Skip to main content Accesibility Help
×
×
Home

Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

  • Ross Stokke (a1)
Abstract

Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

Copyright
Footnotes
Hide All

This research was partially supported by an NSERC grant.

Footnotes
References
Hide All
[1] Arens, R., The adjoint of a bilinear operation . Proc. Amer. Math. Soc. 2(1951), 839848. https://doi.org/10.2307/2031695.
[2] Arsac, G., Sur l’espace de Banach engendré par les coefficients d’une représentation unitaire . Publ. Dép. Math. (Lyon) 13(1976), 1101.
[3] Bekka, M. E. B., Amenable unitary representations of locally compact groups . Invent. Math. 100(1990), 383401. https://doi.org/10.1007/BF01231192.
[4] Berglund, J. F., Junghenn, H., and Milnes, P., Analysis on semigroups: Function spaces, compactifications, representations. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley, New York, 1989.
[5] Conway, J., A course in functional analysis. Second edition. Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
[6]P. C. Curtis, Jr. and A. Figá-Talamanca, Factorization theorems for Banach algebras. In: Function algebras. Scott-Foresman, Chicago, IL, 1966, pp. 169–185.
[7] Dales, H. G., Banach algebras and automatic continuity. London Mathematical Society Monographs, 24. Clarendon Press, New York, 2000.
[8] Dales, H. G. and Lau, A. T.-M., The second duals of Beurling algebras. Mem. Amer. Math. Soc. 177 (2005). no. 836, vi 191 pp. https://doi.org/10.1090/memo/0836.
[9] Daws, M., Dual Banach algebras: representations and injectivity. Studia Math. 178 (2007), no. 3, 231–275. https://doi.org/10.4064/sm178-3-3.
[10] Effros, E. G. and Ruan, Z.-J., Operator spaces. London Mathematical Society Monographs, 23. Oxford University Press, New York, 2000.
[11] Filali, M., Neufang, M., and Sangani Monfared, M., Representations of Banach algebras subordinate to topologically introverted spaces. Trans. Amer. Math. Soc. 367 (2015), no. 11, 8033–8050. https://doi.org/10.1090/tran/6435.
[12] Ghahramani, F., Isometric representation of on . Glasgow Math. J. 23 (1982), no. 2, 119–122. https://doi.org/10.1017/S0017089500004882.
[13] Hu, Z., Neufang, M., and Ruan, Z.-J., Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres. Proc. Lond. Math. Soc. (3) 100 (2010) no. 2, 429–458. https://doi.org/10.1112/plms/pdp026.
[14] Hewitt, E. and Ross, K. A., Abstract harmonic analysis II. Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Grundelehern der Mathematischen Wissenschasften, 152. Springer-Verlag, New York, 1970.
[15] Lau, A. T.-M., Uniformly continuous functionals on the Fourier algebra of any locally compact group . Trans. Amer. Math. Soc. 251(1979), 3959. https://doi.org/10.2307/1998682.
[16] Lau, A. T.-M., Uniformly continuous functionals on Banach algebras . Colloq. Math. 51(1987), 195205. https://doi.org/10.4064/cm-51-1-195-205.
[17] Neufang, M., Abstrakte harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren. Ph.D. Thesis, Universität des Saarlandes, 2000.
[18] Neufang, M., Isometric representation of convolution algebras as completely bounded module homomorphisms and a characterization of the measure algebra. Carleton University, Canada, 2001.
[19] Neufang, M., Ruan, Z.-J., and Spronk, N., Completely isometric representations of and . Trans. Amer. Math. Soc. 360 (2008), no. 3, 1133–1161. https://doi.org/10.1090/S0002-9947-07-03940-2.
[20] Pisier, G., Introduction to operator space theory. London Mathematical Society Lecture Note Series, 294. Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9781107360235.
[21] Spronk, N. and Stokke, R., Matrix coefficients of unitary representations and associated compactifications of locally compact groups. Indiana Univ. Math. J. 62 (2013), no. 1, 99–148. https://doi.org/10.1512/iumj.2013.62.4825.
[22] Stokke, R., Homomorphisms of convolution algebras. J. Funct. Anal. 261 (2011), no. 12, 3665–3695. https://doi.org/10.1016/j.jfa.2011.08.014.
[23] Stokke, R., Amenability and modules for Arens product algebras. Q. J. Math. 66 (2015), no. 1, 295–321. https://doi.org/10.1093/qmath/hau018.
[24] Størmer, E., Regular abelian Banach algebras of linear maps of operator algebras. J. Funct. Anal. 37 (1980), no. 3, 331–373. https://doi.org/10.1016/0022-1236(80)90048-8.
[25] Uygul, F., A representation theorem for completely contractive dual Banach algebras. J. Operator Theory 62 (2009), no. 2, 327–340.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed