- https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/on-ringtheoretic-infiniteness-of-banach-algebras-of-operators-on-banach-spaces/6DE8AE9FCE77A1FC23757CE949749826
Final published version

Research output: Contribution to journal › Journal article › peer-review

Published

<mark>Journal publication date</mark> | 31/01/2003 |
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<mark>Journal</mark> | Glasgow Mathematical Journal |

Issue number | 1 |

Volume | 45 |

Number of pages | 9 |

Pages (from-to) | 11-19 |

Publication Status | Published |

<mark>Original language</mark> | English |

Let script B sign (x) denote the Banach algebra of all bounded linear operators on a Banach space x. We show that script B sign(x) is finite if and only if no proper, complemented subspace of x is isomorphic to x, and we show that script B sign(x) is properly infinite if and only if x contains a complemented subspace isomorphic to x ⊕ x. We apply these characterizations to find Banach spaces x_{1}, x_{2} and x_{3} such that script B sign(x_{1}) is finite, script B sign(x_{2}) is infinite, but not properly infinite, and script B sign(x_{3}) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces η_{1} and η_{2} such that script B sign(η_{1} and script B sign(η_{2}) are infinite without being properly infinite, script B sign(η_{1}) has a continued bisection of the identity, and script B sign(η_{2}) has no continued bisection of the identity. Finally, we exhibit a unital C*-algebra which is finite and has a continued bisection of the identity.