Let $\mathcal{B}_1(\mathcal{H})$ be the set of trace class operators in a Hilbert space $\mathcal{H}$ and $\mathcal{H}^{(d)} = \bigoplus_{i=1}^d \mathcal{H}$ with $1 \leq d \leq \infty$.
If $C \in \mathcal{B}_1(\mathcal{H}^{(d)})$ with matrix representation $C=(C_{jk})$, does the series $\sum_kC_{kk}$ converges in the trace norm?