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2018, Т. 160, кн. 2 С. 287-292
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)
UDK 517.98
ULTRAPRODUCTS OF VON NEUMANN ALGEBRAS AND ERGODICITY
S.G. Haliullin
Kazan Federal University, Kazan, 420008 Russia Abstract
An ultraproduct of any linear spaces with respect of a non-trivial ultrafilter in an index set is generalization of the non-standard expansion *R of the set of real numbers R. The nonstandard mathematical analysis has the objects and methods of a research, which only to some extent depend on laws of the standard mathematical analysis.
In this work, non-standard objects - ultraproducts of von Neumann algebras - have been studied from the point of view of the standard analysis. This approach allows to receive, in particular, a criterion of contiguity of sequences of normal faithful states in terms of the equivalence of states on the corresponding ultraproducts.
We note that the classical ultraproduct of von Neumann algebras, generally speaking, is not a von Neumann algebra. Therefore, in accordance with A. Ocneanu's work, we have considered the changed construction of the ultraproduct of von Neumann algebras.
We have introduced the concept of ergodic action with respect to the normal state of group on an abelian von Neumann algebra. Its properties have been studied. In particular, we have provided the example showing that the ultraproduct of ergodic states is not, generally speaking, ergodic.
Keywords: ultraproducts, actions of group, ergodicity, states on von Neumann algebra
Introduction
Definition 1. Given a non-empty set X, a filter on X is a non-empty family U consisting of subsets of X, such that
1. the empty set is not an element of U;
2. if A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U;
3. if A and B are elements of U, then the intersection of A and B is also element of U.
A filter U on X is a maximal filter or an ultrafilter on X, if for every filter U' on X that contains U we have U' = U.
Note that the filter U on X is an ultrafilter if and only if, for any subset A of X, either A or X \ A belongs to U, but not both.
The family U = {A C X : A contains an element xo G X} is an example of an ultrafilter. This ultrafilter is said to be trivial. In what follows, we will use only nontrivial ultrafilters.
Definition 2. Let us consider a sequence (Xn) of nonempty sets and a nontrivial ultrafilter U in the set N of natural numbers. The ultraproduct (Xn)u is the factorization
of the Cartesian product Xn by the equivalence relation:
n=1
(x)n ~ (y)n {n : Xn = yn} G U.
The linear structure for an ultraproduct of linear spaces is defined in the natural way:
(xn)u + (Vn)u = (xn + Vn)u, c ■ (xn)u = (c ■ xn)u■
The set-theoretic ultraproduct possesses a number of extraordinary properties, for which we refer the reader to the articles by D. Mushtari and S. Haliullin [1] and S. Hal-iullin [2].
Definition 3. Let U be a nontrivial ultrafilter on the set of integers N. Let (xn) be a sequence of points in a metric space (X, d). A point x G X is said to be the limit of the sequence (xn) with respect to the ultrafilter U, denoted x = limu xn, if for every £ > 0 we have {n : d(xn, x) < £} G U.
As is well-known, if K is a compact Hausdorff space and U an arbitrary ultrafilter on the set N, then every sequence (xn)^=i, xn G K, has a unique limit with respect to the ultrafilter U.
Definition 4 (see, for example, [3]). Let (Hn)neN be a sequence of Banach spaces and let U be a nontrivial ultrafilter in the set N of natural integers. Let us put l™(N,Hn) = {(hn),hn G Hn : supn ||M < to} and Nu = {(hn) G l^(N,Hn) : limu llhnll = 0}. The ultraproduct (Hn)u of the sequence of Banach spaces is the quotient l^(N, Hn)/Nu ; here Nu is the closed subspace of l^(N,Hn).
We denote an element of (Hn)u by (hn)u. Then, the formula
ll(hn)ull =lim llhnll
defines a norm on (Hn)u . In this case, (Hn)u is a Banach space.
It is known, (S. Heinrich, [3]), that the class of Banach algebras and the class of C* -algebras are stable under ultraproducts.
In fact, the multiplicative and involutive structures of the ultraproduts are defined in the natural way:
(xn)u ■ (yn)u = (xn ■ yn)u, ((xn)u)* = ((xn)*)u.
