trace on direct sum

Question

Let $\mathcal H_n$ is separable Hilbert space for every $n\in\mathbb N$. By $\mathcal{B(H_n)}$ denote the algebra of all linear bounded operators acting on $\mathcal H_n$. As we know there is canonical trace $\tau_n$ on each $\mathcal{B(H_n)}$. Further we know that every $\mathcal{B(H_n)}$ can be considered as subspace of the direct sum $\bigoplus\limits_{n=1}^{\infty} \mathcal{B(H_n)}$. Is there trace $\tau$ on the direct sum $\bigoplus\limits_{n=1}^{\infty} \mathcal{B(H_n)}$ such that $\tau(A)=\tau_n(A)$ when $A\in \mathcal{B(H_n)}$? In other words can one from traces $\tau_n$ on $\mathcal{B(H_n)}$ construct a trace $\tau$ on the direct sum $\bigoplus\limits_{n=1}^{\infty} \mathcal{B(H_n)}$ such that $\tau |_{\mathcal{B(H_n)}} = \tau_n$?