Nilpotent Ideals in Generalized Amenable Banach Algebras
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Nilpotent Ideals in Generalized Amenable Banach Algebras Mohammad Hossein Sattari1 · Esmaeil Alizadeh2 Received: 31 July 2019 / Accepted: 14 January 2020 © Iranian Mathematical Society 2020
Abstract In this article, some properties such as the approximation property of non-zero nilpotent ideals in generalized amenable Banach algebras are investigated. Also, we show that in pseudo-contractible Banach algebras, non-zero nilpotent ideals cannot have the approximation property. Keywords Amenable Banach algebras · Pseudo-contractible · Approximation property Mathematics Subject Classification 46H201 · 46B28
1 Introduction It is well known that in amenable Banach algebras any finite-dimensional nilpotent ideal must be zero, see proposition VII.2.31 in [4]. In [6], the existence of the approximation property of nilpotent ideals in amenable and biprojective Banach algebras is investigated by Loy and Willis, where the finite dimensionality condition is relaxed to the approximation property of Grothendieck. The approximation property of Grothendieck has some connections with Banach algebra cohomology. For the algebra A(E) of approximable operators on a Banach space E to be amenable, we have to require that E ∗ have the bounded approximation property [7]. In [6], it is shown that under certain conditions in amenable Banach algebras, nonzero nilpotent ideals cannot have the approximation property. The authors of [6] have left open the question as to whether it is true in general that in amenable
Communicated by Massoud Amini.
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Mohammad Hossein Sattari [email protected] Esmaeil Alizadeh [email protected]
1
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751 71379, Iran
2
Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran
123
Bulletin of the Iranian Mathematical Society
Banach algebras, nonzero nilpotent ideals cannot have the approximation property. In Sect. 2, we show that in biflat Banach algebras, essential closed ideals are not L 1 -predual, and so in amenable Banach algebras, nilpotent ideals are not L 1 -predual. Also, another property of such ideals is studied in that section. We conclude the paper in Sect. 3 by obtaining some results about nilpotent ideals in generalized amenable Banach algebras. Indeed, it is proved that in pseudo-contractible Banach algebras, non-zero nilpotent ideals cannot have the approximation property.
2 Nilpotent Ideals in Amenable and Biflat Banach Algebras A Banach space N is said to have the bounded approximation property if there is a constant M such that for each compact subset K of N , and each ε > 0, there is a finite rank operator F on N such that F ≤ M and F(x) − x < ε for any x ∈ K . If this holds with M = 1, then N is said to have the metric approximation. As [6], in this case, for any Banach spaces Y and A such that A contains N as subspace, the ˆ → A⊗Y ˆ is injective, where i is the inclusion map. Furthermore, map i ⊗ IY : N ⊗Y this map has closed range when N is L 1 -predual i.e. N ∗ = L 1 (μ),
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