Let $A$ be something interesting like the C\*-algebra of compact operators on a separable infinite-dimensional Hilbert space and let $A^1$ be an ultrapower of $A$. Then $A^1$ is a primitive C\*-algebra strictly containing $A$ (Ge and Hadwin). Now let $A^2$ be an ultrapower of $A^1$. Then $A^2$ is a primitive C\*-algebra containing $A^1$, presumably strictly. In the same way we may define $A^3$, $A^4$, etc. Since $A^n$ contains $A^{n-1}$ we may define an inductive limit $A^{\omega}$, and then continue transfinitely with $A^{{\omega} +1}$ etc.


My question is, Does the process ever stabilise? If so does it stabilise at some finite step, or at $\omega$ or at the first uncountable ordinal?



iterating ultrapowers of C*-algebras
In the C\*-algebra ultrapower construction, instead of identifying two sequences if they agree on a set in the ultrafilter, you identify two sequences if their difference goes to zero along the ultrafilter. You also throw out those sequences whose norm goes to infinity along the ultrafilter.


I think the simple answer to the question is that *any* infinite dimensional C\*-algebra is properly contained in *any* ultrapower coming from a free (nonprincipal) ultrafilter. In fact this will be true of any infinite dimensional Banach space, as a simple consequence of the fact that its unit ball is not precompact. We can find a sequence of vectors in the unit ball such that the distance between any two of them is at least 1/2, and this will always give rise to a new element in the ultrapower (it is not identified with any constant sequence). So no, the process never stabilizes.


