From a finite von Neumann algebra \mathfrak{F} with a faithful normal trace and its normal injective * endomorphism φ satisfying φ(\mathfrak{F})=φ(1)\mathfrak{F}φ(1), we construct another von Neumann algebra
M(\mathfrak{F}, φ) by a method which reduces to group-measure construction when \mathfrak{F} is commutative and φ is an automorphism. If φ satisfies φ(z)=φ(1)
z for all central elements
z of \mathfrak{F} and φ(1)
{\
atural}=
e−a for a positive number
a, then
M(\mathfrak{F}, φ) has the following 3 properties: (1) It has a faithful normal state ρ whose modular operator Δ
ρ has the spectrum {0}∪{
ena;
n=0, ±1, ...}=
Sea. (2) The set \mathfrak{M}
0 of all elements of
M(\mathfrak{F}, φ), commuting with Δ
ρ is isomorphic to \mathfrak{F}. (3) The center of \mathfrak{M}
0 coincides with the center of \mathfrak{M}.
Conversely, any von Neumann algebra with a faithful normal state ρ such that log Δ
ρ has exclusively an isolated point spectrum and the center of \mathfrak{M}
0 coincides with its center is a direct sum of
M(\mathfrak{F}
j, φ
j),
j=1, ..., and possibly a finite von Neumann algebra, where each φ
j satisfies φ
j(\mathfrak{F})=φ
j(1)\mathfrak{F}φ
j(1), φ
j(
z)=
zφ
j(1) for all central element
z of \mathfrak{F} and φ
j(1)
{\
atural}=
e−aj.
If ρ is a
KMS state under time translation of a
C* algebra, which is asymptotically abelian with respect to (either discrete or continuous) space translation and if the spectrum of generator of time translation has exclusively an isolated point spectrum in the representation associated with ρ, then the associated von Neumann algebra has the above structure where the asymptotic ratio set of \mathfrak{F}
j as well as that of a possible finite summand (if non-zero) is {1} and
r∞(
M(\mathfrak{F}
j, φ
j))=
Sxj where
xj=
eaj. The last result on asymptotic ratio set is limited to the case where the representation space is separable.
A generalization of
M(\mathfrak{F}, φ) for a commutative semigroup of endomor-phisms of a finite von Neumann algebra, instead of one φ, is given.
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