Stable Adams operations I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not come across a nicely written introduction on the stable Adams operations in complex topological K-theory. Could someone kindly explain the spectrum level maps used in defining the stable Adams operations or perhaps redirect me to some literature on the same?
 A: We can define complex K-theory spectrum $K$ looking at even suspension maps
$\Sigma^2:S^2\wedge BU\to BU$. The map $\Sigma^2$ has an expression as
a classifying map for vector bundle $E\otimes \tau$, where $\tau=(\mathcal{O}(1)-1)\in \tilde{K}^0(S^2)$ is a generator and $E$ is the universal bundle over $BU$.
Adams operations $\Psi^k$ are easily defined by universality and the splitting principle, thus giving $\Psi^k:BU\to BU$. To define them on the level of spectrum $K\to K$
we would like to have a commuting square:
$\begin{align*}
&S^2\wedge BU &&\overset{\Sigma^2}{\to}&& BU\\
&\downarrow{id\wedge \Psi^k} && &&\downarrow{\Psi^k}&\\
&S^2\wedge BU&&\overset{\Sigma^2}{\to}&&BU& 
\end{align*}$
Which fails, since $\Psi^k\circ \Sigma^2$ classifiyes $\Psi^k(E\otimes\tau)=\Psi^k(E)\cdot k\tau$, meanwhile $\Sigma^2\circ (id\wedge \Psi^k)$ classifies
$\Psi^k(E)\cdot \tau$.
Hence, to obtain a commutative square we have to adjust maps between corresponding terms of spectrum, i.e. putting
$\Psi^k/k^n:BU\to BU$, where $K_{2n}=BU$ is 2n-th term of the spectrum $K$. Of course this make sense only after localisation of spectra inverting $k$ and does not provides a stable operation otherwise.
This construction is sketched in Adams book on stable homotopy and generalized homology, see $K_*(K)$ computation there.
