If $\sigma (T)'\subseteq \{ 0\} \ \ \forall T $ then are $T$s compact?

Question

In the book: "spectral theory of operators on Hilbert spaces, Carlos. S., (2012). page(46)." it has written:

Is the converse of this theorem true? Namely if for every operator $T\in A(\text{a C}^*-\text{subalgebra of}\ B(H))$ we have $\sigma(T)'\subseteq \{0\} $, can we imply $A\subseteq K(H)$?

Answer 1

No. For instance, if $A$ consists just of scalar multiples of the identity, then the spectrum of every element of $A$ is a single point but $A\not\subseteq K(H)$.