representing tensor-C*-categories in BIM - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:20:47Z http://mathoverflow.net/feeds/question/67590 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67590/representing-tensor-c-categories-in-bim representing tensor-C*-categories in BIM André Henriques 2011-06-12T17:33:01Z 2011-06-13T15:08:10Z <p>Given a factor <i>M</i> (=von Neumann alg. with center &#8450;), let us write <i>BIM</i> for the &otimes;-<i>C</i>*-category of <i>M</i>-<i>M</i>-bimodules.</p> <blockquote> <p>Which &otimes;-<i>C</i>*-categories can one faithfully embed into <i>BIM</i>?</p> </blockquote> <p>⓵ Are there <b><i>necessary</i></b> conditions for a &otimes;-<i>C</i>*-category to be representable in <i>BIM</i>?<br> ⓶ Are there <b><i>sufficient</i></b> conditions for a &otimes;-<i>C</i>*-category to be representable in <i>BIM</i>?<br><br></p> <p><i>Comment:</i><br> I suspect that a lot is known about ⓶ in relation with the theory of planar algebras...<br> I specially care about ⓵: are there examples of &otimes;-<i>C</i>*-categories that don't embed?</p> http://mathoverflow.net/questions/67590/representing-tensor-c-categories-in-bim/67669#67669 Answer by Dave Penneys for representing tensor-C*-categories in BIM Dave Penneys 2011-06-13T15:08:10Z 2011-06-13T15:08:10Z <p>I don't know about necessary conditions, but here are some results concerning sufficient conditions:</p> <ul> <li>In MR1749868, Hayashi and Yamagami realize amenable \$C^*\$-tensor categories in the category of bifinite (Jones index) bimodules of the hyperfinite \$II_1\$-factor.-</li> <li>In arXiv:0811.1764v4, Stefaan Vaes and Sébastien Falguières showed that "the representation category of any compact group is the [bifinite] bimodule category of a \$II_1\$-factor," i.e., given a compact group \$G\$ with representation category \$C\$, there is a \$II_1\$-factor \$M\$ whose category of bifinite bimodules is exactly \$C\$.-</li> <li>Recently, Sven Raum and Sébastien Falguières showed that "all finite \$C^*\$-tensor categories are [bifinite] bimodule categories of a \$II_1\$-factor," i.e., given a finite \$C^*\$-tensor category \$C\$, there is a \$II_1\$-factor \$M\$ whose category of bifinite bimodules is exactly \$C\$. This paper has yet to appear on the arXiv. Here is the <a href="http://genco-2011.institut.math.jussieu.fr/program.html" rel="nofollow">link</a> to the conference at which the talk was given.-</li> </ul> <p>I don't know of such results for type \$III\$ factors.</p>