TY - JOUR
T1 - The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B
AU - Aron, Richard
AU - Choi, Yun Sung
AU - Kim, Sun Kwang
AU - Lee, Han Ju
AU - Martín, Miguel
N1 - Publisher Copyright:
© 2015 American Mathematical Society.
PY - 2015
Y1 - 2015
N2 - We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(ε) such that for every Y , the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c0-, ℓ1- and ℓ∞-sums of Banach spaces.
AB - We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(ε) such that for every Y , the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c0-, ℓ1- and ℓ∞-sums of Banach spaces.
KW - Approximation
KW - Banach space
KW - Bishop-Phelps-Bollobás theorem
KW - Norm-attaining operators
UR - http://www.scopus.com/inward/record.url?scp=84928090410&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2015-06551-9
DO - 10.1090/S0002-9947-2015-06551-9
M3 - Article
AN - SCOPUS:84928090410
SN - 0002-9947
VL - 367
SP - 6085
EP - 6101
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 9
ER -