TY - JOUR
ID - 15512
T1 - Abstract structure of partial function $*$-algebras over semi-direct product of locally compact groups
JO - Sahand Communications in Mathematical Analysis
JA - SCMA
LA - en
SN - 2322-5807
A1 - Ghaani Farashahi, Arash
A1 - Kamyabi-Gol, Rajab Ali
Y1 - 2015
PY - 2015/12/01
VL - 02
IS - 2
SP - 23
EP - 44
KW - Semi-direct products of groups
KW - Left $tau$-convolution ($tau_l$-convolution)
KW - Right $tau$-convolution
($tau_r$-convolution)
KW - $tau$-convolution
KW - $tau$-involution
KW - $tau$-approximate identity
DO -
N2 - This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism. Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-convolution on $L^1(G_tau)$ and we show that, with respect to each of them, the function space $L^1(G_tau)$ is a Banach algebra. We define $tau$-convolution as a linear combination of the left and right $tau$-convolution and we show that the $tau$-convolution is commutative if and only if $K$ is abelian. We prove that there is a $tau$-involution on $L^1(G_tau)$ such that with respect to the $tau$-involution and $tau$-convolution, $L^1(G_tau)$ is a non-associative Banach $*$-algebra. It is also shown that when $K$ is abelian, the $tau$-involution and $tau$-convolution make $L^1(G_tau)$ into a Jordan Banach $*$-algebra. Finally, we also present the generalized notation of $tau$-convolution for other $L^p$-spaces with $p>1$.
UR - http://scma.maragheh.ac.ir/article_15512.html
L1 - http://scma.maragheh.ac.ir/pdf_15512_5770f8eeb189b81deec09dc87fdd4b39.html
ER -