In this paper, we introduce $p$-amenability, bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals for a Banach algebra $\mathfrak{ A}$ where $s$ is a non-zero element of algebraic center of $\mathfrak{ A}$ that is denoted by $Z(\mathfrak{ A})$. We show that if a Banach algebra $\mathfrak{ A}$ is $p$-amenable then it has bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals and by this fact we prove that if the Banach algebra $\mathfrak{ A}$ is unital then $p$-amenability and symmetric amenability are equivalent.
Categories: math.FA