Let $A$ be a $C^\*$-algebra acting on a Hilbert space that admits a cyclic unit vector $\xi \in H$. Pose $S\_\xi = \{\eta \in A \xi: \| \eta \| = 1\}$, and for each $\eta \in S\_\xi$, pose $A\_\eta = \{a \in A : a \xi = \eta\}$, so $A\_\eta \ne \emptyset$ by definition; and let $c\_\eta = \inf\{\| a \| : a \in A\_\eta\}$. Clearly, $c\_\eta \ge 1$ for all $\eta \in S\_\xi$. I am interested in the following questions.


**Question 1**. Is it true that $c\_\eta = 1$ for all $\eta \in S\_\xi$ ?


**Question 1 (weak form)**. Is it true that $S(A)\xi$ is dense in $S(H)$, where $S(X)$ denotes the closed unit ball of a normed space $X$?


**Question 2**. If the answer of Question 1 turns out to be negative, can there be any examples for which $\sup\{c\_\eta : \eta \in S\_\xi\} = +\infty$ ?


**Question 3**. What if we drop the assumption of $\xi$ being cyclic but merely asks $A$ acts non-degenerately on $H$ ?



Controlling norm of operators sending a fixed vector to another
Set $A = C([0,1])$ and $H = L^2([0,1])$ (with respect to Lebesgue measure) with $\xi=1$ the constant function. Then $A\xi$ is the image of $C([0,1])$ in $L^2([0,1])$, which is a norm-decreasing injective but not bounded below inclusion. Then $S\_\xi$ is the intersection of the unit ball of $L^2$ with the continuous functions, and for each $\eta$ we see that $A\_\eta$ is the singleton $\{\eta\}$.


Thus, Q1 (both versions) have a negative answer, this example gives a positive answer to Q2, and Q3 can be reduced to the main question by restricting the action of $A$ to the cyclic subspace generated by $\xi$.


However, if $A$ were the compact operators on $H$ then we'd have a positive answer to Q1.


