This is a question from page 552 of Atiyah and Singer's paper "The index of elliptic operators III". The line is "We suppose first that $X$ is compact so that $K(X)$ has an identity element."
Even if $X$ is noncompact, doesn't $K(X)$ always have an identity element, coming from the trivial line bundle?
Here I am understanding $K(X)$ to denote the group arising as a quotient of the free group generated by equivalence classes of complex vector bundles over $X$. I am understanding the ring structure to arise from tensor product. I don't see how compactness of $X$ is relevant.