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

摘要

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?X. We apply these characterizations to find Banach spaces X_1, X_2, and X_3 such that B(X_1) is finite, 98(3:2) is infinite, but not properly infinite, and B(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 D_1 and D_2 sucn that B(D_1) and B(D_2) are infinite without being properly infinite, B(D_2) has a continued bisection of the identity, and B(D_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.