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

  • Sorin Popa

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

We undertake here a more detailed study of the structure and basic properties of the symmetric enveloping algebra \disM\bteNM\o\p\dis M\bt_{e_N}M^{\o\p} associated to a subfactor NMN\subset M, as introduced in [Po5]. We prove a number of results relating the amenability properties of the standard invariant of NM,\CalGN,MN\subset M,{\Cal G}_{N,M}, its graph ΓN,M\Gamma_{N,M} and the inclusion MM\o\pM\bteNM\o\pM\vee M^{\o\p} \subset M\bt_{e_N}M^{\o\p}, notably showing that \disM\bteNM\o\p\dis M\bt_{e_N}M^{\o\p} is amenable relative to its subalgebra MM\o\pM\vee M^{\o\p} iff ΓN,M(orequivalently\CalGN,M)\Gamma_{N,M} (or equivalently {\Cal G}_{N,M}) is amenable, i.e., ΓN,M2=[M:N]\|\Gamma_{N,M}\|^2=[M:N]. We then prove that the hyperfiniteness of \disM\bteNM\o\p\dis M\bt_{e_N}M^{\o\p} is equivalent to MM being hyperfinite and ΓN,M\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 QPQ\subset P is embedded into an inclusion of hyperfinite factors NMN\subset M with amenable graph, then its graph ΓQ,P\Gamma_{Q,P} follows amenable as well. Finally, we use the symmetric enveloping algebra to introduce a notion of property T for inclusions NMN\subset M, by requiring \disM\bteNM\o\p\dis M\bt_{e_N}M^{\o\p} to have the property T relative to MM\o\pM\vee M^{\o\p}. We prove that this property doesn't in fact depend on the inclusion NMN\subset M but only on its standard invariant \CalGN,M\Cal G_{N,M}, thus defining a notion of property T for abstract standard lattices \CalG\Cal G.

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

DOI 10.4171/DM/71