Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property $T$

Abstract

We undertake here a more detailed study of the structure and basic properties of the symmetric enveloping algebra $M{⊠}_{e_N}{M}^{\mathrm{op}}$ associated to a subfactor $N\subset M$, as introduced in [Po5]. We prove a number of results relating the amenability properties of the standard invariant of $N\subset M,{\mathcal{G}}_{N,M}$, its graph ${\Gamma }_{N,M}$ and the inclusion $M\vee M^{\mathrm{op}}\subset M{⊠}_{e_N}{M}^{\mathrm{op}}$, notably showing that $M{⊠}_{e_N}{M}^{\mathrm{op}}$ is amenable relative to its subalgebra $M\vee M^{\mathrm{op}}$ iff ${\Gamma }_{N,M}$ (or equivalently ${\mathcal{G}}_{N,M})$ is amenable, i.e., $\Vert {\Gamma }_{N,M}{\Vert }^2=[M:N]$. We then prove that the hyperfiniteness of $M{⊠}_{e_N}{M}^{\mathrm{op}}$ is equivalent to $M$ being hyperfinite and ${\Gamma }_{N,M}$ being amenable. We derive from this a hereditarity property for the amenability of graphs of subfactors showing that if an inclusion of factors $Q\subset P$ is embedded into an inclusion of hyperfinite factors $N\subset M$ with amenable graph, then its graph ${\Gamma }_{Q,P}$ follows amenable as well. Finally, we use the symmetric enveloping algebra to introduce a notion of property T for inclusions $N\subset M$, by requiring $M{⊠}_{e_N}{M}^{\mathrm{op}}$ to have the property T relative to $M\vee M^{\mathrm{op}}$. We prove that this property doesn't in fact depend on the inclusion $N\subset M$ but only on its standard invariant ${\mathcal{G}}_{N,M}$, thus defining a notion of property T for abstract standard lattices $\mathcal{G}$.