[Elsewhere](https://mathoverflow.net/questions/95699/iterating-ultrapowers-of-c-algebras) I asked about ultrapowers of the C\*-algebra $A$ of compact operators on separable infinite-dimensional Hilbert space. My question was whether the process of taking ultrapowers of ultrapowers ever stabilizes. 


It was pointed out that the definition of an ultrapower of a C\*-algebra is slightly different from the usual definition of an ultrapower; that the process never stabilizes in the sense that the canonical embedding is never an isomorphism; but that nevertheless (assuming CH) that if $A^1$ is an ultrapower of $A$ then the ultrapowers of $A^1$ are isomorphic as C\*-algebras to $A^1$.


This last statement depends on the fact $A$ is separable. Here I would like to ask about the corresponding question for the C\*-algebra $B$ of bounded operators on Hilbert space and also for the Calkin algebra $C=B/A$. Does the process of taking ultrapowers of ultrapowers of $B$ ever stabilize in the sense of producing isomorphic C\*-algebras? And the same question for $C$?



iterating ultrapowers of C*-algebras: the Calkin algebra
The statement does not depend on separability of A. Actually in the Ge-Hadwin paper they explicitly state the result for C\*-algebras of cardinality continuum. So under CH you have
that the ultrapowers of C^1 are isomorphic to C^1. 
It is not completely trivial that C is not isomorphic to C^1. 
This was proved in 
 arXiv:1112.3898 Countable saturation of corona algebras. Ilijas Farah, Bradd Hart.
Also, if CH fails then every C\*-algebra has 2^c nonisomorphic ultrapowers, so the 
sequence does not stabilize. This is in 
 arXiv:0908.2790 Model theory of operator algebras I: Stability. Ilijas Farah, Bradd Hart, David Sherman.


