My questions are
whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say.
If such a ring structure exists then how to multiply two elements? For example, if we take $K^{-1}(X):=[X, GL(\mathbb{C})]$, where $GL(\mathbb{C})$ is the infinite general linear group, then for $f, g\in K^{-1}(X)$, I could have guessed that the product structure is given by $f\otimes g$, and I would think of it as the tensor product of two matrices of finite but not equal size.
Moreover, if a ring structure for $K^{-1}$ really exists, may you provide me a reference?
I know $K^0(X)$ is a ring and $K^0(X)\oplus K^{-1}(X)$ is a ring. But I don't know whether $K^{-1}(X)$ along is a ring or not.
Thanks a lot.
