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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 - Maximal left ideals of the Banach algebra of bounded operators on a Banach space
AU - Dales, H.G.
AU - Kania, Tomasz
AU - Kochanek, Tomasz
AU - Koszmider, Piotr
AU - Laustsen, Niels
PY - 2013
Y1 - 2013
N2 - We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.
AB - We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.
KW - Finitely-generated, maximal left ideal
KW - Banach algebra
KW - bounded operator
KW - inessential operator
KW - Banach space
KW - Argyros-Haydon space
KW - Sinclair-Tullo theorem
U2 - 10.4064/sm218-3-3
DO - 10.4064/sm218-3-3
M3 - Journal article
VL - 218
SP - 245
EP - 286
JO - Studia Mathematica
JF - Studia Mathematica
SN - 0039-3223
IS - 3
ER -