{"id":4161,"date":"2022-08-01T18:33:37","date_gmt":"2022-08-01T18:33:37","guid":{"rendered":"https:\/\/www.hnjournal.net\/?page_id=4161"},"modified":"2022-08-01T18:33:40","modified_gmt":"2022-08-01T18:33:40","slug":"3-8-13","status":"publish","type":"page","link":"https:\/\/www.hnjournal.net\/ar\/3-8-13\/","title":{"rendered":"On regularizing nets with inequalities and equality between weights"},"content":{"rendered":"

Isam Eldin Ishag Idris1<\/sup> Fadool Abass Fadool2<\/sup> Aisha Yousif Mustafa3<\/sup><\/strong><\/p>\n

University of Kordofan-Faculty of Education-Department of Mathematics<\/p>\n

E. mail: E.mail:isamishag018@gmail.com<\/p>\n

2<\/sup> University of Kordofan- Faculty of Education- Department of Mathematics<\/p>\n

E.mail:fadoolabass1984@gamail.com<\/p>\n

3<\/sup> Sudan University of Science and Technology- Faculty of Education- Department of Mathematics<\/p>\n

E.mail:yousifkasala2015@gmail.com<\/p>\n

HNSJ, 2022, 3(8); https:\/\/doi.org\/10.53796\/hnsj3813<\/p>\n

Download<\/a><\/p>\n

Published at 01\/08\/2022 Accepted at 05\/07\/2022 <\/strong><\/p>\n

Abstract <\/strong><\/p>\n

We determine and verify regularizing nets with inequalities and equality between weights, <\/strong>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.<\/p>\n

Key Words: <\/strong>regularizing, nets , inequality, equality, weights.<\/p>\n

Introduction <\/strong><\/p>\n

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 .<\/p>\n

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<\/p>\n

    \n
  1. and for each compact ;<\/li>\n
  2. in the -topology for all .<\/li>\n<\/ol>\n

    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 \u2018\u2018mollified\u2019\u2019, for modle, by the mollifier , that is passing to the net then<\/p>\n

    The verification of (i) is straightforward, more troublesome is to verify the inclusion and the convergence (ii).<\/p>\n

    Concerning the verification of (ii), if the net would be increasing, we could proceed as in the proof of [13] by using Dini\u2019s 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].<\/p>\n

    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<\/p>\n

    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].<\/p>\n

    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.<\/p>\n

    Lemma 1. <\/strong>Let be a faithful, semi-finite, normal weight on a -algebra . If<\/p>\n

    and , then<\/p>\n

    Let be a faithful, semi-finite, normal weight on a -algebra and<\/p>\n

    We define the linear operator as follows: the pair belongs to its graph whenever the map has a -continuous extension on the closed strip ,<\/p>\n

    Analytic in the interior and taking the value at It is easily seen (see e.g. [17],Theorem 1.6) that, for each<\/p>\n

    for every<\/p>\n

    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<\/p>\n

    \n

    (see [3],Theorem 6.2 or [2], Theorem 2.3).<\/p>\n

    Here we determine and verify the form of an element of hence to<\/p>\n

    Lemma 2. <\/strong>Let be a faithful, semi-finite, normal weight on a -algebra<\/p>\n

    and and , that is<\/p>\n

    Proof. <\/strong>Let be arbitrary. Then<\/p>\n

    Application of (2) with yields and so, applying (3) to and, we deduce<\/p>\n

    \n

    By (4) and (5) we conclude:<\/p>\n

    .<\/p>\n

    By the aboves<\/p>\n

    applying [2], Lemma 2.6 (1) to deduce that [19].<\/p>\n

    Taking into account that and , and using [2], (5), as well as the above (3) with, we deduce:<\/p>\n

    .<\/p>\n

    Since is -dense in , it follows the equality .<\/p>\n

    The above two lemmas can be used to produce elements of the Tomita algebra<\/p>\n

    by \u2018\u2018regularizing\u2019\u2019 elements of (not only elements of as customary : (see in [15], the comments after the proof of Theorem 10.20 on page 347) :<\/p>\n

