  We prove the regularity of the $\eta$ function for classical
pseudodifferential operators with Shubin symbols. We recall the construction of
complex powers and of the Wodzicki and Kontsevich-Vishik functionals for
classical symbols on $\mathbb{R}^{n}$ with these symbols. We then define the
$\zeta$ and $\eta$ functions associated to suitable elliptic operators. We
compute the $K_{0}$ group of the algebra of zero-order operators and use this
knowledge to show that the Wodzicki trace of the idempotents in the algebra
vanishes. From this, it follows that the $\eta$ function is regular at 0 for
any self-adjoint elliptic operator of positive order.
