  The main result of this paper is to calculate the Batalin-Vilkovisky
structure of $HH^*(C^*(\mathbf{K}P^n;R);C^*(\mathbf{K}P^n;R))$ for $
\mathbf{K}=\mathbb{C}$ and $\mathbb{H}$, and $R=\mathbb{Z}$ and any field; and
shows that in the special case when $M=\mathbb{C}P^1=S^2$, and $R=\mathbb{Z}$,
this structure can not be identified with the BV-structure of
$\mathbb{H}_*(LS^2;\mathbb{Z})$ computed by Luc Memichi in \cite{menichi2}.
However, the induced Gerstenhaber structures are still identified in this case.
Moreover, according to a recent work of Y.Felix and J.Thomas
\cite{felix--thomas}, the main result of the present paper eventually
calculates the BV-structure of the rational loop homology,
$\mathbb{H}_*(L\mathbb{C}P^n;\mathbb{Q})$ and
$\mathbb{H}_*(L\mathbb{H}P^n;\mathbb{Q})$, of projective spaces.
