  The infinite matrix `Schwartz' group $G^{-\infty}$ is a classifying group for
odd K-theory and carries Chern classes in each odd dimension, generating the
cohomology. These classes are closely related to the Fredholm determinant on
$G^{-\infty}.$ We show that while the higher (even, Schwartz) loop groups of
$G^{-\infty},$ again classifying for odd K-theory, do \emph{not} carry
multiplicative determinants generating the first Chern class, `dressed'
extensions, corresponding to a star product, do carry such functions. We use
these to discuss Bott periodicity for the determinant bundle and the eta
invariant. In so doing we relate two distinct extensions of the eta invariant,
to self-adjoint elliptic operators and to elliptic invertible suspended
families and show that the corresponding $\tau$ invariant is a determinant in
this sense.
