Let us view topological K-theory as a functor $K$ from the cateory of compact pairs (that is, a compact Hausdorff spaces with a distinguished closed subset) to the category of $\mathbb Z/2$-graded Abelian groups. We could also restrict to second countable spaces and thus countable groups.


An additive operation on topological K-theory is just a natural transformation from $K$ to $K$. These natural transformations form an Abelian group under addition.


*Question:* What is the isomorphism class of this group?


Examples of operations are discussed in Efton Park's book and in Max Karoubi's book, but I cannot find discussion of the collection of all (additive) operations.


*Edit:* After looking at Karoubi's book again, I have to state that it actually contains a very satisfactory treatment of operations in K-theory.



What is the group of additive operations on topological K-theory?
The most explicit answers are in work of Sarah Whitehouse and her collaborators. You could start with this paper and its references:


<http://www.ams.org/journals/proc/2010-138-06/S0002-9939-10-10237-8/home.html>


