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If $\xi$ is a virtual bundle of virtual dimension $1$ in the ring $K(X)$, where $X$ is a compact Hausdorff topological space and $K$ stays for complex topological $K$-theory, then is $\xi$ invertible in the ring $K(X)$ ?

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Yes. Any rank zero element x in K(X) is nilpotent by https://ncatlab.org/nlab/show/virtual%20vector%20bundle, hence 1+x is invertible.

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