### Symmetric operator amenability of operator algebras

#### Abstract

Using the notion of a symmetric virtual diagonal for a Banach algebra, we prove that a Banach algebra is symmetrically amenable if its second dual is symmetrically amenable. We introduce symmetric operator amenability in the category of completely contractive Banach algebras as an operator algebra analogue of symmetric amenability of Banach algebras. We give some equivalent formulations of symmetric operator amenability of completely contractive Banach algebras and investigate some hereditary properties of symmetric operator amenable algebras. We show that amenability of locally compact groups is equivalent to symmetric operator amenability of its Fourier algebra. Finally, we discuss about Jordan derivation on symmetrically operator amenable algebras.

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