Let us consider the notion of the ultraproduct for a sequence of von Neumann algebras using Ocneanu's definition (see [4], [5]).
Definition 5. Let (Mn) be a sequence of a -finite von Neumann algebras, and let n be a normal faithful state on Mn for each n G N. Let us put
l™(N, Mn) = {(xn),xn G Mn : sup lxnll < to},
n
Nu(Mn,?n) = {(xn) G l™(N, Mn) : lim Vn(x*nxn + xnx*J1/2 = 0}.
We let
Mu(Mn,^n) = {(xn) G l™(N, Mn) :
(xn)Nu(Mn,<Pn) cNu(Mn,<Pn), Nu (Mn,Vn)(xn) cNu(Mn,^n)}.
Then, we define the ultraproduct for the sequence of von Neumann algebras with normal faithful states as the quotient
(Mn, vn)u = Mu(Mn, Vn)/Nu(M n 7 ^n ) .
Finally, we define a state vu on (Mn, vn)u as follows:
vu ((xn)u) = lim Vn(xn).
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It is known (see [5]) that if the state yn is normal and faithful, then (Mn, yn)u is a von Neumann algebra with the normal faithful state yu.
There are various definitions of the absolute continuity of positive linear functionals on *-algebras (see, for example, the papers by S. Gudder [6] and by E. Chetcutti and J. Hamhalter [7].
Definition 6. Let M be a von Neumann algebra, and let y and 0 be normal states on M. The state y is said to be absolutely continuous with respect to the state 0 if the equation 0(x*x) = 0 implies y(x*x) = 0, x G M. The states y and 0 are called equivalent if they are mutually absolutely continuous. The states y and 0 are called singular if there is an operator x G M, such that y(x*x) = 0 and 0(x*x) = 1.
Let us notice that it is enough to set an equivalence and singularity of states on projections in the case of von Neumann algebra. Now, we need the following concept introduced in [8].
Definition 7. Let (Mn) be a sequence of a -finite von Neumann algebras, let yn and 0n be normal states on Mn, n G N. The sequence (yn) is said to be contigual with respect to the sequence (0n) if
0n(x*nxn) ^ 0 implies yn(x*nxn) ^ 0, xn G Mn, (n ^ m);
If the sequence (yn) is contigual with respect to the sequence (0n) and the sequence (0n) is contigual with respect to the sequence (yn), then the sequences (yn) and (0n) are said to be mutually contigual; the sequences (yn) and (0n) are said to be entirely separable if there is a subsequence (n') and operators xn' G Mn>, such that
yn' (x*nxn' ) ^ 0, 0n' (x*n,xn' ) ^ 1, (n'
These notions generalize the concepts of equivalence and singularity of states.
Theorem 1 [8]. Let (Mn ) be a sequence of a -finite von Neumann algebras, let yn and 0n be normal faithful states on Mn, n G N.
i) The sequences (yn) and (0n) are mutually contigual if and only if the states yu and 0u are equivalent for every nontrivial ultrafilter U on N.
ii) The sequences (yn) and (0n) are entirely separable if and only if there is a nontrivial ultrafilter U on N, such that the states yu and 0u are singular.
1. Ultraproducts and ergodicity
Let G be a separable locally compact group, and let (Q,j) be a a-finite standard measure space. By an action of G on (Q, j) , we mean a Borel map T : (s,w) € G x Q ^ Ts(w) € Q, such that
1) for each fixed s € G, the map w ^ Ts(u) is a non-singular bijection of Q;
2) Ts(Tt(w)) = Tst(w), s,t € G, w € Q;
3) Te(w) = w, where e is the unit of G.
We also say that (Q, j) is a G-measure space and we will designate (G, Q, j) .
Definition 8. ( [9]) The action G on (Q,j) is said to be non-singular (with respect to the measure j) if E € F j(E) =0 j(Ts(E)) = 0 for any s € G; The action G on (Q, j) is said to be free (with respect to the measure j) if for any compact subset K of G, such that e € K, and any Borel subset E of Q with j(E) > 0, there exists a Borel subset F C E, such that j(F) > 0 and j(Ff)Ts(F)) = 0 for every s € K; The action G on (Q,j) is said to be ergodic (with respect to the measure j) if j(EATs(E)) = 0 for every s € G implies j(E) =0 or j(Q \ E) = 0.
