Continuous functional calculus, C*-subalgebra

Question

Let $A$ be a unital C*-Algebra and $U\subseteq A$ a C*-subalgebra (not necessarily unital). If I take $T \in U$, is there any criterion as to when $f(T) \in U$? I mean, if $f=1$, and U is not unital, then it's $f(T)\not\in U$, isn't it? Thank you.

Answer 1

If $U$ is not unital, then the criterion is that $f(0)=0$ .

In more detail: we assume that $T$ is normal; then (a) the spectrum of $T$ in $A$ and in $U$ is the same (this follows from Pedersen "$C^*$-algebras and their isomorphism groups", Sec. 1.1.8) (b) for each continuous function $f$ on the spectrum of $T$ in $U$ vanishing at zero, $f(T)\in U$ (Pedersen "$C^*$-algebras and their isomorphism groups", Sec. 1.1.9). Now if you take $f$ continuous on the spectrum of $T$ with $f(0)\ne0$, then $f(T)=f(0)+g(T)$, where $g(x)=f(x)-f(0)$ vanishes at $0$ and hence $g(T)\in U$; since $f(0)\notin U$, we see that $f(T)\notin U$ either.