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Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$. Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$ Let $\phi: S \rightarrow B_2.$ Given that the ...