The zero-undetected-error capacity of discrete memoryless channels with feedback

We show that the zero-undetected-error capacity (a.k.a. erasures-only capacity and zero-error erasure capacity) of a discrete memoryless channel with feedback is equal to its ordinary capacity whenever its zero-undetected-error capacity without feedback is positive, i.e., whenever it has an output that is reachable from some but not all inputs, and that otherwise its zero-undetected-error capacity is zero. We then show that feedback can increase the zero-undetected-error capacity. Finally, the result is extended to multiple-access and broadcast channels.