Hidden Markov Model initial probability reestimate: Why $\pi^*_i = \gamma_i(1)$ instead of $\pi^*_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$
It is defined to be a probability.  A probability is by definition already normalized.  In particular, we are guaranteed that

$$\sum_{j=1}^N \gamma_j(1) = 1,$$

as there are only $N$ possibilities for the state that you're in at time $1$, and these $N$ cases have no overlap.