Weakly compact bilinear forms and Arens regularity

Ülger, A.

doi:10.1090/S0002-9939-1987-0911036-X

Weakly compact bilinear forms and Arens regularity
HTML articles powered by AMS MathViewer

by A. Ülger PDF

Proc. Amer. Math. Soc. 101 (1987), 697-704 Request permission

Abstract:

Let $X$ and $Y$ be two Banach spaces. We show that a bounded bilinear form $m:X \times Y \to {\mathbf {C}}$ is Arens regular iff it is weakly compact. This result permits us to find very short proofs of some known results as well as some new results. Some of them are: Any ${C^*}$-algebra, the disk algebra and the Hardy class ${H^\infty }$ are Arens regular under every possible product. We also characterize the Arens regularity of certain bilinear mappings.

References

C. A. Akemann, P. G. Dodds, and J. L. B. Gamlen, Weak compactness in the dual space of $C^{\ast }$-algebra, J. Functional Analysis 10 (1972), 446–450. MR 0344898, DOI 10.1016/0022-1236(72)90040-7
Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
Nilgün Arikan, Arens regularity and reflexivity, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 128, 383–388. MR 635587, DOI 10.1093/qmath/32.4.383
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
J. Bourgain, New Banach space properties of certain spaces of analytic functions, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 945–951. MR 804748
Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870. MR 143056, DOI 10.2140/pjm.1961.11.847
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094, DOI 10.1007/BFb0082079
J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309–325. MR 559675, DOI 10.1017/S0308210500017170

Linear operators

A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
S. L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math. 18 (1966), 121–137. MR 194912, DOI 10.2140/pjm.1966.18.121
Uffe Haagerup, The Grothendieck inequality for bilinear forms on $C^\ast$-algebras, Adv. in Math. 56 (1985), no. 2, 93–116. MR 788936, DOI 10.1016/0001-8708(85)90026-X
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4

Topological vector spaces

Aleksander Pełczyński, Structural theory of Banach spaces and its interplay with analysis and probability, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 237–269. MR 804684

Grothendieck’s theorem for noncommutative

-algebras with an application on Grothendieck’s constants

29

John Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31–33. MR 780872
John S. Pym, The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc. (3) 15 (1965), 84–104. MR 173152, DOI 10.1112/plms/s3-15.1.84
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
Pak Ken Wong, On the Arens products and certain Banach algebras, Trans. Amer. Math. Soc. 180 (1973), 437–448. MR 318885, DOI 10.1090/S0002-9947-1973-0318885-4
N. J. Young, Periodicity of functionals and representations of normed algebras on reflexive spaces, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 2, 99–120. MR 435849, DOI 10.1017/S0013091500010610
N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59–62. MR 320756, DOI 10.1093/qmath/24.1.59