    Lemma 3.<\/strong> Let be a faithful, semi-finite, normal weight on a -algebra<\/p>\n

    For each .<\/p>\n

    belongs to and<\/p>\n

    \n

    We assume that , we get<\/p>\n

    Proof.<\/strong> If<\/p>\n

    allows the entire extension<\/p>\n

    we have and (6) holds true.<\/p>\n

    By assuming if we have<\/p>\n

    Using (6) it is easy to see that<\/p>\n

    so<\/p>\n

    and<\/p>\n

    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<\/p>\n

    By using the integrals of equation (6) and Lemma 4 we can prove the dominated convergence theorem for integrals and nets.<\/p>\n

    Lemma 4.<\/strong> Take as a faithful, semi-finite, normal weight on a -algebra<\/p>\n

    and a net in the closed unit ball of such that in the -topology. Let the net be defined by the equation<\/p>\n

    Then<\/p>\n

      \n
    1. for all<\/li>\n
    2. for all and<\/li>\n
    3. in the -topology for all .<\/li>\n<\/ol>\n

      Proof. <\/strong>(i) is immediate consequence of Lemma 3.<\/p>\n

      For (ii), let and be arbitrary. By Lemma 3 we have<\/p>\n

      Since for all it follows<\/p>\n

      .<\/p>\n

      The more involved issue is (iii). For fixed we have to show that<\/p>\n

      in the -topology. Since the -topology is definded by the semi-norms<\/p>\n

      a normal positive form on , then<\/p>\n

      for every a normal positive form on<\/p>\n

      For let be any a normal positive form on Since, according to [19], equation (3),<\/p>\n

      , if we prove the convergence the proof will be complete.<\/p>\n

      ,<\/p>\n

      that is consequence of<\/p>\n

      \n

      because and<\/p>\n

      \n

      The proof will be complete by using verifying (9) [19].<\/p>\n

      Since in the -topology and for all we have that<\/p>\n

      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-\u0160mulian 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<\/p>\n

      for every weakly compact<\/p>\n

      Now let be arbitrary. Choose some , then<\/p>\n

      Since is a weakly compact subset of (10) holds true with .Then there exists some such that<\/p>\n

      for all (11) implies<\/p>\n

      ,<\/p>\n

      while using (12) we deduce for every<\/p>\n

      .<\/p>\n

      Consequently for every<\/p>\n

      \n

      Theorem 5.<\/strong> 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<\/p>\n

      Then<\/p>\n

        \n
      1. for all<\/li>\n
      2. for all and<\/li>\n
      3. in the -topology for all .<\/li>\n<\/ol>\n

        Futhermore, if for all hence belongs to for every and therefore is a regularizing net for<\/p>\n

        For determing and verifying criteria for inequalities and equalities between weights we use the generalization of [18], Lemma 2.1.<\/p>\n

        By recalling that -subalgebra of a -algebra is called facial subalgebra or hereditary subalgebra whenever is a face, that is a convex cone satisfying<\/p>\n

        and is the linear span of it (see e.g. [15], Section 3.21).<\/p>\n

        Theorem 6.<\/strong> 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<\/p>\n

        Then, there exists a – invariant, -subalgebra of such that ,<\/p>\n

        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<\/p>\n

          \n
        1. is semi-finite and – invariant and<\/li>\n
        2. is contained already in (which of course, according to [13], Theorem 3.6, is a subset of ).<\/li>\n<\/ol>\n

          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<\/p>\n

          Proof.<\/strong> Let be arbitrary. Since , we have and therefore and are normal positive forms on . We notice that and is -dense in , we deduce that<\/p>\n

          \n

          By the density theorem of Kaplansky there exists a net in such that for all and . Set, for each ,<\/p>\n

          Clearly, for all . According to Lemma 4, for all and<\/p>\n

          in the-topology for all . Since , also<\/p>\n

          belongs to for each Furthermore, yields<\/p>\n

          hence We apply Lemma 3 and (17) we deduce that for all<\/p>\n

          Let and be arbitrary. Since and is \u2013invariant, application of (14) yields for every and :<\/p>\n