With (G, Q, (), we consider an action a of G on the abelian von Neumann algebra A = Ьж (Q) given by the following:
as(f)(ш) = f (Ts-1ш), s e G, f eA, ш e Q.
Definition 9. Let p be a normal state on the von Neumann algebra A = Ьж (Q). We claim that an action a of G on A is non-singular with respect to the state p if p(f) = 0 ^ p(as(f)) = 0, f e A for any s e G; we say that an action a of G on A is free with respect to the state p if for any compact subset K of G, such that e e K, and for any projection g e A, g = 0, there exists a non-zero projection f e A, such that f < g and p(fas(f)) = 0 for every s e K ; we say that a non-singular action a of G on A is ergodic with respect to the state p if for any projection f e A, as(f) = f for every s e G implies p(f) =0 or p(1 — f) = 0.
At the same time, the state p is called quasi-invariant, free, and ergodic with respect to the action a, respectively.
Theorem 2. Let p and ф be normal states on the abelian von Neumann algebra A = LX(Q), the action a of G on A is ergodic with respect to the states p and ф. Then, the states p and ф are either equivalent or singular.
Proof. Let p and q be the supports of the states p and ф, respectively. It is clear that the states p and ф are equivalent if and only if p = q, and the states p and ф are singular if and only if p L q. Let us assume the opposite, i.e., that the states p and ф are non-equivalent and non-singular. We put r = pq. Then, r = p,r = q,r = 0, 0 < p(r) < 1 and, at the same time, as(r) = as(pq) = pq(TfT1(ш)) = p(q(TfT1(ш)) = p(as(q)) = pq = r for all s e G .It contradicts to an ergodicity of the state p. □
We designate for a projection f: pas (f) = p(as (f)).
We consider the ultraproduct ( Am pn)u of the sequence of the abelian von Neumann algebras (Ari)neN with the actions an, on An and normal faithful states pn. Let U be a non-trivial ultrafilter on the set of natural numbers N. Put
au((fn)u) = (a.n(fn))u-
Theorem 3. If the transformation (an)sn is non-singular and free with respect to the state pn, sn e Gn,n e N, and the sequences of the states (pn) and ((pn)(an)s ) are mutually contigual, then the action au is non-singular and free on (An,pn)u .
Proof. It is clear that, generally, the action a can not be non-singular on the group (Gn)u . We put G = {s = (sn)u e (Gn)u : the sequences (pn) and ((pn)(an)sn) are mutually contigual}. Then, non-singularity of the action as, s e G, is provided with the first statement of theorem 1. The freeness of the action as follows from definition 9 and the first statement of theorem 1. So, the action au is non-singular and free. □
Let the action an be ergodic with respect to the normal faithful state pn on the algebra An, n e N. Generally, au in not ergodic with respect to the normal faithful state pu. It follows from the example given below.
Example 1. Let ün = Rn, in be Gaussian measure N(О,1п) , Gn is the group of shifts on the elements of Rn, n G N. It is known, (see [1]), that the ultraproduct lu is quasi-invariant with respect to action on the group G = {x = (xn)u : supn 11 xn \ < ж}. In other words, the action on the group G is non-singular with respect to measure lu .
Let us put An = Lx(Qn,in) Vn(fn) = J fn(x)d^n. It is clear that the state vn on the algebra An is normal and faithful. We consider the action а.Г1, of the group Gn
ULTRAPRODUCTS OF VON NEUMANN ALGEBRAS
29l
on the algebra An as in the above. It is obvious that the action an is non-singular and free with respect to the state yn .
Now, we show that the measure ^u is not ergodic with respect to action au on the group G. In the paper [1], it is shown that for any element x G G we have /i,u(BA(B — x)) = 0, where B = (Bn)u, Bn is the ball on the Q,n of the radius i/n and with the center at zero. It is clear that the measure f^u(B) = 1/2. Therefore, the nu is not ergodic.
Further, we put p = Ib . Then, ax(p) = p for all x G G, at the same time we have y(p) = 0 and y(p) = 1. □
This result can be compared to Ando and Haagerup's results, see [5].
