Geodesic
ON RING-THEORETIC (IN)FINITENESS OF BANACH ALGEBRAS OF OPERATORS ON BANACH SPACES
Glasgow mathematical journal, Tome 45 (2003) no. 1, pp. 11-19

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DOI

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.
DOI : 10.1017/S0017089502008947
Mots-clés : 47L10, 16B99, 46B15, 46L05 
LAUSTSEN, NIELS JAKOB. ON RING-THEORETIC (IN)FINITENESS OF BANACH ALGEBRAS OF OPERATORS ON BANACH SPACES. Glasgow mathematical journal, Tome 45 (2003) no. 1, pp. 11-19. doi: 10.1017/S0017089502008947
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     title = {ON {RING-THEORETIC} {(IN)FINITENESS} {OF} {BANACH} {ALGEBRAS} {OF} {OPERATORS} {ON} {BANACH} {SPACES}},
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