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If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence $$ \mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}. $$ My question is whether $K(\mathbb{P}X)$ is known to be a $\lambda$-ring. If so, is there a reference? Thanks.

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