Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a topological group.
It has long been known that $M$ is injective if and only if $U(M)$ is an amenable group [de la Harpe, 1979].
Q1: Could we replace $U(M)$ with another semigroup $S\subseteq M$ so that "$M$ is injective if and only if $S$ is an amenable semigroup"?
Perhaps an obvious note: a bounded weak$^*$ closed semigroup with the weak$^*$ topology is automatically a compact semitopological semigroup.
Building on [de la Harpe, 1979], Paterson had shown that a $C^*$-algebra $A$ is amenable if and only if $U(A)$ with the relative weak topology $\sigma(A,A^*)$ is an amenable group. Indeed, $A$ is amenable iff $A^{**}$ is an injective vN-algebra iff $U(A^{**})$ with the relative $\sigma(A^{**},A^*)$ topology is amenable iff $U(A)$ with the relative $\sigma(A,A^*)$ topology is amenable (since $U(A)$ is a dense subgroup of $U(A^{**})$).
Q2: Is there an analogous result for nonselfadjoint operator algebras in the literature?
