Approximate Identities in Banach Algebras of Compact Operators
Canadian mathematical bulletin, Tome 36 (1993) no. 1, pp. 45-53
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Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.
Grønbæk, Niels; Willis, George A. Approximate Identities in Banach Algebras of Compact Operators. Canadian mathematical bulletin, Tome 36 (1993) no. 1, pp. 45-53. doi: 10.4153/CMB-1993-008-8
@article{10_4153_CMB_1993_008_8,
author = {Gr{\o}nb{\ae}k, Niels and Willis, George A.},
title = {Approximate {Identities} in {Banach} {Algebras} of {Compact} {Operators}},
journal = {Canadian mathematical bulletin},
pages = {45--53},
year = {1993},
volume = {36},
number = {1},
doi = {10.4153/CMB-1993-008-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1993-008-8/}
}
TY - JOUR AU - Grønbæk, Niels AU - Willis, George A. TI - Approximate Identities in Banach Algebras of Compact Operators JO - Canadian mathematical bulletin PY - 1993 SP - 45 EP - 53 VL - 36 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1993-008-8/ DO - 10.4153/CMB-1993-008-8 ID - 10_4153_CMB_1993_008_8 ER -
%0 Journal Article %A Grønbæk, Niels %A Willis, George A. %T Approximate Identities in Banach Algebras of Compact Operators %J Canadian mathematical bulletin %D 1993 %P 45-53 %V 36 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1993-008-8/ %R 10.4153/CMB-1993-008-8 %F 10_4153_CMB_1993_008_8
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