There is now an extensive literature dealing with ......
For a comprehensive treatment and for references to the extensive literature on the subject one may refer to the book [M] by Markov.
However, no extension in this direction has appeared in the literature.
Incidentally, the question of whether $K(E)$ is amenable for specific Banach spaces $E$ seems to have received almost no attention in the literature.
Unfortunately, the connection with Gromov's work has been obscured in recent years by an emphasis (in the algebraic topology literature) on configuration spaces.
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