I'm going through the topic $C^*$ algebra and facing few questions. It would be great if you people could help me to clear the doubts.
Q3. Does there exists some $X$ belonging to a $C^*$ algebra such that resolvent set is countable.
Ans. If suppose I consider the $n \times n$ matrix $M_n(C)$ as $C^*$ algebra then as the spectrum set is the set of Eigenvalues for any element in $M_n(C)$ , by definition resolvent set = C( complex number) - n ..so uncountable for all $x$ belongs to $M_n(C)$.
The question is, am I sounding correct ? If not please suggest a suitable example.
The spectrum of any element of a Banach algebra is compact, so the resolvent set is always nonempty and open therefore uncountable.