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

Sorin Popa

Documenta mathematica, Tome 4 (1999), pp. 665-744

Résumé

We undertake here a more detailed study of the structure and basic properties of the symmetric enveloping algebra M⊠eN​​Mop 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,GN,M​, its graph ΓN,M​ and the inclusion M∨Mop⊂M⊠eN​​Mop, notably showing that M⊠eN​​Mop is amenable relative to its subalgebra M∨Mop iff ΓN,M​ (or equivalently GN,M​) is amenable, i.e., ∥ΓN,M​∥2=[M:N]. We then prove that the hyperfiniteness of M⊠eN​​Mop is equivalent to M being hyperfinite and ΓN,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 ΓQ,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⊠eN​​Mop to have the property T relative to M∨Mop. We prove that this property doesn't in fact depend on the inclusion N⊂M but only on its standard invariant GN,M​, thus defining a notion of property T for abstract standard lattices G.

Détail

DOI : 10.4171/dm/71

Classification : 46L37, 46L40