TY - JOUR
T1 - Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices
AU - Kim, Jaegil
AU - Lee, Han Ju
PY - 2009/8/15
Y1 - 2009/8/15
N2 - Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb (K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of P (nX), then the set of all strongly norm attaining elements in P (nX) is dense. In particular, the set of all points at which the norm of P (nX) is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to ℓ∞n. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.
AB - Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb (K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of P (nX), then the set of all strongly norm attaining elements in P (nX) is dense. In particular, the set of all points at which the norm of P (nX) is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to ℓ∞n. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.
KW - Peak functions
KW - Peak points
KW - Polynomial numerical index
UR - http://www.scopus.com/inward/record.url?scp=67349193616&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2008.11.024
DO - 10.1016/j.jfa.2008.11.024
M3 - Article
AN - SCOPUS:67349193616
SN - 0022-1236
VL - 257
SP - 931
EP - 947
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 4
ER -