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C. Schochet shows K\"unneth theorem for the $C^*$-algebras in the smallestclass of nuclear $C^*$-algebras which contains the separable Type I algebrasand is closed under some operations. We calculate the $K$-theory for thecrossed product of the infinite tensor product of a unital $C^*$-algebra inthis class by the shift. In particular, we calculate the $K$-theory of thelamplighter group $C^*$-algebra.

$K$-theory for the crossed products of infinite tensor product of $C^*$-algebras by the shift