Abstract of a paper by Spiros A. Argyros, Irene Deliyanni, and Andreas G. Tolias
This is an announcement for the paper "Hereditarily indecomposable Banach algebras of diagonal operators" by Spiros A. Argyros, Irene Deliyanni, and Andreas G. Tolias. Abstract: We provide a characterization of the Banach spaces $X$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to $\mathcal{L}_{diag}({\mathfrak X}_D)$ with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every $T\in \mathcal{L}_{diag}({\mathfrak X}_D)$ is of the form $T=\lambda I+K$, where $K$ is a compact operator. Archive classification: math.FA Mathematics Subject Classification: 46B28, 47L10, 46B20, 46B03. Remarks: 35 pages, submitted for publication to Israel J. Math The source file(s), HI_DIAG.tex.bak: 124849 bytes, is(are) stored in gzipped form as 0902.1646.gz with size 33kb. The corresponding postcript file has gzipped size 188kb. Submitted from: sargyros@math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.1646 or http://arXiv.org/abs/0902.1646 or by email in unzipped form by transmitting an empty message with subject line uget 0902.1646 or in gzipped form by using subject line get 0902.1646 to: math@arXiv.org.
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