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Arens regularity of the algebra $ A\hat\otimes B$


Author: A. Ülger
Journal: Trans. Amer. Math. Soc. 305 (1988), 623-639
MSC: Primary 46H05; Secondary 46J15, 46M05
DOI: https://doi.org/10.1090/S0002-9947-1988-0924772-1
Erratum: Trans. Amer. Math. Soc. 355 (2003), 3839-3839.
MathSciNet review: 924772
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Abstract: Let $ A$ and $ B$ be two Banach algebras. On the projective tensor product $ A\hat \otimes \,B$ of $ A$ and $ B$ there exists a natural algebra structure. In this note we study Arens regularity of the Banach algebra $ A\hat \otimes \,B$.


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  • [1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 939-948. MR 0045941 (13:659f)
  • [2] N. Arikan, Arens regularity and reflexivity, Quart. J. Math. Oxford Ser. 32 (1981), 383-388. MR 635587 (83a:46055)
  • [3] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin, 1973. MR 0423029 (54:11013)
  • [4] J. Bourgain, New Banach space properties of certain spaces of analytic functions, Proc. Internat. Congr. Math., Warszawa, 1983, pp. 945-951. MR 804748 (86m:46050)
  • [5] -, Bilinear forms on $ H$ and bounded bianalytic functions, Trans. Amer. math. Soc. 286 (1984), 313-337. MR 756042 (86c:46060)
  • [6] R. D. Bourgin, Geometric aspects of convex sets with Radon-Nikodym property, Lecture Notes in Math., vol. 993, Springer-Verlag, Berlin and New York, 1983. MR 704815 (85d:46023)
  • [7] F. Delbean, The Dunford-Pettis property for certain uniform algebras, Pacific J. Math. 65 (1976), 29-33. MR 0425617 (54:13571)
  • [8] J. Diestel, A survey of results related to the Dunford-Pettis property, Contemporary Math., vol. 2, Amer. Math. Soc., Providence, R. I., 1980, pp. 15-60. MR 621850 (82i:46023)
  • [9] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [10] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309-325. MR 559675 (81f:46057)
  • [11] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [12] R. Edwards, Functional analysis, Holt, Rinehart and Winston, New York, 1965. MR 0221256 (36:4308)
  • [13] B. R. Gelbaum, Tensor products of Banach algebras, Canad. J. Math. 11 (1959), 297-310. MR 0104162 (21:2922)
  • [14] J. Gil de Lamadrid, Uniform cross norms and tensor products of Banach algebras, Bull. Amer. Math. Soc. 69 (1967), 797-803. MR 0162154 (28:5353)
  • [15] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 1955. MR 0075539 (17:763c)
  • [16] -, Critéres de compacité dans les espaces functionels généraux, Amer. J. Math. 74 (1952), 168-186.
  • [17] -, Sur les applications linéaires faiblement compactes d'espaces du type $ C(K)$, Canad. J. Math. 5 (1953), 129-173. MR 0058866 (15:438b)
  • [18] -, Résumé de la théorie métriques des produits tensoriels topologiques, Bol. Soc. Math. Sao Paulo 8 (1956), 1-79.
  • [19] S. Kaijer, Some results in the metric theory of tensor products, Studia Math. 63 (1978), 157-170. MR 511301 (80c:46069)
  • [20] G. Köthe, Topological vector spaces. II, Springer-Verlag, Berlin and New York, 1979.
  • [21] E. O. Oshobi and J. S. Pym, Banach algebras whose duals consist of multipliers (to appear). MR 906623 (88j:46044)
  • [22] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. MR 0126145 (23:A3441)
  • [23] A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III, Studia Math. 18 (1959), 211-222. MR 0107806 (21:6528)
  • [24] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conf. Ser. in Math., no. 60, Amer. Math. Soc., Providence, R. I., 1984. MR 829919 (88a:47020)
  • [25] J. S. Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31-33. MR 780872 (86g:46072)
  • [26] -, The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc. 15 (1965), 84-104. MR 0173152 (30:3367)
  • [27] H. P. Rosenthal, A characterization of Banach spaces containing $ {l^1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413 MR 0358307 (50:10773)
  • [28] R. Schatten, A theory of cross spaces, Ann. of Math. Studies, no. 26, Princeton Univ. Press, Princeton, N. J., 1950. MR 0036935 (12:186e)
  • [29] J. Tomiyama, Tensor products of commutative Banach algebras, Tôhoku Math. J. 12 (1960), 147-154. MR 0115108 (22:5910)
  • [30] A. Ülger, Weakly compact bilinear forms and Arens regularity, Proc. Amer. Math. Soc. 101 (1987), 697-704. MR 911036 (88k:46056)
  • [31] -, Geometry of Banach spaces applied to Arens regularity problems (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0924772-1
Keywords: Arens regularity, tensor product
Article copyright: © Copyright 1988 American Mathematical Society

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