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

### Sorin Popa

## Abstract

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

## Cite this article

Sorin Popa, Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property $T$. Doc. Math. 4 (1999), pp. 665–744

DOI 10.4171/DM/71