Characterizations of $${*}$$ and $${*}$$ -left derivable mappings on some algebras
Tóm tắt
A linear mapping
$$\delta $$
from a
$${*}$$
-algebra
$$\mathcal {A}$$
into a
$${*}$$
-
$$\mathcal {A}$$
-bimodule
$$\mathcal {M}$$
is a
$${*}$$
-derivable mapping at
$$G\in \mathcal {A}$$
if
$$A\delta (B)^{*}+\delta (A)B=\delta (G)$$
for each A, B in
$$\mathcal {A}$$
with
$$AB^{*}=G$$
. We prove that every (continuous)
$${*}$$
-derivable mapping at G from a (unital
$$C^{*}$$
-algebra) factor von Neumann algebra into its Banach
$${*}$$
-bimodule is a
$${*}$$
-derivation if and only if G is a left separating point. A linear mapping
$$\delta $$
from a
$${*}$$
-algebra
$$\mathcal {A}$$
into a
$${*}$$
-left
$$\mathcal {A}$$
-module
$$\mathcal {M}$$
is a
$${*}$$
-left derivable mapping at
$$G\in \mathcal {A}$$
if
$$A\delta (B)^{*}+B\delta (A)=\delta (G)$$
for each A, B in
$$\mathcal {A}$$
with
$$AB^{*}=G$$
. We prove that every continuous
$${*}$$
-left derivable mapping at a left separating point from a unital
$$C^{*}$$
-algebra or von Neumann algebra into its Banach
$${*}$$
-left
$$\mathcal {A}$$
-module is identical with zero under certain conditions.
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