Isam Eldin Ishag Idris1 Fadool Abass Fadool2 Aisha Yousif Mustafa3
University of Kordofan-Faculty of Education-Department of Mathematics
E. mail: E.mail:isamishag018@gmail.com
2 University of Kordofan- Faculty of Education- Department of Mathematics
E.mail:fadoolabass1984@gamail.com
3 Sudan University of Science and Technology- Faculty of Education- Department of Mathematics
E.mail:yousifkasala2015@gmail.com
HNSJ, 2022, 3(8); https://doi.org/10.53796/hnsj3813
Published at 01/08/2022 Accepted at 05/07/2022
Abstract
We determine and verify regularizing nets with inequalities and equality between weights, we used the deductive method and we found that for the equality of two normal positive forms on a -algebra it is enough that they coincide on a -dense subset. And there are typically many weights which are of little importance in regularizing nets with inqualities and equality between weights.
Key Words: regularizing, nets , inequality, equality, weights.
Introduction
Suppose be semi-finite, normal weights on a -algebra faithful and -invariant. If for all in a -dense subset –subalgebra of , then This criterion was further extended in [18] as follows: Let be as above, and a positive element of the centralizer of If for in a -dense subset –subalgebra of then .
Regularizing nets are useful in the modular theory of faithful, semi-finite, normal weight. Suppose be -algebra, and a faithful, semi-finite, normal weight on . We call regularizing net for any net in such that
- and for each compact ;
- in the -topology for all .
In the modular theory of faithful, semi-finite, normal weights the regularizing nets are useful. Ususlly they are constructed starting with a bounded net in such that in the -topology and then letting it ‘‘mollified’’, for modle, by the mollifier , that is passing to the net then
The verification of (i) is straightforward, more troublesome is to verify the inclusion and the convergence (ii).
Concerning the verification of (ii), if the net would be increasing, we could proceed as in the proof of [13] by using Dini’s theorem. But there are situations in which we cannot restrict us to the case of increasing. For example, it is not clear whether every -dense, -invariant -subalgebra of contains some increasing net with in the -topology as used in the proof of [13].
By the other side, if the net would be a sequence, we can use the dominated convergence theorem of Lebesgue, similarly as, for example, in the proof of [14],Theorem 2.16. But again, unless is countably decomposable (and so its unit ball -metrizable), the unit ball of not every –subalgebra of contains a sequence -convergent to Here we notice that Lebesgue theorem of convergence is very useful in this case also we can cover the other case of non
countable nets to determine and verify (ii) directly, using advantage of the particularites of the situation. Here we will prove that, starting with a bounded net even in , equation (1) furnishes a regularizing net[19].
The next lemma is [2] equation (2.27) it is also another type of the modular theory of faithful semi-finite, normal weights concerns some facts.
Lemma 1. Let be a faithful, semi-finite, normal weight on a -algebra . If
and , then
Let be a faithful, semi-finite, normal weight on a -algebra and
We define the linear operator as follows: the pair belongs to its graph whenever the map has a -continuous extension on the closed strip ,
Analytic in the interior and taking the value at It is easily seen (see e.g. [17],Theorem 1.6) that, for each
for every
We recall that belongs to if and only if the operator is defined and bounded on a core of in which case and that is
(see [3],Theorem 6.2 or [2], Theorem 2.3).
Here we determine and verify the form of an element of hence to
Lemma 2. Let be a faithful, semi-finite, normal weight on a -algebra
and and , that is
Proof. Let be arbitrary. Then
Application of (2) with yields and so, applying (3) to and, we deduce
By (4) and (5) we conclude:
.
By the aboves
applying [2], Lemma 2.6 (1) to deduce that [19].
Taking into account that and , and using [2], (5), as well as the above (3) with, we deduce:
.
Since is -dense in , it follows the equality .
The above two lemmas can be used to produce elements of the Tomita algebra
by ‘‘regularizing’’ elements of (not only elements of as customary : (see in [15], the comments after the proof of Theorem 10.20 on page 347) :
Lemma 3. Let be a faithful, semi-finite, normal weight on a -algebra
For each .
belongs to and
We assume that , we get
Proof. If
allows the entire extension
we have and (6) holds true.
