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Let $A$ be a unital C*-algebra and let $U(A)$ be its unitary group. Let $U_2(A)$ be the unitary group of the C*-algebra $M_2(A)$ of 2-by-2 matrices over $A$.

Let $u \in U(A)$. Suppose there is a (norm-continuous) path from $\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix} \right)$ to $\left( \begin{smallmatrix} 1 & 0 \\ 0 & u \end{smallmatrix} \right)$ in $U_2(A)$. Does it necessarily follow that there is a norm continuous path from $1$ to $u$ in $U(A)$?

In the case where $A$ is commutative, one can use the determinant to give a positive answer, but I'm not sure what to do in general, or even whether this is true...

Mike F
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In general, this is false. Since you could always replace $A$ by $M_{2}(A)$, this would imply that a unitary $u\in U(A)$ is homotopic to $1_A$ if and only if $\operatorname{diag}(u,1_{k-1})\in U(M_k(A))$ is homotopic to $1_{M_k(A)}$ for any $k>0$. In other words, the canonical map $$U(A)/U_0(A)\quad\longrightarrow\quad K_1(A).$$ would always be injective, where $U_0(A)$ denotes all the unitaries homotopic to $1_A$. But this is false in general. C*-algebras satisfying this condition are called $K_1$-injective.

Gabor Szabo
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