Laustsen, Niels Jakob
(2003)
*On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces.*
Glasgow Mathematical Journal, 45 (1).
pp. 11-19.
ISSN 0017-0895

## Abstract

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 x1, x2 and x3 such that script B sign(x1) is finite, script B sign(x2) is infinite, but not properly infinite, and script B sign(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 η1 and η2 such that script B sign(η1 and script B sign(η2) are infinite without being properly infinite, script B sign(η1) has a continued bisection of the identity, and script B sign(η2) 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.