We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $Sigma X$. We prove several structural properties about different notions of stability, corresponding to different levels of invariance required of $Xotimes A$, and establish stability in important cases, such as complex and real periodic topological K-theory, $KU$ and $KO$.
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