Let
S
S
be an inverse semigroup with the set of idempotents
E
E
. In this article, we find necessary and sufficient conditions for the weighted semigroup algebra
l
1
(
S
,
ω
)
{l}^{1}\left(S,\omega )
to be module approximately amenable (contractible) and module character amenable (as
l
1
(
E
)
{l}^{1}\left(E)
-module).
In this paper, for a complex Hilbert space ℋ with dim ℋ ≥ 2, we study the linear maps on ℬ(ℋ), the bounded linear operators on ℋ, that preserves projections and idempotents. As a result, we characterize the linear maps on ℬ(ℋ) that preserves involutions in both directions.
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