Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T03:55:39.957Z Has data issue: false hasContentIssue false
  • English
  • Français

ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

Part of: Topological algebras, normed rings and algebras, Banach algebras Selfadjoint operator algebras Abstract harmonic analysis

Published online by Cambridge University Press:  04 December 2019

RUKI MATSUI Open the ORCID record for RUKI MATSUI [Opens in a new window]  and
YUJI TAKAHASHI Open the ORCID record for YUJI TAKAHASHI [Opens in a new window]
RUKI MATSUI
Affiliation:
Department of Mathematics,Hokkaido University of Education, Asahikawa, 070-8621, Japan email rouge.pine317@gmail.com
YUJI TAKAHASHI*
Affiliation:
Department of Mathematics,Hokkaido University of Education, Asahikawa, 070-8621, Japan email takahashi.yuji@a.hokkyodai.ac.jp
*
✉takahashi.yuji@a.hokkyodai.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Keywords

Arens productArens regularitysecond dual Banach algebraradicallocally compact groupgroup algebrameasure algebraC∗ -algebraFourier and Fourier–Stieltjes transforms

MSC classification

Primary: 43A20: $L^1$-algebras on groups, semigroups, etc.
Secondary: 46H05: General theory of topological algebras 46L05: General theory of $C^*$-algebras 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 138 - 150
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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.)

References

Arens, R., ‘Operations induced in function classes’, Monatsh. Math. 55(1951) 1–19.10.1007/BF01300644CrossRefGoogle Scholar
Arens, R., ‘The adjoint of a bilinear operation’, Proc. Amer. Math. Soc. 2(1951) 839–848.10.1090/S0002-9939-1951-0045941-1CrossRefGoogle Scholar
Bonsall, F. F. and Duncan, J., Complete Normed Algebras (Springer, Berlin–Heidelberg–New York, 1973).10.1007/978-3-642-65669-9CrossRefGoogle Scholar
Civin, P. and Yood, B., ‘The second conjugate space of a Banach algebra as an algebra’, Pacific J. Math. 11(1961) 847–870.10.2140/pjm.1961.11.847CrossRefGoogle Scholar
Dales, H. G., Banach Algebras and Automatic Continuity (Clarendon Press, Oxford, 2000).Google Scholar
Dales, H. G. and Lau, A. T.-M., ‘The second duals of Beurling algebras’, Mem. Amer. Math. Soc. 177(2005) 1–191.Google Scholar
Diestel, J., ‘A survey of results related to the Dunford–Pettis property’, Contemp. Math. 2(1980) 15–60.10.1090/conm/002/621850CrossRefGoogle Scholar
Duncan, J. and Hosseiniun, S. A. R., ‘The second dual of a Banach algebra’, Proc. Roy. Soc. Edinburgh Sect. A 84(1979) 309–325.10.1017/S0308210500017170CrossRefGoogle Scholar
Fell, J. M. G. and Doran, R. S., Representations of ∗-Algebras, Locally Compact Groups, and Banach ∗-Algebraic Bundles I (Academic Press, Boston, MA, 1988).Google Scholar
Friedberg, S. H., ‘The Fourier transform is onto only when the group is finite’, Proc. Amer. Math. Soc. 27(1971) 421–422.Google Scholar
Graham, C. C., ‘The Fourier transform is onto only when the group is finite’, Proc. Amer. Math. Soc. 38(1973) 365–366.10.1090/S0002-9939-1973-0313716-6CrossRefGoogle Scholar
Granirer, E. E., ‘On amenable semigroups with a finite-dimensional set of invariant means I’, Illinois J. Math. 7(1963) 32–48.10.1215/ijm/1255637480Google Scholar
Granirer, E. E., ‘The radical of L ∞(G)∗’, Proc. Amer. Math. Soc. 41(1973) 321–324.Google Scholar
Gulick, S. L., ‘Commutativity and ideals in the biduals of topological algebras’, Pacific J. Math. 18(1966) 121–137.10.2140/pjm.1966.18.121CrossRefGoogle Scholar
Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis II (Springer, Berlin–Heidelberg–New York, 1970).Google Scholar
Palmer, T. W., Banach Algebras and the General Theory of ∗-Algebras I (Cambridge University Press, Cambridge, 1994).10.1017/CBO9781107325777CrossRefGoogle Scholar
Paterson, A. L. T., Amenability (American Mathematical Society, Providence, RI, 1988).CrossRefGoogle Scholar
Rajagopalan, M., ‘Fourier transform in locally compact groups’, Acta Sci. Math. (Szeged) 25(1964) 86–89.Google Scholar
Sakai, S., ‘Weakly compact operators on operator algebras’, Pacific J. Math. 14(1964) 659–664.10.2140/pjm.1964.14.659CrossRefGoogle Scholar
Segal, I. E., ‘The class of functions which are absolutely convergent Fourier transforms’, Acta Sci. Math. (Szeged). 12(1950) 157–161.Google Scholar
Ülger, A., ‘Weakly compact bilinear forms and Arens regularity’, Proc. Amer. Math. Soc. 101(1987) 697–704.10.1090/S0002-9939-1987-0911036-XCrossRefGoogle Scholar
Ülger, A., ‘Arens regularity sometimes implies the RNP’, Pacific J. Math. 143(1990) 377–399.10.2140/pjm.1990.143.377CrossRefGoogle Scholar
Young, N. J., ‘The irregularity of multiplication in group algebras’, Quart. J. Math. Oxford. 24(1973) 59–62.10.1093/qmath/24.1.59CrossRefGoogle Scholar