#### DOCUMENTA MATHEMATICA, Vol. 4 (1999), 665-744

Sorin Popa

Some Properties of the Symmetric Enveloping Algebra of a Subfactor, with Applications to Amenability and Property T

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\$.

1991 Mathematics Subject Classification: Primary: 46L37, secondary: 46L40

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