Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$.

Two embeddings $\epsilon_1$ and $\epsilon_2$ are weakly equivalent if $\;u \epsilon_1 (\cdot)u^* = \epsilon_2(\cdot)$ for some unitary $u \in B(H)/K$ and strongly equivalent if $u$ has a unitary lift to $B(H)$.

Question Apparently weak equivalence classes are strictly larger in general.

  • Is there an obvious $K$-theoretic reason for this? In particular, does the unitary group of the Calkin algebra has something to do with this?

  • Strong and weak equivalence are the same for abelian $A$. What makes abelian algebras special here?


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