I have an infinite dimensional unital $\mathrm{C}^*$-algebra $A\subset B(H)$ and a subspace $U\subset H$ such that for all $f\in A$, $f$ restricted to $U$ is scalar (there exists $\lambda\in\mathbb{C}$ such that:):
$$f(u)=\lambda u.$$
Can we conclude that $U$ is (zero or) one dimensional?
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Follow up in the linked question.
No. Take for instance $H = U \oplus V$ where $U$ and $V$ are Hilbert spaces with $U$ more than 1-dimensional and $V$ infinite-dimensional. Take $A$ to be the C*-algebra of operators of the form $\begin{bmatrix} \lambda \cdot \mathrm{id}_U & 0 \\ 0 & T \\ \end{bmatrix}$ where $\lambda \in \mathbb{C}$ and $T \in B(V)$.