Taiwanese Journal of Mathematics

Approximate Cyclic Amenability of $T$-Lau Product of Banach Algebras

Hossein Javanshiri

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Associated with two Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a norm decreasing homomorphism $T: \mathcal{B} \to \mathcal{A}$, there is a certain Banach algebra product $\mathcal{M}_T := \mathcal{A} \times_T \mathcal{B}$, which is a splitting extension of $\mathcal{B}$ by $\mathcal{A}$. In this paper, we investigate approximate cyclic amenability of $\mathcal{M}_T$ which has been introduced and studied by Esslamzadeh and Shojaee in [5]. In particular, apart from the characterization of all cyclic derivations on the Banach algebra $\mathcal{M}_T$, we improve the results of [1, 2] for cyclic amenability of $\mathcal{M}_T$. These results paves the way for obtaining new results for (approximate) cyclic amenability of the Banach algebra $\mathcal{A} \times \mathcal{B}$ equipped with the coordinatewise product algebra and $\ell^1$-norm.

Article information

Taiwanese J. Math., Volume 20, Number 5 (2016), 1139-1147.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 46H05: General theory of topological algebras 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
Secondary: 46H99: None of the above, but in this section

Banach algebra $T$-Lau product cyclic derivation approximate cyclic amenability


Javanshiri, Hossein. Approximate Cyclic Amenability of $T$-Lau Product of Banach Algebras. Taiwanese J. Math. 20 (2016), no. 5, 1139--1147. doi:10.11650/tjm.20.2016.7154. https://projecteuclid.org/euclid.twjm/1498874521

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  • F. Abtahi and A. Ghafarpanah, A note on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism, Bull. Aust. Math. Soc. 92 (2015), no. 2, 282–289.
  • S. J. Bhatt and P. A. Dabhi, Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism, Bull. Aust. Math. Soc. 87 (2013), no. 2, 195–206.
  • Y. Choi, Triviality of the generalized Lau product associated to a Banach algebra homomorphism, Preprint in Bulletin of the Australian Mathematical Society.
  • H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series 24, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000.
  • G. H. Esslamzadeh and B. Shojaee, Approximate weak amenability of Banach algebras, Bull. Belg. Math. Soc. Simon Stevin. 18 (2011), no. 3, 415–429.
  • N. Grønbæk, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinb. Math. Soc. (2) 35 (1992), no. 2, 315–328.
  • H. Javanshiri and M. Nemati, On a certain product of Banach algebras and some of its properties, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 15 (2014), no. 3, 219–227.
  • A. T. M. Lau, Analysis on a class of Banach algebras with applications to Harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161–175.
  • M. Nemati and H. Javanshiri, Some homological and cohomological notions on $T$-Lau product of Banach algebras, Banach J. Math. Anal. 9 (2015), no. 2, 183–195.
  • ––––, The multiplier algebra and BSE–functions for certain product of Banach algebras, Preprint, see arXiv:1509.00895.
  • B. Shojaee and A. Bodaghi, A generalization of cyclic amenability of Banach algebras, Math. Slovaca 65 (2015), no. 3, 633–644.