Let $B(X)$ denote the space of bounded operators on a Banach space $X$ and $K(X)$ the compact operators. Let $\pi : B(X) \to B(X) / K(X)$ denote the quotient map. Let $u$ denote the right shift operator.
My question is: why does the equality $$ e^{\pi(w')}= \pi (e^{w'})$$
hold? It is on the third last line in the following example:
For any two Banach algebras $ A $ and $ B $, any bounded homomorphism $ \pi: A \to B $ and any element $ a \in A $, we have \begin{align} e^{\pi(a)} & = \sum_{n = 0}^{\infty} \frac{1}{n!} \cdot [\pi(a)]^{n} \\ & = \sum_{n = 0}^{\infty} \frac{1}{n!} \cdot \pi(a^{n}) \\ & = \sum_{n = 0}^{\infty} \pi \! \left( \frac{1}{n!} \cdot a^{n} \right) \\ & = \pi \! \left( \sum_{n = 0}^{\infty} \frac{1}{n!} \cdot a^{n} \right) \\ & = \pi(e^{a}). \end{align}