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Владикавказский математический журнал Апрель-июнь, 2003, Том 5, Выпуск 3
УДК 517.98
ON THE WEAK CONVERGENCE OF OPERATORS ITERATIONS IN VON NEUMANN ALGEBRAS
A. A. Katz
Equivalent conditions are obtained for the weak convergence of iterations of the positive contractions in
the pre-conjugate spaces of von Neumann algebras.
1. Introduction
This paper is devoted to ergodic type properties of the weak convergence of operators iterations in von Neumann algebras.
The first results in the field of non-commutative ergodic theory were obtained by Sinai and Anshelevich [18] and Lance [14]. Developments of the subject are reflected in the monographs of Jajte [9] and Krengel [13] (see also [4]—[7], [10], [11], [16]).
We will use facts and the terminology from the general theory of von Neumann algebras ([1], И, [15], [17], [19]).
Let M be a von Neumann algebra, acting on a separable Hilbert space H, Л/„, is a preconjugate space of Л/. which always exists according to the Sakai theorem [17].
Recall some standard terminology ([4], [5], [6], [10], [11], [13]).
DEFiNiTiON 1. A linear mapping T from Л/„, in itself is called a contraction if its norm is not greater then one.
Definition 2. A contraction T is said to be positive if TM*+ с M*+.
We will consider two topologies on the space M*: the weak topology, or the cr(M*,M) topology, and the strong topology of the M*-space norm convergence.
Definition 3. A matrix (anyi), i,n = 1,2,... of real numbers is called uniformly regular,
if:
lim sup |an i\ = 0, lim у an * = 1.
n—»oo Л ’ n—»oo 4 ’
* i
2. Main Result
The following theorem is valid:
Theorem 1. The following conditions for a positive contraction T in the pre-conjugate space of a von Neumann algebras M are equivalent:
(i) The sequence {T*}j=1,2,... converges weakly;
supjj
© 2003 Katz A. A.
Author is thankful to Professor Michael S. Goldstein (University of Toronto, Canada) for helpful discussions.
(ii) For each strictly increasing sequence of natural numbers {fcj}j=i,2,...,
-1 'y ' rpki
n
i<n
converges strongly;
(iii) For any uniformly regular matrix (an>j), the sequence {An(T)}n=
An(T) = YJanAT')>
converges strongly.
<1 We first prove the following lemma:
Lemma 1. Let there exists a uniformly regular matrix (aHji) such that for each strictly increasing sequence {ki}i=1,2,... of natural numbers,
(1)
converges strongly. Then the sequence {T*}j=1,2,... converges weakly.
<1 Let (aUji) be a matrix with the aforementioned properties. Then the limit Bn is not dependent upon the choice of the sequence {&;*}*=1,2,...- In fact, let {ki}i=i,2,... and {h}i=1,2,... be the sequences for which the limits Bn are different. This means that for x G A/„,.
and
as n —>• 00. For a matrix (an^) let us build increasing sequences {ij}j=1,2,... and {rij}j=1,2,..., such that
lim
j-too
= 0.
Let
Then
= hi for i G and m* = k for i G [i2j, *2j+i), j = 1,2,
lim
i
= 0, lim j
= 0,
which contradicts (1), and therefore x\ = ®2- Let now y G M is such that
(T”® — ®i, y) —?• 0,
when n —>• 00. Let us choose a subsequence {fcj} such that
(Tfci® — xi ,y) —>• 7 7^ 0,
where 7 is a real number. Then, from the uniform regularity of the matrix (aUji) it follows that
lim ^an,iTkix -xi=7,
which contradicts the choice of the matrix (an^). >
[Proof of the Theorem 1 (cont.).] Because the implication (ii) (iii) is obvious, the implications (ii) =>• (iii) =>• (i) immediately follow from the lemma 1.
The implication (iii) (ii) is trivial, because the matrix (a„5j), an,i = ^ $j-M *s
i<n
uniformly regular.
Applying the above with lemma 1, an,i = i ^ n and an,i = 0 for i > n, we get the implication (ii) (i).
To prove the implication (i) (ii), we would need the following lemma:
Lemma 2. Let Q be a contraction in the Hilbert space H. Then the weak convergence of
Qnx in H, where x G H, implies the strong convergence of
X
for any uniformly regular matrix (anyi).
