I wonder if there is a nice and short proof that the $K$-theory of a topological space is a *special* $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly that $\lambda^k(V \otimes W)$ and $P_k(\lambda^1(V),...,\lambda^k(V),\lambda^1(W),...,\lambda^k(W))$ are stabily equivalent, without making unnatural choices? The same question for $\lambda^i(\lambda^k(V))$ and $P_{ik}(\lambda^1(V),...,\lambda^{ik}(V))$. It would be nice if there is some natural isomorphism on the level of vector spaces, which then may be glued to an isomorphism between vector bundles, perhaps with some extra summands on both sides which vanish in K-theory. Note that an affirmative answer would, in particular, answer this question by Darij Grinberg.

I hope that my question is clear enough although it is not precise. On the other hand, there is a precise generalization: Let $A$ be a topological ring, perhaps a Banach algebra. Is the Grothendieck ring of topological $A$-module bundles over a fixed $X$ a *special* $\lambda$-ring?