The canonical endomorphism for infinite index inclusions (Q1284525)

The paper extends some results of \textit{R. Longo} [Commun. Math. Phys. 159, 133-150 (1994; Zbl 0802.46075)]. Given are necessary and sufficient algebraic conditions on an endomorphism \(\gamma\) of a von Neumann algebra \(M\) which guarantee the existence of a subalgebra \(N\subset M\) for which \(\gamma\) is the associated canonical endomorphism. This is done in the cases (dual to each other) when there is a normal faithful conditional expectation either from \(M\) to \(N\) or from \(N'\) to \(M'\). The result is then applied when compact and discrete Woronowicz algebras act alternately on the factors in the various levels of Jones' tower. The authors explain that their motivation to study such canonical endomorphisms \(\gamma\) of the inclusion \(N\subset M\) is that the inclusion can be interpreted as being generated by means of a cross product by the action on \(N\) of an implicitly defined ``quantum object'' and \(\gamma\) can be regarded in some sense as the ``regular representation'' to this object.