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# Quantum Topology

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**Volume 2, Issue 3, 2011, pp. 301–337**

**DOI: 10.4171/QT/23**

Published online: 2011-07-06

The embedding theorem for finite depth subfactor planar algebras

Vaughan F. R. Jones^{[1]}and David Penneys

^{[2]}(1) UC, Berkeley, USA

(2) UC, Berkeley, USA

We define a canonical planar *-algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.

*Keywords: *Subfactors, planar algebras

Jones Vaughan, Penneys David: The embedding theorem for finite depth subfactor planar algebras. *Quantum Topol.* 2 (2011), 301-337. doi: 10.4171/QT/23