TY - JOUR
T1 - On Banach spaces whose group of isometries acts micro-transitively on the unit sphere
AU - Cabello Sánchez, Félix
AU - Dantas, Sheldon
AU - Kadets, Vladimir
AU - Kim, Sun Kwang
AU - Lee, Han Ju
AU - Martín, Miguel
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, inherited by one-complemented subspaces, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces. In particular, they are uniformly convex and uniformly smooth, and form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an Lp(μ) space is micro-transitive or uniformly micro-semitransitive only when p=2.
AB - We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, inherited by one-complemented subspaces, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces. In particular, they are uniformly convex and uniformly smooth, and form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an Lp(μ) space is micro-transitive or uniformly micro-semitransitive only when p=2.
KW - Banach space
KW - Bishop-Phelps-Bollobás property
KW - Mazur rotation problem
KW - Micro-transitivity
KW - Norm attaining operators
UR - http://www.scopus.com/inward/record.url?scp=85081966405&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2020.124046
DO - 10.1016/j.jmaa.2020.124046
M3 - Article
AN - SCOPUS:85081966405
SN - 0022-247X
VL - 488
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
M1 - 124046
ER -