Hostname: page-component-6b989bf9dc-vmcqm Total loading time: 0 Render date: 2024-04-13T15:34:32.370Z Has data issue: false hasContentIssue false

The second dual of a Banach algebra*

Published online by Cambridge University Press:  14 November 2011

J. Duncan
Department of Mathematics, University of StirlingTabriz, Iran
S. A. R. Hosseiniun
Department of Mathematics, The University, Tabriz, Iran


We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

Research Article
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



1Alexander, F. E... The dual and bidual of certain A*-algebras. Proc. Amer. Math. Soc. 38 (1973), 571576.Google Scholar
2Alexander, F. E.. Some algebraic properties of F(X) and K(X). Proc. Edinburgh Math. Soc. 19 (1975), 353361.Google Scholar
3Alexander, F. E.. The bidual of A*-algebras of the first kind. J. London Math. Soc. 12 (1975), 16.CrossRefGoogle Scholar
4Arens, R.. Operations induced in function classes. Monatsh. Math. 55 (1951), 119.CrossRefGoogle Scholar
5Arens, R.. The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2 (1951), 839848.CrossRefGoogle Scholar
6Baker, J. W. and Butcher, R. J.. The Stone-Čech compactification of a topological semigroup. Math. Proc. Camb. Philos. Soc. 80 (1976), 103107.CrossRefGoogle Scholar
7Berglund, M. C. F.. Ideal C*-algebras. Duke Math. J. 40 (1973), 241257.Google Scholar
8.Bonsall, F. F. and Duncan, J.. Numerical ranges of operators on normed spaces and elements of normed algebras. London Math. Soc. Lecture Note Series 2 (Cambridge Univ. Press, 1971).Google Scholar
9Bonsall, F. F. and Duncan, J.. Numerical ranges II. London Math. Soc. Lecture Note Series 10 (Cambridge Univ. Press, 1973).Google Scholar
10Bonsall, F. F. and Duncan, J.. Complete normed algebras (New York: Springer, 1973).CrossRefGoogle Scholar
11Civin, P.. Extensions of homomorphisms. Pacific J. Math. 11 (1961), 12231233.Google Scholar
12Civin, P.. Ideals in the second conjugate algebra of a group algebra. Math. Scand. 11 (1962), 161174.CrossRefGoogle Scholar
13Civin, P.. Annihilators in the second conjugate algebra of a group algebra. Pacific J. Math. 12 (1962), 855862.CrossRefGoogle Scholar
14Civin, P. and Yood, B.. The second conjugate space of a Banach algebra as an algebra. Pacific J. Math. 11 (1961), 847870.Google Scholar
15Craw, I. G. and Young, N. J.. Regularity of multiplication in weighted group and semigroup algebras. Quart. J. Math. Oxford 25 (1974), 351358.Google Scholar
16Davenport, J. W.. Multipliers on a Banach algebra with a bounded approximate identity. Pacific J. Math. 63 (1976), 131135.Google Scholar
17Day, M. M.. Amenable semigroups. Illinois J. Math. 1 (1957), 509544.CrossRefGoogle Scholar
18Dixmier, J.. Les C*-algèbres et leurs représentations. Cahiers Scientifiques Fac. 29 (Paris: Gauthier-Villars, 1964).Google Scholar
19Fell, J. M. G.. The dual spaces of Banach algebras. Trans. Amer. Math. Soc. 114 (1965), 227250.Google Scholar
20Granirer, E. and Rajagopalan, M.. A note on the radical of the second conjugate algebra of a semigroup algebra. Math. Scand. 15 (1964), 163166.Google Scholar
21Granirer, E. E.. The radical of L(G)*. Proc. Amer. Math. Soc. 41 (1973), 321324.Google Scholar
22Gulick, S. L.. The bidual of a locally multiplicatively-convex algebra. Pacific J. Math. 17 (1966), 7196.CrossRefGoogle Scholar
23Gulick, S. L.. Commutativity and ideals in the biduals of topological algebras. Pacific J. Math. 18 (1966), 121137.CrossRefGoogle Scholar
24Hennefeld, J.. Duals of Banach algebras (Columbia Univ. Ph.D. Thesis, 1967).Google Scholar
25Hennefeld, J.. A note on the Arens product. Pacific J. Math. 26 (1968), 115119.Google Scholar
26Hennefeld, J.. The Arens products and an embedding theorem. Pacific J. Math. 29 (1969), 551563.Google Scholar
27Hosseiniun, S. A. R.. The second dual of a Banach algebra (Stirling Univ. Ph.D. Thesis, 1978).Google Scholar
28Macri, N.. The continuity of Arens product on the Stone-Čech compactification of semigroups. Trans. Amer. Math. Soc. 191 (1974), 185193.Google Scholar
29Màté, L.. Embedding multiplier operators of a Banach algebra B into its second conjugate space B**. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 809812.Google Scholar
