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The only other variant that I have seen is for periodic algebraic cobordism and algebraic K-theory, in Gepner-Snaith 08.
Concerning notation for periodic $MU$: how about $PMU$? I have changed it to that in the entry.
In
is a tower of Snaith theorems for Morava $E$-theories. Have added a pointer to this to the entry.
I have added at bit more of a remark along these lines:
Snaith-like theorem for Morava E-theories
also added a brief pointer along these lines to Morava E-theory itself.
I have expanded the section Snaith theorem – For complex topological K-theory by adding a fair amount of the basic details that go into the construction of the ring spectrum homomorphism
$\Sigma^\infty(B U(1)_+)[\beta^{-1}] \longrightarrow KU$itself.
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