Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail isometry $v\in A$ such that $vv^{*} = p$ and $v^{*}v = q$.

I know that with tools of K-theory is can be easily proved. Since $p \sim q$ (as Murray-von Neumann equivalence) and $Tr(p) = Tr(q)$ are equivalent for $p, q \in Proj(\mathbb{M}_n(\mathbb{C}))$. So we can calculate its k-theory and so on.

But my question is: Are there any solutions or ideas for proving this not using k-thoery?


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