          We apply [19], equation (1.2) with<\/p>\n

          it follows for :<\/p>\n

          .<\/p>\n

          Since, by (15),<\/p>\n

          we conclude that<\/p>\n

          Next let be arbitrary. Using (18) and applying [6], Lemmme 7 (b) or [18], Proposition 1.1. we deduce for every<\/p>\n

          .<\/p>\n

          Since and in the -topology, and is lower semicontinuous in the -topology, we get<\/p>\n

          Applying now [18], Corollary 1.2, we conclude:<\/p>\n

          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.<\/p>\n

          For every , since and , (19) yields<\/p>\n

          .<\/p>\n

          being -dense in , it follows , what means .<\/p>\n

          For we consider the projection where this depends on characteristic function of Then We consider also the inverse of in the reduced algebra with<\/p>\n

          \n

          Now let be arbitrary. Then , so<\/p>\n

          \n

          Applying (19) and [18], Corollary 1.2, we obtain for every<\/p>\n

          Since and is lower semicontinuous in the -topology, it follows<\/p>\n

          We apply [18], Corollary 1.2 again, we conclude:<\/p>\n

          Taking a -, -subalgebra of so, the proof of the theorem will completed, then<\/p>\n

          We notice that:<\/p>\n

            \n
          1. is a face.<\/li>\n
          2. .<\/li>\n<\/ol>\n

            Since is a convex cone, for (i) we have only to verify the implication<\/p>\n

            It follows surely by using<\/p>\n

            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:<\/p>\n

              \n
            • for all by the assumption on<\/li>\n
            • for all by applying (2.8) with and<\/li>\n
            • by the normality of and .<\/li>\n<\/ul>\n

              Now we set<\/p>\n

              ,<\/p>\n

              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<\/p>\n

              Remark 7. <\/strong>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.<\/p>\n

              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 ).<\/p>\n

              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<\/p>\n

                \n
              • is a faithful, semi-finite, normal trace on ,<\/li>\n
              • is a positive, self-adjoint operator to<\/li>\n
              • is a -, faithful, semi-finite, normal weight on ,<\/li>\n
              • for where is a -subalgebra of<\/li>\n<\/ul>\n

                but because otherwise it would follow , hence in contradiction to .<\/p>\n

                Remark 8. <\/strong>If in Theorem 6 we assume that belongs to the \u2013closure of (that happens, for example, if because ), then it follows also the equality<\/p>\n

                Since trivially, we have to verify that for any with , that is with , we have<\/p>\n

                By (16), by the lower semicontinuity of in the -topology, and by (18), we obtain .<\/p>\n

                Using now the inequalities<\/p>\n

                and as in [6], Lemme 7 (b) or [18], Proposition 1.1, we have<\/p>\n

                Since, by (16), in the -topology, we conclude that<\/p>\n

                what is equivalent to [19].<\/p>\n

                The next theorem is a slight extension of [18], Theorem 2.3:<\/p>\n

                Theorem 9.<\/strong> 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<\/p>\n

                Proof.<\/strong> By Theorem 6 we have In particular, is semi-finite.<\/p>\n

                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.<\/p>\n

                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.<\/p>\n

                Theorem 10.<\/strong> 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<\/p>\n

                Proof. <\/strong>Since is -invariant and , the normal weight is still -invariant : we have for every and<\/p>\n

                Hence we applying Theorem 9 with replaced by An immediate consequence of Theorem 2.4 an 2.5 is [13], Proposition 5.9 :<\/p>\n

                Corollary 11. <\/strong>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 .<\/p>\n

                Theorem 12. <\/strong> 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,<\/p>\n

                Proof. <\/strong>We applying here twice Theorem 7. An immediate consequence of Theorem 12 are :<\/p>\n

                Theorem 13. <\/strong>Let be a -algebra, faithful, semi-finite, normal weights on and . We assuming that there exists a -dense, both and — subalgebra of then<\/p>\n

