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Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

  • Ross Stokke (a1)

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.

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This research was partially supported by an NSERC grant.

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[1] Arens, R., The adjoint of a bilinear operation . Proc. Amer. Math. Soc. 2(1951), 839848.
[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.
[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.
[9] Daws, M., Dual Banach algebras: representations and injectivity. Studia Math. 178 (2007), no. 3, 231–275.
[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.
[12] Ghahramani, F., Isometric representation of on . Glasgow Math. J. 23 (1982), no. 2, 119–122.
[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.
[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.
[16] Lau, A. T.-M., Uniformly continuous functionals on Banach algebras . Colloq. Math. 51(1987), 195205.
[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.
[20] Pisier, G., Introduction to operator space theory. London Mathematical Society Lecture Note Series, 294. Cambridge University Press, Cambridge, 2003.
[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.
[22] Stokke, R., Homomorphisms of convolution algebras. J. Funct. Anal. 261 (2011), no. 12, 3665–3695.
[23] Stokke, R., Amenability and modules for Arens product algebras. Q. J. Math. 66 (2015), no. 1, 295–321.
[24] Størmer, E., Regular abelian Banach algebras of linear maps of operator algebras. J. Funct. Anal. 37 (1980), no. 3, 331–373.
[25] Uygul, F., A representation theorem for completely contractive dual Banach algebras. J. Operator Theory 62 (2009), no. 2, 327–340.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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