  We introduce and study a new notion of amenability called symmetric
pseudo-amenability. We obtain some properties of symmetrically pseudo-amenable
Banach algebras and with examples, we compare this type of amenability with
some other types of amenability. We also provide some special classes of
symmetrically pseudo-amenable Banach algebras. Finally, Jordan and Lie
derivations from a class of Banach algebras into appropriate Banach bimodules
are investigated using the notion of symmetric pseudo-amenability.
