The problem of stability of approximate homomorphisms was first posed by S. Ulam in the context of groups equipped with a metric. If $G$ and $H$ are groups and $H$ is equipped with a metric $d$, then $\varphi\colon G\to H$ is an $\varepsilon$-homomorphism if $d(\varphi(xy), \varphi(x)\varphi(y))\leq \varepsilon$ for all $x,y\in G$. Ulam’s well-studied problem asks how closely such a map can be approximated by a true homomorphism.
Analogous questions have been investigated in many algebraic and analytic settings. For C*-algebras, the notion of an $\varepsilon$-*-homomorphism admits several possible formalizations. The variant I will discuss, while perhaps not the most immediate, turns out to be particularly interesting, because its associated Ulam stability problem is closely related to rigidity for corona C*-algebras. Namely, Ulam stability of $\varepsilon$-*-isomorphisms between C*-algebras in a certain class (e.g., AF algebras) is equivalent to the rigidity question for coronas of direct sums of C*-algebras in this class.

 

TBA