TY - JOUR
ID - 4052
TI - Completely Continuous Banach Algebras
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN - 2008-6822
AU - Hayati, Bahman
AD - Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran
Y1 - 2019
PY - 2019
VL - 10
IS - 1
SP - 55
EP - 62
KW - Amenability
KW - Completely continuous
KW - Banach algebra
DO - 10.22075/ijnaa.2019.1184.1268
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_4052.html
L1 - https://ijnaa.semnan.ac.ir/article_4052_98d6bd83d5d5695668206e7b12bb7f92.pdf
ER -