By assuming if we have
Using (6) it is easy to see that
so
and
For each applying Lemma 1 with we deduce that Since is here arbitrary, also holds true. But by (7) we get so Applying now Lemma 2, we conclude that belongs also to hence
By using the integrals of equation (6) and Lemma 4 we can prove the dominated convergence theorem for integrals and nets.
Lemma 4. Take as a faithful, semi-finite, normal weight on a -algebra
and a net in the closed unit ball of such that in the -topology. Let the net be defined by the equation
Then
- for all
- for all and
- in the -topology for all .
Proof. (i) is immediate consequence of Lemma 3.
For (ii), let and be arbitrary. By Lemma 3 we have
Since for all it follows
.
The more involved issue is (iii). For fixed we have to show that
in the -topology. Since the -topology is definded by the semi-norms
a normal positive form on , then
for every a normal positive form on
For let be any a normal positive form on Since, according to [19], equation (3),
, if we prove the convergence the proof will be complete.
,
that is consequence of
because and
The proof will be complete by using verifying (9) [19].
Since in the -topology and for all we have that
is a bounded net, convergent to 0 in the -topology. According to a theorem due to Akemann (see [1], Theorem II.7 or [16], Corollary 8.17), on bounded subsets of the -topology coincides with the Mackey topology associated to the the -topology, that is with the topology of the uniform convergence on the weakly compact absolutely convex subsets of the predual Since, by the classical Krein-Šmulian theorem (see e.g. [9], Theorem V.6.4), the closed absolutely convex hull of every weakly compact set in Banach space is still weakly compact, is actually the topology of the uniform convergence on the weakly compact subsets of Therefore
for every weakly compact
Now let be arbitrary. Choose some , then
Since is a weakly compact subset of (10) holds true with .Then there exists some such that
for all (11) implies
,
while using (12) we deduce for every
.
Consequently for every
Theorem 5. Let be a faithful, semi-finite, normal weight on a -algebra and a net in the closed unit ball of such that in the -topology. Let the net we define it by the equation
Then
- for all
- for all and
- in the -topology for all .
Futhermore, if for all hence belongs to for every and therefore is a regularizing net for
For determing and verifying criteria for inequalities and equalities between weights we use the generalization of [18], Lemma 2.1.
By recalling that -subalgebra of a -algebra is called facial subalgebra or hereditary subalgebra whenever is a face, that is a convex cone satisfying
and is the linear span of it (see e.g. [15], Section 3.21).
Theorem 6. Let be a -algebra, a faithful, semi-finite, normal weight on and a normal weight on . Assume that there exists a -dense, — subalgebra of such that Then
Then, there exists a – invariant, -subalgebra of such that ,
The difference between the above Theorem 6 and [18], Lemma 2.1 consists in the fact that in [18], Lemma 2.1 is additionally assumed that
- is semi-finite and – invariant and
- is contained already in (which of course, according to [13], Theorem 3.6, is a subset of ).
However the proof of [18], Lemma 2.1 does not use assumption (i) and, by the other side, we can adapt it to work with the assumption
Proof. Let be arbitrary. Since , we have and therefore and are normal positive forms on . We notice that and is -dense in , we deduce that
By the density theorem of Kaplansky there exists a net in such that for all and . Set, for each ,
Clearly, for all . According to Lemma 4, for all and
in the-topology for all . Since , also
belongs to for each Furthermore, yields
hence We apply Lemma 3 and (17) we deduce that for all
Let and be arbitrary. Since and is –invariant, application of (14) yields for every and :
We apply [19], equation (1.2) with
it follows for :
.
Since, by (15),
we conclude that
Next let be arbitrary. Using (18) and applying [6], Lemmme 7 (b) or [18], Proposition 1.1. we deduce for every
.
Since and in the -topology, and is lower semicontinuous in the -topology, we get
Applying now [18], Corollary 1.2, we conclude:
To have (13) proved, we must show that (19) actually holds for every . This follows by the proof of [18], Lemma 2.1. We report it for sake of completeness.
For every , since and , (19) yields
.
being -dense in , it follows , what means .
For we consider the projection where this depends on characteristic function of Then We consider also the inverse of in the reduced algebra with
Now let be arbitrary. Then , so
Applying (19) and [18], Corollary 1.2, we obtain for every
Since and is lower semicontinuous in the -topology, it follows
We apply [18], Corollary 1.2 again, we conclude:
Taking a -, -subalgebra of so, the proof of the theorem will completed, then
We notice that:
- is a face.
