TY - JOUR
ID - 383
TI - Completely continuous Banach algebras
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN -
AU - Hayati, Bahman
AD - Department of Mathematics, Malayer University, P.O. Box 16846-13114, Malayer, Iran
Y1 - 2016
PY - 2016
VL - 7
IS - 1
SP - 301
EP - 308
KW - Amenability
KW - Completely continuous
KW - Banach algebra
DO - 10.22075/ijnaa.2016.383
N2 - For a Banach algebra $\mathfrak{A}$, we introduce ~$c.c(\mathfrak{A})$, the set of all $\phi\in \mathfrak{A}^*$ such that $\theta_\phi:\mathfrak{A}\to \mathfrak{A}^*$ is a completely continuous operator, where $\theta_\phi$ is defined by $\theta_\phi(a)=a\cdot\phi$~~ for all $a\in \mathfrak{A}$. We call $\mathfrak{A}$, a completely continuous Banach algebra if $c.c(\mathfrak{A})=\mathfrak{A}^*$. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gale, T.J. Ransford and M. C. White: Is there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.
UR - https://ijnaa.semnan.ac.ir/article_383.html
L1 - https://ijnaa.semnan.ac.ir/article_383_3bb2ce040cb0b5b1b2133ec62d0d7465.pdf
ER -