# Hochschild coboundary on the space of alternative forms

Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is an element $\phi \in C^{k}(A)$ with $\phi(a_{\sigma(1))},a_{\sigma(2)},\ldots,a_{\sigma(n)})=sign(\sigma)\phi(a_1,a_2,\ldots,a_k)$ for every permutation $\sigma$. The standard Hochschild coboundary $b:C^n(A) \to C^{n+1}(A)$ is defined by $$b\phi(a_0,a_1,\ldots,a_n)=\sum_{j=0}^{n-1}{(-1)}^{j}\phi(a_0,a_1,\ldots a_ja_{j+1},\ldots a_n)+{(-1)}^n\phi(a_na_0,a_1,\ldots,a_{n-1})$$

Question: Is there a unital complex algebra $A$ with the following property?:

For every alternative $k-$form $\phi: A^k\to \mathbb{C}$, $b\phi:A^{k+1}\to \mathbb{C}$ is an alternative $(k+1)-$ form