Conclusions
If to consider the results of this paper and paper [8], we can conclude that the ultra-products of von Neumann algebras are characterized by unusual properties. We hope to obtain other interesting results in the our further investigations of the problem.
References
1. Mushtari D.H., Haliullin S.G. Linear spaces with a probability measure, ultraproducts and contiguity. Lobachevskii J. Math., 2014, vol. 35, no. 2, pp. 138-146. doi: 10.1134/S1995080214020097.
2. Haliullin S. Orthogonal decomposition of the Gaussian measure. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 436-438. doi: 10.1134/S1995080216040090.
3. Heinrich S. Ultraproducts in Banach space theory. J. Reine Angew. Math., 1980, vol. 313, pp. 72-104.
4. Ocneanu A. Actions of Discrete Amenable Groups on von Neumann Algebras. Berlin, Heidelberg, Springer, 1985. vii, 114 p. doi: 10.1007/BFb0098579.
5. Ando H., Haagerup U. Ultraproducts of von Neumann algebras. J. Funct. Anal., 2014, vol. 266, no. 12, pp. 6842-6913. doi: 10.1016/j.jfa.2014.03.013.
6. Gudder S.P. A Radon-Nikodym theorem for *-algebras. Pac. J. Math., 1979, vol. 80, no. 1, pp. 141-149.
7. Chetcutti E., Hamhalter J. Vitali-Hahn-Saks theorem for vector measures on operator algebras. Q. J. Math.., 2006, vol. 57, pp. 479-493. doi: 10.1093/qmath/hal006.
8. Haliullin S. Contiguity and entire separability of states on von Neumann Algebras. Int. J. Theor. Phys, 2017, vol. 56, no. 12, pp. 3889-3894. doi: 10.1007/s10773-017-3373-z.
9. Takesaki M. Theory of Operator Algebras III. Berlin, Heidelberg, Springer, 2003. xxii, 548 p. doi: 10.1007/978-3-662-10453-8.
Received
November 13, 2017
Haliullin Samigulla Garifullovich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical Statistics Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia E-mail: Samig.Haliullin@kpfu.ru
УДК 517.98
Ультрапроизведения алгебр фон Неймана и эргодичность
С.Г. Халиуллин
Казанский (Приволжский) федеральный университет, г. Казань, 420008, Россия
Аннотация
Ультрапроизведение произвольных линейных пространств по некоторому нетривиальному ультрафильтру в индексном множестве есть ни что иное, как обобщение нестандартного расширения *К множества действительных чисел К. Нестандартный математический анализ имеет свои объекты и методы исследования, которые лишь в определённой степени зависят от законов стандартного математического анализа.
В работе «нестандартные» объекты - ультрапроизведения алгебр фон Неймана - изучаются с точки зрения стандартного анализа. Такой подход позволяет, в частности, получить критерий «контигульности» последовательностей точных нормальных состояний в терминах «эквивалентности» состояний на соответствующих ультрапроизведениях.
Известно, что классическое ультрапроизведение алгебр фон Неймана, вообще говоря, не является алгеброй фон Неймана, поэтому мы рассматриваем специальную конструкцию ультрапроизведений алгебр фон Неймана, следуя работам А. Окненеу. Мы вводим понятие эргодического относительно некоторой группы преобразований состояния на алгебре фон Неймана и изучаем его свойства. Рассмотрено ультрапроизведение таких состояний и приведены их свойства. В частности, приведён пример, показывающий, что ультрапроизведение эргодических состояний не является, вообще говоря, эргодическим.
Ключевые слова: действие группы, эргодичность, состояния на алгебре фон Неймана
Поступила в редакцию 13.11.17
Халиуллин Самигулла Гарифуллович, кандидат физико-математических наук, доцент кафедры математической статистики
Казанский (Приволжский) федеральный университет
ул. Кремлевская, д. 18, г. Казань, 420008, Россия E-mail: Samig.Haliullin@kpfu.ru
I For citation: Haliullin S.G. Ultraproducts of von Neumann algebras and ergodicity. ( Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, \ vol. 160, no. 2, pp. 287-292.
I Для цитирования: Haliullin S.G. Ultraproducts of von Neumann algebras ( and ergodicity // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. - 2018. - Т. 160, \ кн. 2. - С. 287-292.