                Then<\/p>\n

                Corollary 14. <\/strong>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<\/p>\n

                  \n
                1. for all if and only if has a continuous extension , which is analytic in the interior and such that is isometric.<\/li>\n<\/ol>\n

                  References: <\/strong><\/p>\n

                  [1] C.A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc.126 (1967), 286- 302.<\/p>\n

                  [2] H. Araki, L. Zsid\u00f3, Extension of the structure theorem of Borchers and its application to half- sided modular inclusions, Rev, Math. Phys. 17 (2005), 1- 53.<\/p>\n

                  [3] I. Cior\u00e1nescu, L. Zsid\u00f3, Analytic generators for one- parameter groups, T\u00f4hoku Math. J. 28 (1976), 311- 346.<\/p>\n

                  [4] D. L. Cohn, Measure Theory, Birkh\u0201user, Boston. Basel. Stuttgart, 1980.<\/p>\n

                  [5] A. Connes, Une classification des facteurs de type III, Ann. Sci. \u00c9cole Norm. Sup. 6 (1973), 133- 252.<\/p>\n

                  [6] A. Connes, Sur le theorem de Radon- Nikodym pour les poids normaux fid\u00e9les semi- finis, Bull. Sc. Math., s\u00e9rie, 97 (1973), 253- 258.<\/p>\n

                  [7] A. Connes, M Takesaki The flow of weights on factors of type III, T\u00f4hoku Math. J. 29 (1977), 473- 575.<\/p>\n

                  [8] J. Dixmier, Les alg\u00e9bres d\u2019op\u00e9rateurs dans l\u2019espace hilbertien (Alg\u00e9bres de von Neumann), edition, Gauthier-Villars, Paris, 1969.<\/p>\n

                  [9] N. Dunford, J. T. Schwartz, Linear Operators, Part 1, Interscience Publishers, New York, 1958.<\/p>\n

                  [10] U. Haagerup, Normal weights on -algebras, J. Funct. Anal. 19 (1975), 302- 317.<\/p>\n

                  [11] U. Haagerup, Operator valued weights in von Neumann algebras I, J. Func. Anal. 32 (1979), 175- 206.<\/p>\n

                  [12] U. Haagerup, Operator valued weights in von Neumann algebras II, J. Func. Anal. 33 (1979), 339- 361.<\/p>\n

                  [13] G. K. Pedersen, M. Takesaki, The Radon- Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53- 87.<\/p>\n

                  [14] S. Str\u0103tila\u0103, Modular theory in operator algebras, Abacus Press, Tunbridge Wells, Kent, 1981.<\/p>\n

                  [15] S. Str\u0103tila\u0103, L. Zsid\u00f3, Lectures on von Neumann algebras, 2nd<\/sup> edition, Cambridge Uninv. Press, 2019.<\/p>\n

                  [16] S. Str\u0103tila\u0103, L. Zsid\u00f3, Operator Algebras, INCREST Prepublication (1977-1979), 511 p, to appear at The Theta Foundation, Bucuresti, Romania.<\/p>\n

                  [17] L. Zsid\u00f3, Spectral and ergodic properties of the analytic generator, J. Appr. Theory 20 (1977), 77- 138.<\/p>\n

                  [18] L. Zsid\u00f3, On the equality of two weights, Rev. Roum. Math Pures et Appl. 23 (1978), 631- 646.<\/p>\n

                  [19] L. Zsid\u00f3, On the equality of operator valued weights, J. Funct. Anal. Vol. 282, 9 (2022), 1- 20.<\/p>\n

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                  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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_joinchat":[],"footnotes":""},"class_list":["post-4161","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/pages\/4161","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/comments?post=4161"}],"version-history":[{"count":1,"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/pages\/4161\/revisions"}],"predecessor-version":[{"id":4186,"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/pages\/4161\/revisions\/4186"}],"wp:attachment":[{"href":"https:\/\/www.hnjournal.net\/ar\/wp-json\/wp\/v2\/media?parent=4161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}