Abstract
For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.
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References
Allen S. D., Sinclair A. M., Smith R. R.: The ideal structure of the Haagerup tensor product of C*-algebras. J. Reine Angew. Math. 442, 111–148 (1993)
Arikan N.: Arens regularity and reflexivity. Quart. J. Math. 32, 383–388 (1981)
Blecher D. P.: Tensor products which do not preserve operator algebras. Math. Proc. Camb. Philos Soc. 108, 395–403 (1990)
Civin P., Bertram Y.: The second conjugate space of a Banach algebra as an algebra. Pacific J. Math. 11, 847–870 (1961)
Effros E. G., Ruan Z. J.: On approximation properties for operator spaces. International J. Math. 1, 163–187 (1990)
E. G. Effros and Z. J. Ruan, Operator spaces, Claredon Press-Oxford, New York, 2000.
Forrest B.: Arens regularity and discrete groups. Pacific J. Math. 151, 217–227 (1991)
Forrest B.: Arens regularity and the A p (G) groups. Proc. Amer. Math. Soc. 119, 595–598 (1993)
Freedman W., Ülger A.: The Phillips properties. Proc. Amer. Math. Soc. 128, 2137–2145 (2000)
A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74(1952), 168–186.
Haagerup U., Musat M.: The Effros-Ruan conjecture for bilinear forms on C*-algebras. Invent. Math. 174, 139–163 (2008)
Jain R., Kumar A.: Ideals in operator space projective tensor products of C*-algebras. J. Aust. Math. Soc. 91, 275–288 (2011)
Lau A.T.-M., Wong J.C.S.: Weakly almost periodic elements in \({L^\infty(G)}\) of a locally compact group G. Proc. Amer. Math. Soc. 107, 1031–1036 (1989)
Lau A.T.-M., Ülger A.: Some geometric properties on the Fourier and Fourier Stieltjes algebras of locally compact Groups Arens regularity and related problems. Trans. Amer. Math. Soc. 337, 321–359 (1993)
Pisier G., Shlyakhtenko D.: Grothendieck’s theorem for operator spaces. Invent. Math. 150, 185–217 (2002)
G. Pisier, Introduction to operator space theory, Cambridge University Press, Cambridge, 2003.
Pym J. S.: The convolution of functionals on spaces of bounded functions. Proc. London Math. Soc. 15, 84–104 (1965)
Ülger A.: Arens regularity of the algebra \({A\otimes_{\gamma} B}\). Trans. Amer. Math. Soc. 305, 623–639 (1988)
Ülger A.: Arens regularity sometimes implies the RNP. Pacific J. Math. 143, 377–399 (1990)
Ülger A.: Erratum to Arens regularity of the algebra \({A\otimes_{\gamma} B}\). Trans. Amer. Math. Soc. 355, 3839 (2003)
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Rajpal, V., Kumar, A. Arens regularity of projective tensor products. Arch. Math. 107, 531–541 (2016). https://doi.org/10.1007/s00013-016-0942-y
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DOI: https://doi.org/10.1007/s00013-016-0942-y