<1 If the weak limit Qnx exists and is equal to x\, then
Qxi = Q( lim Qnx) = xi,
Tl—tOo
where the limit is considered in the weak topology, i. e. x\ is Q-invariant. Therefore
2
y'J0’N,iQ%X
< 53 " V./" V,,((/•'•• Qjx) < IaN,iaN,j{Q%x., Qjx) I
3
3
Let us fix e > 0. Because Q is a contraction, the limit ||Q”a;|| does exist. Now, we can find K > 0, such that for k > K and j > 0,
Qkx
Qk+jz
and
(Qkx,x)
< £.
Then,
(Qkx,x)^(Qk^x,Q^x) = (Qkx,x)^(Q^Qk^x,x)
Qkx - Q*jQk+jx
€
Qkx
Qkx
Qk+3x Qk+jc
x\\ =
+
Qkx - Q*jQk+j Q*iQk+iX x\\ ^ e ■ ||a:|| ,
i
2 \ 2
\X\
and therefore
(Qfc+J®, Q^x) < e ■ (1 + ||®| for all k > K and j > 0, or for \i — j\ > k the inequality
\(Q'lx,Q:’x)\ <e-(l + ||a:||).
is valid. We will fix rj > 0, and let N be such a natural number with max* \an,i\ < 'q, for
N. Then the expression (1) for n ^ N could be estimated the following way:
53 53 "a./"a,,((/•'•• Q‘-r) 53 \an,ian,j(Qtx,Qjx)\ + 53 \an,ian,j(QKx,Qjx)\
i j |j—j\^k \i—j\>k
Ojn,iO"n,j I
i i j
< C-ri • |M|2 • (2k - 1) + C2 ■ e • (1 + \\x\\).
From the arbitrarity of the values of e and r] it follows that the strong convergence is present
and the lemma is proven. >
[Proof of the Theorem 1 (cont.).] Let us prove the implication (i) (iii). Let x G M*+ and the sequence {Tlx}i=converges weakly. Without loss of generality we can consider ||a;|| ^ 1, and let
x = lim Tnx,
Tl—YQQ
where the limit is understood in the weak sense. Let us consider
oo
v 532
n=0
The series that defines y is convergent in the norm of the space M*. From the positivity
of x and the properties of the operator T it follows that Ty ^ 2y, and, therefore, for all
k = 1,2,..., s(Tky) ^ s(y), where by s(z) we denote the support of the normal functional z.
Lemma 3. Let u G M*+ and s(u) ^ s(y). Then s(u) ^ s(x), where
u = lim Tnu.
n—>oo
<1 Indeed, let us fix e > 0. From the density of the set
£ = {w G M*+, w ^ Ay, for someA > 0},
in the set
6 = {wG M*+, s(w) < s(y)}, in the norm of the space A/„, it follows that there are A > 0 and w G £ such that
||w — «|| ^ £ and w ^ Ay.
Let w = limn^oo Tnw. Then
w(l—s(x)) = lim (Tn(w))(l^s(x)) ^ A • lim (Tny)(l^s(x))
n—too n—too
f OO \
< A • lim y 2 u • (Tn+kx)(l^s(x))
n—too \ ^J J
\k=0 /
oo
= A • V 2-k lim (Tn+kx)(l^s(x)) = 0.
n-4-OO
n—>■ OO fe=0
Because the operator T does not increase the norm of the functionals from Л/*. we get that й(1—«(ж)) = lim (Tnu)(l^s(x)) ^ lim (Tnw)(l^s(x)) + lim \\Tn(w — -u)|| ^ є,
п—УСЮ n—УСЮ n—УСЮ
The needed inequality follows, since є is arbitrary. >
[Proof of the Theorem 1 (cont.).] Let now ^ Є M*. We will denote by /i.E, where E is a projection from the algebra Л/. the functional
(p,.E)(A) = fj,(EAE),
where . І Є M. Let us fix є > 0. We will find a number N, such that
(Tnx)(l^s(x)) < є2
for n > N. Thus,
\\TNx.s(x) — Т^жЦ = sup \(TN x)((l^s(x))A(l^s(x)))
Plloo^l
+(TNx)((s(x))A(l^s(x))) + (TNx)((l^s(x))A(s(x))) \ ^ є • (є + 2 ||ж||2),
because
|/і(ЛВ)|2 < /і(А*Л) • ім{В*В),
where [Л Є М*+ and А, В Є М.
