Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces
AU - Laustsen, Niels Jakob
PY - 2003/1/31
Y1 - 2003/1/31
N2 - 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.
AB - 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.
U2 - 10.1017/S0017089502008947
DO - 10.1017/S0017089502008947
M3 - Journal article
AN - SCOPUS:0037263066
VL - 45
SP - 11
EP - 19
JO - Glasgow Mathematical Journal
JF - Glasgow Mathematical Journal
SN - 0017-0895
IS - 1
ER -