30Màté, L.. The Arens product and multiplier operators. Studia Math. 28 (1967), 227234.Google Scholar
31McKilligan, S. A.. Duality in B*-algebras. Proc. Amer. Math. Soc. 38 (1973), 8688.Google Scholar
32McKilligan, S. A.. Arens regularity of certain Banach algebras which are ideals of a Banach algebra. Proc. Amer. Math. Soc. 50 (1975), 223229.CrossRefGoogle Scholar
33McKilligan, S. A. and White, A. J.. Representations of L-algebras, Proc. London Math. Soc. 25 (1972), 655674.Google Scholar
34Palmer, T. W.. Arens multiplication and a characterization of W*-algebras. Proc. Amer. Math. Soc. 44 (1974), 8187.Google Scholar
35Pym, J. S.. The convolution of functionals on spaces of bounded functions. Proc. London Math. Soc. 15 (1965), 84104.Google Scholar
36Pym, J. S.. Convolution and the second dual of a Banach algebra. Proc. Camb. Philos. Soc. 65 (1969), 597599.Google Scholar
37Pym, J. S.. Convolution measure algebras are not usually Arens regular. Quart. J. Math. Oxford 25 (1974), 235240.Google Scholar
38Rennison, J. F.. Arens products and measure algebras. J. London Math. Soc. 44 (1969), 369377.CrossRefGoogle Scholar
39Rennison, J. F.. Arens products and measure algebras—a supplement. J. London Math. Soc. 1 (1969), 232236.Google Scholar
40Sherman, S.. The second adjoint of a C*-algebra. Proc. Int. Congr. Math. Cambridge, Vol. 1 (1950), 470.Google Scholar
41Stafney, J. D.. Arens multiplication and convolution. Pacific J. Math. 14 (1964), 14231447.CrossRefGoogle Scholar
42Takeda, Z.. Conjugate spaces of operator algebras. Proc. Japan Acad. 30 (1954), 9095.Google Scholar
43Tomita, M.. The second dual of a C*-algebra. Mem. Fac. Sci. Kyushu Univ. Ser. A 21 (1967), 185193.Google Scholar
44Tomiuk, B. J. and Wong, P. K.. The Arens product and duality in B*-algebras. Proc. Amer. Math. Soc. 25 (1970), 529535.Google Scholar
45Watanabe, S.. A Banach algebra which is an ideal in the second dual space. Sci. Rep. Niigata Univ. Ser. A 11 (1974), 95101.Google Scholar
46Watanabe, S.. A Banach algebra which is an ideal in the second dual space II. Sci. Rep. Niigata Univ. Ser. A 13 (1976), 4348.Google Scholar
47Wong, P. K.. The Arens product and duality in B*-algebras II. Proc. Amer. Math. Soc. 27 (1971), 535538.Google Scholar
48Wong, P. K.. Modular annihilator A*-algebras. Pacific J. Math. 37 (1971), 825834.CrossRefGoogle Scholar
49Wong, P. K.. On the Arens product and annihilator algebras. Proc. Amer. Math. Soc. 30 (1971), 7983.Google Scholar
50Wong, P. K.. On the Arens product and commutative Banach algebras. Proc. Amer. Math. Soc. 37 (1973), 111113.Google Scholar
51Wong, P. K.. On the Arens product and certain Banach algebras. Trans. Amer. Math. Soc. 180 (1973), 437448.Google Scholar
52Wong, P. K.. The second conjugate of certain Banach algebras. Canad. J. Math. 27 (1975), 10291035.Google Scholar
53Wong, P. K.. A minimax formula for dual B*-algebras. Trans. Amer. Math. Soc. 224 (1976), 281289.Google Scholar
54Wong, P. K.. On certain subalgebras of a dual A*-algebra. J. Austral. Math. Soc. Ser. A 23 (1977), 105111.Google Scholar
55Young, N. J.. Separate continuity and multilinear operations. Proc. London Math. Soc. 26 (1973), 289319.Google Scholar
56Young, N. J.. The irregularity of multiplication in group algebras. Quart. J. Math. Oxford 24 (1973), 5962.Google Scholar
57Young, N. J.. Semigroup algebras having regular multiplication. Studia Math. 47 (1973), 191196.Google Scholar
58Young, N. J.. Periodicity of functionals and representations of normed algebras on reflexive spaces. Proc. Edinburgh Math. Soc. 20 (1976), 99120.Google Scholar

Additional References

59Alexander, J. C.. Compact Banach algebras. Proc. London Math. Soc. 18 (1968), 118.Google Scholar
60Barnes, B. A. and Duncan, J.. The Banach algebra l1(S). J. Functional Analysis 18 (1975), 96113.Google Scholar
61Bonsall, F. F. and Duncan, J.. Dually irreducible representations of Banach algebras. Quart. J. Math. Oxford 19 (1968), 97111.CrossRefGoogle Scholar
62Burckel, R. B.. Weakly almost periodic functions on semigroups (New York: Gordon and Breach, 1970).Google Scholar
63Dunford, N. and Schwartz, J. T.. Linear Operators, Part I (New York: Interscience, 1958).Google Scholar
64McGregor, C. M.. A representation for l1(S). Bull. London Math. Soc. 8 (1976), 156160.Google Scholar