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ON RING-THEORETIC (IN)FINITENESS OF BANACH ALGEBRAS OF OPERATORS ON BANACH SPACES

Published online by Cambridge University Press:  01 May 2003

NIELS JAKOB LAUSTSEN
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5 DK-2100 Copenhagen Ø, Denmark e-mail: [email protected]
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Abstract

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Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and we show that B(X) is properly infinite if and only if X contains a complemented subspace isomorphic to X[oplus ]X. We apply these characterizations to find Banach spaces X1, X2, and X3 such that B(X1) is finite, B(X2) is infinite, but not properly infinite, and B(X3) 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 D1 and D2 such that B(D1) and B(D2) are infinite without being properly infinite, B(D1) has a continued bisection of the identity, and B(D2) has no continued bisection of the identity. Finally, we exhibit a unital $C^\ast$-algebra which is finite and has a continued bisection of the identity.

Type
Research Article
Copyright
2003 Glasgow Mathematical Journal Trust