elements in $C^*$ algebra

Question

If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$?

My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?

Answer 1

Let's first assume that $A$ is unital. Then let's define, for $x \in A$ \begin{equation} \exp(x) = \lim_{N \rightarrow \infty} \sum_{n=0}^{N} x^{n}/n! \end{equation} The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence \begin{equation} a_{N} = \sum_{n=0}^{N} x^{n}/n! \end{equation} is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)