Derivations on universally complete \(f\)-algebras. (English) Zbl 1319.46037
I. M. Singer and J. Wermer [Math. Ann. 129, 260–264 (1955; Zbl 0067.35101)] showed that every bounded derivation on a commutative Banach algebra maps into the radical and they conjectured that the condition of boundedness for the derivation was not necessary. This conjecture has finally been proved by M. P. Thomas [Ann. Math. (2) 128, No. 3, 435–460 (1988; Zbl 0681.47016)]. Motivated by the interest in range inclusion results for derivations on Banach algebras, the authors of this paper give a non-Banach Singer-Wermer version for derivations acting on universally complete \(f\)-algebras. As results, they prove that, if \(A\) is a universally complete \(f\)-algebra and \(M\) is a uniformly closed maximal ring ideal of \(A\), then \(M\) is invariant under any derivation \(D\) on \(A\) and \(D\) maps into \(\mathrm{Clorad}(A),\) where \(\mathrm{Clorad}(A)\equiv \bigcap_{M \in m(A)} M,\) and \(m(A)\) denotes the set of all uniformly closed maximal ring ideals of \(A\). Furthermore, the authors present some additional systematic theorems concerned with the study of weakly locally one-dimensional spaces by using the range inclusion theorems for derivations acting on universally complete \(f\)-algebras.
Reviewer: Hark-Mahn Kim (Daejeon)
References:
| [1] | Abramovich, Yu. A.; Veksler, A. I.; Koldunov, A. V., On disjointness preserving operators, Dokl. Akad. Nauk SSSR, 289, 5, 1033-1036 (1979) · Zbl 0445.46017 |
| [2] | Abramovich, Yu. A.; Wickstead, A. W., The regularity of order bounded operators into \(C(K)\) II, Q. J. Math. Oxford, Ser. 2, 44, 257-270 (1993) · Zbl 0991.47023 |
| [3] | Aliprantis, C. D.; Burkinshaw, O., Positive Operators (1985), Academic Press: Academic Press Orlando · Zbl 0567.47037 |
| [4] | Basly, M.; Triki, A., On Uniformly Closed Ideals in \(f\)-algebras, 2nd Conference, Functions Spaces, 29-33 (1995), Marcel Dekker · Zbl 0841.46032 |
| [5] | Ben Amor, F., On orthosymmetric bilinear maps, Positivity, 14, 1, 123-134 (2010) · Zbl 1204.06010 |
| [6] | Boulabiar, K., Positive derivations on Archimedean almost \(f\)-rings, Order, 19, 4, 385-395 (2002) · Zbl 1025.06015 |
| [7] | Colville, P.; Davis, G.; Keimel, K., Positive derivations on \(f\)-rings, J. Aust. Math. Soc. A, 23, 371-375 (1977) · Zbl 0376.06021 |
| [9] | Duhoux, M.; Meyer, M., Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Brux., 98, 3-18 (1984) · Zbl 0559.46003 |
| [10] | Gutman, A. E., Locally one-dimensional K-spaces and \(\sigma \)-distributive Boolean algebras, Sib. Adv. Math., 5, 2, 99-121 (1995) · Zbl 0854.46007 |
| [13] | Henriksen, M.; Smith, F. A., Some properties of positive derivations on \(f\)-rings, Contemp. Math., 8, 175-184 (1982) · Zbl 0491.06016 |
| [14] | Huijsmans, C. B., Lattice-ordered division algebras, Proc. R. Ir. Acad. A, 92, 2, 239-241 (1992) · Zbl 0802.06019 |
| [15] | Huijsmans, C. B.; de Pagter, B., Ideal theory in \(f\)-algebras, Trans. Amer. Math. Soc., 269, 225-245 (1982) · Zbl 0483.06009 |
| [16] | Johnson, B. E., Continuity of derivations on commutative algebras, Amer. J. Math., 91, 1-10 (1969) · Zbl 0181.41103 |
| [17] | Johnson, B. E.; Sinclair, A. M., Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90, 1067-1073 (1968) · Zbl 0179.18103 |
| [18] | Kaplansky, I., (Some Aspects of Analysis and Probability. Some Aspects of Analysis and Probability, Surveys in Applied Mathematics, vol. 4 (1958), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York), 1-34 · Zbl 0086.00102 |
| [19] | Kusraev, A. G., Automorphisms and derivations on a universally complete complex \(f\)-algebra, Sibirsk. Mat. Zh., 47, 1, 97-107 (2006) · Zbl 1113.46043 |
| [20] | Luxemburg, W. A.; Moore, L. C., Archimedean quotient Riesz spaces, Duke Math. J., 34, 725-739 (1967) · Zbl 0171.10501 |
| [21] | McPolin, P. T.N.; Wickstead, A. W., The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cambridge Philos. Soc., 97, 3, 481-487 (1985) · Zbl 0564.47017 |
| [22] | Sakai, S., (Operator Algebras in Dynamical Systems, The Theory of Unbounded Derivations in \(C^{\text{*}} \)-algebras. Operator Algebras in Dynamical Systems, The Theory of Unbounded Derivations in \(C^{\text{*}} \)-algebras, Encyclopedia of Mathematics and its Applications, vol. 41 (1991), Cambridge University Press) · Zbl 0743.46078 |
| [23] | Silov, G. E., On a property of rings of functions, Dokl. Akad. Nauk SSSR, 58, 985-988 (1947) · Zbl 0030.26004 |
| [24] | Singer, I. M.; Wermer, J., Derivations on commutative normed algebras, Math. Ann., 129, 260-264 (1955) · Zbl 0067.35101 |
| [25] | Thomas, M. P., The image of a derivation is contained in the radical, Ann. Math. (2), 128, 3, 435-460 (1988) · Zbl 0681.47016 |
| [26] | Toumi, M. A., Continuous generalized \((\theta, \phi)\)-separating derivations on archimedean almost \(f\)-algebras, Asian-Eur. J. Math., 5, 3, 17 (2012), Paper No. 1250045 · Zbl 1314.06021 |
| [28] | Toumi, A.; Toumi, M. A., Order bounded derivations on Archimedean almost \(f\)-algebras, Positivity, 14, 2, 239-245 (2010) · Zbl 1213.06014 |
| [29] | Toumi, M. A.; Toumi, N., The Wickstead problem on Dedekind \(\sigma \)-complete vector lattices, Positivity, 14, 1, 135-144 (2010) · Zbl 1193.06017 |
| [30] | Wickstead, A. W., Representation and duality of multiplication operators on Archimedean Riesz spaces, Compos. Math., 35, 3, 225-238 (1977) · Zbl 0381.47021 |
| [31] | Wielandt, H., Uber die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann., 121, 21 (1949) · Zbl 0035.19903 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
