On a specific C*-algebra

Question

How to describe the C*-algebra $C^*(\eta, \eta^*)/<\eta^*\eta=\mu \eta \eta^*>$ where $\mu$ is a positive real number less than $1$? What happens if $\mu$ is replaced by a complex number? Any hint/suggestion/reference will be highly appreciated.

Answer 1

If a C*-algebra $A$ is generated by an element $\eta $ satisfying $\eta^*\eta=\mu \eta \eta^*$, where $\mu$ is a positive real number less than $1$, then $\eta =0$, and hence $A=\{0\}$. The reason is that $$ \|\eta \|^2 = \|\eta ^*\eta \| = \mu \|\eta \eta^*\| = \mu \|\eta\|^2, $$ which imply that $\|\eta \|=0$.

If $\mu $ is any complex number not equal to $1$, one also deduces that $\eta =0$, because of comparison of spectra: $$ \sigma (\eta \eta ^*) \cup \{0\} = \sigma (\eta ^*\eta ) \cup \{0\} = \sigma (\mu \eta \eta ^*) \cup \{0\} = \mu \sigma (\eta \eta ^*) \cup \{0\}, $$ while the only nonempty subset $S\subseteq {\mathbb R}_+$ such that $\mu S=S$ is the set $S=\{0\}$.