′
,
μ
′
)
as isometric lattice automorphisms that leave the constants fixed, then G acts on
$$\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })$$
L
p
(
X
′
,
μ
′
)
in a similar fashion for all
$$1\le p<\infty $$
1
≤
p
<
∞
. Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If
$$({{X}}^\prime ,\mu ^\prime )$$
(
X
′
,
μ
′
)
is separable, the representation of G on
$$\mathrm {L}^p(X^\prime ,\mu ^\prime )$$
L
p
(
X
′
,
μ
′
)
can then be disintegrated into order indecomposable representations. The notions of
$${\mathrm {L}^p}$$
L
p
-direct integrals of Banach spaces and representations that are developed extend those in the literature.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngPositivitySpringer Journalshttp://www.deepdyve.com/lp/springer-journals/disintegration-of-positive-isometric-group-representations-on-varvec-sPaY2am1nr

Disintegration of positive isometric group representations on
$$\varvec{\mathrm {L}^{p}}$$
L
p
-spaces

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