- .
Since is a convex cone, for (i) we have only to verify the implication
It follows surely by using
For (ii) let be arbitrary. Without loss of generality we can assume that Denoting we obtain an increasing sequence which is -convergent to the support of (see e.g. [15], Section 2.22). Since all belong to the commutative -subalgebra of generated by , the sequence is still increasing and it is -convergent to . Therefore we deduce:
- for all by the assumption on
- for all by applying (2.8) with and
- by the normality of and .
Now we set
,
Then is a face, is -subalgebra of , and is the linear span of (see e.g. [15], Proposition 3.21).Thus is a -subalgebra of and Since is – also and therefore is – Finally, the above (ii) and (i) imply that we have for all
Remark 7. If is assumed only affiliated to and not necessarily bounded, the statement of Theorem 6 is not more true. Counterexamples can be obtained using [13], Proposition 7.8 or [6], Example 8.
Two faithful, semi-finite, normal weights are constructed on such that and , but for where is a -subalgebra of (in [6], Example 8, the construction delivers ).
Now let be a faithful, semi-finite, normal trace on By [13], Theorem 5.12 there exists a positive, self-adjoint operator on , necessarily affiliated to such that . Then
- is a faithful, semi-finite, normal trace on ,
- is a positive, self-adjoint operator to
- is a -, faithful, semi-finite, normal weight on ,
- for where is a -subalgebra of
but because otherwise it would follow , hence in contradiction to .
Remark 8. If in Theorem 6 we assume that belongs to the –closure of (that happens, for example, if because ), then it follows also the equality
Since trivially, we have to verify that for any with , that is with , we have
By (16), by the lower semicontinuity of in the -topology, and by (18), we obtain .
Using now the inequalities
and as in [6], Lemme 7 (b) or [18], Proposition 1.1, we have
Since, by (16), in the -topology, we conclude that
what is equivalent to [19].
The next theorem is a slight extension of [18], Theorem 2.3:
Theorem 9. Let be a -algebra, a faithful, semi-finite, normal weight on and a -, normal weight on If there exists a -dense, and — subalgebra of such that then
Proof. By Theorem 6 we have In particular, is semi-finite.
Addition to that, by [13],Theorem 5.12 there exists a positive, self-adjoint operator , affiliated to such that . Since [18], Lemma 2.2) yields . In particular, is bounded.
Since is the linear span of is the linear span of and we have . If we applying Theorem 6 again this leads us to deduce that Theorem 10 is an equivalent and symmetric form of Theorem 9.
Theorem 10. Let be a -algebra, a faithful, semi-finite, normal weight on and a -, normal weight on If there exists a -dense, and — subalgebra of then then
Proof. Since is -invariant and , the normal weight is still -invariant : we have for every and
Hence we applying Theorem 9 with replaced by An immediate consequence of Theorem 2.4 an 2.5 is [13], Proposition 5.9 :
Corollary 11. Let be a -algebra, a faithful, semi-finite, normal weight on and a -invariant, normal weight on If there exists a -dense, — subalgebra of such that then .
Theorem 12. Let be a -algebra, a faithful, semi-finite, normal weights on . By assuming that there are a -dense, — subalgebra of and a -dense, — subalgebra of then So,
Proof. We applying here twice Theorem 7. An immediate consequence of Theorem 12 are :
Theorem 13. Let be a -algebra, faithful, semi-finite, normal weights on and . We assuming that there exists a -dense, both and — subalgebra of then
Then
Corollary 14. Let be a -algebra, a faithful, semi-finite, normal weight on If there exists a -dense, both and — subalgebra of such that then There exist also criteria of different kind for equality and inequalities between faithful, semi-infinite, normal weights, due to [5]. They are in trems of the Connes cocycle (see [5], Section 1.2 or [15], Theorem 10.28 and C.10.4): if and are faithful, semi-finite, normal weights a -algebra, the Connes cocycle of with respect to will be denoted by , it is analytic in the interior and satisfies
- for all if and only if has a continuous extension , which is analytic in the interior and such that is isometric.
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