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We investigate the sum capacity of Block Diagonalization precoding Multiple Input Multiple Output Broadcast Channels (BD MIMO BC) with imperfect Channel State Information (CSI) at the base station. Since it is difficult to obtain the exact expression, a lower and an upper bounds of the sum capacity under Gaussian channel estimation errors are drived instead. Analyses show that the gap between two bounds is considerably tight at all Signal to Noise Ratio (SNR) region. From the lower bound of the sum capacity, we can see that the multiplexing gain tends to be zero at high SNR region, which indicates that the BD MIMO BC system with channel estimation errors is interference-limited at high SNR.