Let w Є £ is such that w ^ Хх for some А > 0 and ЦТ^ж.^ж) — -ш|| ^ є. Then, for п> N, the following is valid:
||Tnx^Tn-Nw\\ < ||Тп-я(Тяж^Тяж.5(ж))||
+ \\Tn-N(TNx.s(%) -w)|| < 4-е. ^
By taking the weak limit in the inequality (2) and because the unit ball of Л/* is weakly closed, we will get ||ж — го|| ^ 4 • є, where w = lim^^oo Tnw.
Let us now consider the algebra Ms^xy The functional ж is faithful on the algebra Msyxy We will consider the representation ttx of the algebra Ms^ constructed using the functional x [2]. Because the functional ж is faithful, we can conclude that the representation ttx is faithful on the algebra Ms^xy and therefore ttx is an isomorphism of the algebra Ms^ and some algebra 21. The algebra 21 is a von Neumann algebra, and its pre-conjugate space 21* is isomorphic to the space ([17]). Let us note now that
TM.,.s(x) C M.,.s(x).
In fact, T£ C £, and therefore, by taking the norm closure we get TS C S'; by taking now the linear span we will get
TM*.s{x) C M*.e(®).
Denote by T the isomorphic image of the operator T, acting on the space 21*. Let u Є 2l*+ and u ^ Аж for some A > 0. Then there exists the operator В Є 21/, where 21' is a commutant of 21, such that (. I />’<?. fl) = u(A) for all А Є 21. Note, that from the lemma 2
(Ти) (A) = u((T)*A) = (((Т)*А)ВП,П) = (A((T*)'B)fi,fi).
Also, from it follows that
T2l*+ C 21*+, llTull ^ ||ti|| and Tx = x
for all A € 21. Based on the lemma we now conclude that
(T*B) < HSII^ ; ¥*'2l'+ C 21+; T*'l < 1
OO
for all B (E 21'.
The space 2l'tt is a pre-Hilbert space of the self adjoint operators from 21' with the scalar product (B,C)x = (CBfl^fl), and, using the Kadison inequality fl] we have
((T*' B)(T*' B)Sl,Sl) < (T*'(B2)n,n) < (BQ, BQ),
__jj,f
i. e. the operator T is a contraction in the pre-Hilbert space (21'tt, (•, -)x)-
We will identify M*.s(af) and 21*. Because w € £, i. e. w ^ Xx for some A > 0, then w ^ Xx as well. Let
w(A) = (BAn,n) and w(A) = (BAn,n)
for all A € 21, where B, B £ 21'.
Let now (a„5j) be a uniformly regular matrix. Using lemma 2 we will find fee N so that
= sup
Aea
IWIl =1
j OO OO \ 2
<( • SUP (Aft, Aft) *
*=1 '
.*=i
<(*(!))’■
* = 1
< £
for k > K, where by (a'ni) we will denote a matrix with the elements
Q’n.j,
'y ] "/i./ | an,j+N-
i>N
It is easy to see that the matrix (a'ni) will be uniformly regular as well. Then, for a big enough k > K we will have
-iV
w
+ ^ ] |afc,*| ' 1 f ^ ] ak,t i>N i>N
i<^N
1
i>N
T'-Nw II +
53 ak,j+N • ( 53 "kj) 'l'J j=1 'i>N
W — W
+
53
i<N
W
+
53 "kj
i>N
\w — x\\
^ 53 2 ' AT + 53 • 4e + 53 lafc,*l (! — (! + e) X) • 2 + 2 • Tf + (1 + £) • 4g
i^N i>N i>N i^N
^ 2e + (1 + e) • 4e + e ■ 2 • (1 + e) + e + 2e + (1 + e) ■ 4e ^ 25e,
The arbitrarity of e proves the needed statement. The proof of the theorem is now
completed. >
Remark. Extention of the discussed properties of iterations to non-commutative Lp-spaces will be separately presented in the forthcoming paper [12].
References
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Monographs in Physics.—New York — Heidelberg: Springer-Verlag, 1979.—500 p.
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Статья поступила 14 июля 2003 г.
Prof. Alexander A. Katz, Ph.D.
Department of Mathematics and Computer Science, St. John’s University, NY 10301, USA E-mail: [email protected]
