Derivations, multipliers and topological centers of certain Banach algebras related to locally compact groups
Loading...
Date
2017
Authors
Malekzadeh Varnosfaderani, Davood
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We introduce certain Banach algebras related to locally compact groups and study
their properties. Speci cally, we prove that L1(G) is an ideal of L1
0(G) if and only
if G is compact. We also demonstrate that the left topological centers of L1
0(G) and
(M(G)
0) are L1(G) and M(G) respectively. Next, we turn our attention to various
derivation and left multiplier problems. Speci cally, we show that for every weak-star
continuous derivation D : L1(G) ! L1(G) there is 2 M(G) such that D = ad .
We also prove that every derivation from L10
(G) into L1(G) is inner. Next, we focus
on weakly compact derivations and left multipliers and show that for every weakly
compact derivation D on M(G) there is f 2 L1(G) such that D = adf . We also prove
that there exists a non-zero weakly compact derivation on L1(G) ( or L10
(G) for the
special case where there is a unique right invariant mean on L1(G) ) if and only if G
is a non-abelian compact group. We present necessary and su cient conditions for
the existence of non-zero weakly compact left multiplier on L10
(G) . We also show
that for the special case where there is a unique right invariant mean on L1(G), every
weakly compact derivation D on L1(G) is of the form adh where h is in L1(G). We
introduce the concepts of quasi-Arens regularity, quasi topological center and quasiweakly
almost periodic functionals and show that 2 QWAP(A) if and only if ad
is weakly compact. Finally, for a particular G, we construct a continuous non-weakly
compact derivation D : L1(G) ! L1(G) such that D(L1(G)) WAP(G).
ii
Description
Keywords
Banach algebras, Derivations, Left multipliers, Locally compact groups, Arens products, Topological centers