Let S be a discrete semigroup, m (S) the space of all bounded real functions on S with the usualsupremum norm. Let Acm (S) be a uniformly closed left invariant subalgebra of m (S) with 1 c A. We say that A is extremely left amenable if there isamultiplicative left invariant meanon A. Let P = {h ?A: h =|g-1,g | forsome g ?A, s ?S}. It isshown that . A is extremely left amenable if and only ...

In the present paper, we study surfaces invariant under the 1-parameter subgroup in Sol space $rm Sol_3$. Also, we characterize the surfaces in $rm Sol_3$ whose coordinate functions of an immersion of the surface are eigenfunctions of the Laplacian $Delta$ of the surface.

using a generalized spherical mean operator, we obtain a generalization of titchmarsh&apos;s theorem for the dunkl transform for functions satisfying the (&apos;; p)-dunkl lipschitz condition in the space lp(rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.