Rights statement: The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 428 (1), 2015, © ELSEVIER.
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TY - JOUR
T1 - Banach spaces whose algebra of bounded operators has the integers as their K0-group
AU - Kania, Tomasz
AU - Koszmider, Piotr
AU - Laustsen, Niels
N1 - The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 428 (1), 2015, © ELSEVIER.
PY - 2015/8/1
Y1 - 2015/8/1
N2 - Let X and Y be Banach spaces such that the ideal of operators which factor through Y has codimension one in the Banach algebra B(X) of all bounded operators on X, and suppose that Y contains a complemented subspace which is isomorphic to Y⊕Y and that X is isomorphic to X⊕Z for every complemented subspace Z of Y. Then the K0-group of B(X) is isomorphic to the additive group Z of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that K0(B(C([0,ω1])))≅Z, where C([0,ω1]) denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal ω1, endowed with the order topology.
AB - Let X and Y be Banach spaces such that the ideal of operators which factor through Y has codimension one in the Banach algebra B(X) of all bounded operators on X, and suppose that Y contains a complemented subspace which is isomorphic to Y⊕Y and that X is isomorphic to X⊕Z for every complemented subspace Z of Y. Then the K0-group of B(X) is isomorphic to the additive group Z of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that K0(B(C([0,ω1])))≅Z, where C([0,ω1]) denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal ω1, endowed with the order topology.
KW - K0-group
KW - Banach algebra
KW - Bounded
KW - linear operator
KW - Banach space
KW - Continuous functions on the first uncountable ordinal interval
U2 - 10.1016/j.jmaa.2015.03.021
DO - 10.1016/j.jmaa.2015.03.021
M3 - Journal article
VL - 428
SP - 282
EP - 294
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -