On discontinuity of derivations, inducing inequivalent complete metric topologies
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https://doi.org/10.17398/2605-5686.39.1.1Palabras clave:
Fréchet algebra of power series in infinitely many indeterminates, derivation, (in)equivalent Fréchet algebra topologies, Loy’s questionResumen
We give an elementary method for constructing commutative Fréchet algebras with non-unique Fréchet algebra topology. The result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined among other useful applications. We give an affirmative answer to a question of Loy (1974) for Fréchet algebras. We also obtain the uniqueness of the Fréchet algebra topology of certain Fréchet algebras with finite dimensional radicals.
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G.R. Allan, Fréchet algebras and formal power series, Studia Math. 119 (1996), 271 – 278.
R. Arens, The space L ω and convex topological rings, Bull. Amer. Math. Soc. 52 (1946), 931 – 935.
W.G. Bade, P.C. Curtis Jr., The continuity of derivations of Banach algebras, J. Functional Analysis 16 (1974), 372 – 387.
S.J. Bhatt, S.R. Patel, On Fréchet algebras of power series, Bull. Austral. Math. Soc. 66 (2002), 135 – 148.
R.L. Carpenter, Continuity of derivations on F -algebras, Amer. J. Math. 93 (1971), 500 – 502.
R.L. Carpenter, Uniqueness of topology for commutative semisimple F -algebras, Proc. Amer. Math. Soc. 29 (1971), 113 – 117.
H.G. Dales, The uniqueness of the functional calculus, Proc. London Math. Soc. (3) 27 (1973), 638 – 648.
H.G. Dales, “ Banach Algebras and Automatic Continuity ”, London Math. Soc. Monogr. (N.S.) 24, Clarendon Press, Oxford, 2000.
H.G. Dales, R.J. Loy, Uniqueness of the norm topology for Banach algebras with finite-dimensional radical, Proc. London Math. Soc. 74 (1997), 633 – 661.
H.G. Dales, J.P. McClure, Continuity of homomorphisms into certain commutative Banach algebras, Proc. London Math. Soc. (3) 26 (1973), 69 – 81.
H.G. Dales, J.P. McClure, Completion of normed algebras of polynomials, J. Austral. Math. Soc. Ser. A 20 (1975), 504 – 510.
H.G. Dales, S.R. Patel, C.J. Read, Fréchet algebras of power series, in: “ Banach Algebras 2009 ”, Banach Center Publi. 91, Polish Acad. Sci., Warsaw, 2010, 123 – 158.
Y. Domar, A Banach algebra related to the disk algebra, in: “ Radical Banach Algebras and Automatic Continuity 1981 ”, Lecture Notes in Math. 975, Springer-Verlag, Berlin-New York, 1983, 309 – 311.
C. Feldman, The Wedderburn principal theorem in Banach algebras, Proc. Amer. Math. Soc. 2 (1951), 771 – 777.
M. Fragoulopoulou, M. Weigt, I. Zarakas, Derivations of locally convex ∗-algebras, Extracta Math. 26 (2011), 45 – 60.
R.J. Loy, Commutative Banach algebras with non-unique complete norm topology, Bull. Austral. Math. Soc. 10 (1974), 409 – 420.
S.R. Patel, Fréchet algebras, formal power series, and automatic continuity, Studia Math. 187 (2008), 125 – 136.
S.R. Patel, Uniqueness of the Fréchet algebra topology on certain Fréchet algebras, Studia Math. 234 (2016), 31 – 47.
S.R. Patel, On avatars of the Singer-Wermer conjecture in Fréchet algebras, in: “ Proceedings of the ICTAA 2021 ”, Mathematics Studies 8, Tartu, 2021, 121 – 137.
S.R. Patel, On functional analytic approach for Gleason’s problem in the theory of SCV, Banach J. Math. Anal. 16 (2022), article no. 47, 17 pp.
H.L. Pham, Automatic continuity for Banach algebras with finite-dimensional radicals, J. Austral. Math. Soc. 81 (2006), 279 – 295.
C.J. Read, Derivations with large separating subspace, Proc. Amer. Math. Soc. 130 (2002), 3671 – 3677.
S. Rolewicz, Example of semisimple m-convex B 0 -algebra, which is not a projective limit of semisimple B-algebras, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 11 (1963), 459 – 462.
M.P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435 – 460.
D. Vogt, Non-natural topologies on spaces of holomorphic functions, Ann. Polon. Math. 108 (2013), 215 – 217.
M. Weigt, I. Zarakas, A note on derivations of Pro-C ∗ -algebras into complete locally convex bimodules, New Zealand J. Math. 41 (2011), 143 – 152.
M. Weigt, I. Zarakas, Derivations of generalized B ∗ -algebras, Extracta Math. 28 (2013), 77 – 94.
M. Weigt, I. Zarakas, Spatiality of derivations of Fréchet GB ∗ -algebras, Studia Math. 231 (2015), 219 – 239.
M. Weigt, I. Zarakas, Derivations of Fréchet nuclear GB ∗ -algebras, Bull. Austral. Math. Soc. 92 (2015), 290 – 301.
M. Weigt, I. Zarakas, Unbounded derivations of GB ∗ -algebras, in Oper. Theory Adv. Appl., 247, Birkhäuser/Springer, Cham, 2015, 69 – 82.
M. Weigt, I. Zarakas, On domains of unbounded derivations of generalized B ∗ -algebras, Banach J. Math. Anal. 12 (2018), 873 – 908.
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Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial 4.0.
