More properties of the classes of hereditarily \(l^p\) Banach sequence spaces (Q2901509)

In [Pac. J. Math. 122, 287--297 (1986; Zbl 0609.46012)], \textit{P. Azimi} and \textit{J. Hagler} constructed, for every sequence \(\alpha\) of real numbers fulfilling certain conditions, a Banach space \(X_{\alpha,1}\) which is hereditarily \(\ell^1\) but fails the Schur property. Later \textit{P. Azimi} generalised this definition to hereditarily \(\ell^p\) spaces \(X_{\alpha,p}\) for \(1\leq p<\infty\) [Bull. Iran. Math. Soc. 28, No. 2, 57--68, 84 (2002; Zbl 1035.46006)]. All the spaces \(X_{\alpha,p}\) are dual spaces.NEWLINENEWLINE In [\textit{V. ,M. Kadets, R. V. Shvydkoy, G. G. Sirotkin} and \textit{D. Werner}, Trans. Am. Math. Soc. 352, No. 2, 855--873 (2000; Zbl 0938.46016)] the notion of (acs) spaces (short for alternatively convex or smooth) was introduced: a Banach space \(X\) is said to be (acs) if for all \(x,y\in X\) with \(\|{x}\|=\|{y}\|=1=\|\left(x+y\right)/2\|\) and every norm-one functional \(x^*\in X^*\) with \(x^*(x)=1\) one also has \(x^*(y)=1\). This obviously includes all strictly convex and all smooth Banach spaces.NEWLINENEWLINE Uniform (uacs) and locally uniform versions (luacs) of (acs) spaces were also introduced in the same paper and related to the anti-Daugavet property: if \(\mathcal{M}\) is a subset of the space of all bounded linear operators on \(X\), then \(X\) is said to have the anti-Daugavet property with respect to \(\mathcal{M}\) provided that for every \(T\in \mathcal{M}\) which satisfies the so called Daugavet equation \(\|{\text{id} +T}\|=1+\|{T}\|\), one has that \(\|{T}\|\) belongs to the spectrum of \(T\) (the converse is always true, as is easily seen).NEWLINENEWLINE Then \(X\) has the anti-Daugavet property for rank-1-operators if and only if \(X\) has the anti-Daugavet property for compact operators if and only if \(X\) is (luacs).NEWLINENEWLINE In the paper under review it is proved that the spaces \(X_{\alpha,1}\) are not (acs) [Theorem 2.2 (a)] and thus they do not have the anti-Daugavet property for rank-1-operators. It should be mentioned that the separate proofs of the parts (b) and (c) of said theorem (stating that \(X_{\alpha,1}\) is not (uacs) respectively (luacs)) are redundant since (b) and (c) follow directly from part (a) that was proved first.NEWLINENEWLINE Some other observations on the spaces \(X_{\alpha,p}\) are also made, for example that the preduals of the spaces \(X_{\alpha,1}\) fail the weak Phillips property and that none of the spaces \(X_{\alpha,p}\) (\(1\leq p<\infty\)) has the Dunford-Pettis* property (the latter fact is simply deduced from the proof of the stronger statement that these spaces do not even have the Dunford-Pettis property, cf. Lemma 3.15 in [\textit{P. Azimi}, Taiwanese J. Math. 10, No. 3, 713--722 (2006; Zbl 1108.46009)]).