Solution
= Solution
With rows indexed by outputs $A,B,\star$ and columns by inputs $A,B$, the channel matrix is
$$
\boxed{
\begin{pmatrix}
1/2&0\\
0&1/2\\
1/2&1/2
\end{pmatrix}}.
$$
This is a <binary erasure channel> with erasure probability $1/2$. Conditional on no erasure, the input is recovered exactly, whereas an erasure supplies no information. Thus
$$
I(X;Y)=\frac12H(X),
$$
which is maximized by the uniform input distribution. Therefore
$$
\boxed{C=\frac12\text{ bit}}.
$$