The diagonal functors
In the first part of the paper, new categorial proofs are obtained that generalize the diagonal principles introduced by \textit{J. M. F. Castillo} and \textit{Y. Moreno} [Isr. J. Math., 140, 253-270 (2004; Zbl 1063.46003)] to study the automorphic character of a Banach space \(X\). In the second part, the author introduces the automorphy index \(\mathfrak{a}(X)\) and shows that \(\mathfrak{a}(\ell_{\infty})=\aleph_0\) and \(\mathfrak{a}(C[0,1])\geq\aleph_1\). The definition of \(\mathfrak{a}(X)\) is the following: (Isomorphic) embeddings \(i\) and \(j\) of a Banach space \(Y\) into \(X\) are \textit{equivalent} if there exists an automorphism \(\tau\) of \(X\) such that \(\tau i=j\). Let \(\alpha\) be an infinite cardinal and \(\mathcal{C}^{\alpha}_Y\) the set of all equivalence classes of embeddings \(i:Y\to X\) with \(\mathrm{dens}\,X/iY=\alpha\). Then \(\mathfrak{a}(X):=\sup_{\alpha,Y}\mathrm{card}\,\mathcal{C}^{\alpha}_Y\).
scientific article; zbMATH DE number 5367271
scientific article