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 $\dis M\bt_{e_N}M^{\o\p}$ 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 $N\subset M,{\Cal G}_{N,M}$, its graph $\Gamma_{N,M}$ and the inclusion $M\vee M^{\o\p} \subset M\bt_{e_N}M^{\o\p}$, notably showing that $\dis M\bt_{e_N}M^{\o\p}$ is amenable relative to its subalgebra $M\vee M^{\o\p}$ iff $\Gamma_{N,M} (or equivalently {\Cal G}_{N,M})$ is amenable, i.e., $\|\Gamma_{N,M}\|^2=[M:N]$. We then prove that the hyperfiniteness of $\dis M\bt_{e_N}M^{\o\p}$ 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 $\dis M\bt_{e_N}M^{\o\p}$ to have the property T relative to $M\vee M^{\o\p}$. We prove that this property doesn't in fact depend on the inclusion $N\subset M$ but only on its standard invariant $\Cal G_{N,M}$, thus defining a notion of property T for abstract standard lattices $\Cal G$.