  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 $X\otimes A$, and establish
stability in important cases, such as complex and real periodic topological
K-theory, $KU$ and $KO$.
