2015/07/05 by Tomasz Kania, Kania, Tomasz, Niels Jakob Laustsen +1 · 2 citations
Mathematics · Computer Science · #Advanced Banach Space Theory #Advanced Topics in Algebra #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.1507.01213
We construct a Banach space Z such that the lattice of closed two-sided\nideals of the Banach algebra mathscrB(Z) of bounded operators on Z is as\nfollows:
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mathscrB(Z)\n\n We then determine which kinds of approximate identities (bounded/left/right),\nif any, each of the four non-trivial closed ideals of mathscrB(Z) contain,\nand we show that the maximal ideal mathscrM1 is generated as a left ideal\nby two operators, but not by a single operator, thus answering a question left\nopen in our collaboration with Dales, Kochanek and Koszmider (\Studia\nMath. 2013). In contrast, the other maximal ideal mathscrM2 is not\nfinitely generated as a left ideal.\n The Banach space Z is the direct sum of Argyros and Haydon's Banach space\nX\AH which has very few operators and a certain subspace Y of\nX\AH. The key property of~Y is that every bounded operator from\nY into X\AH is the sum of a scalar multiple of the inclusion\nmapping and a compact operator.\n