  In this paper, we extend the classical Bohr's inequality to the setting of
the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann
algebra. As a consequence, we obtain Bohr's inequality for operators in the von
Neumann-Schatten class $\cl C_1$ and square matrices of any finite order.
Interestingly, we establish that the optimal bound for $r$ in the above
mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3
whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to
$\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a
generalization of our above-mentioned Bohr's inequality for finite matrices
where we show that the optimal bound for $r$, unlike above, remains 1/3 for
every fixed order $n\times n,\ n\ge 2$.
