No abstract

As an application of the general theory established in the first part, we determine the structure of Longo–Rehren inclusions for several systems of sectors arising from endomorphisms of the Cuntz algebras. The E6 subfactor and the Haagerup subfactor are included among these examples, and the dual principal graphs and the S and T-matrices for their Longo–Rehren inclusions are obtained. We also construct several new subfactors using endomorphisms of the Cuntz algebras, and determine their tube algebra structure.

No abstract

We investigate the structure of the Longo-Rehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the Longo-Rehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the Longo-Rehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and Evans-Kawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors.

DEDICATED TO PROFESSOR YOSHIOMI NAKAGAMI ON THE OCCASION OF HIS SIXTIETH BIRTHDAYWe discuss some relationships between two different fields, a non-commutative version of the Poisson boundary theory of random walks and the infinite tensor product (ITP) actions of compact quantum groups on von Neumann algebras. In contrast to the ordinary compact group case, the ITP action of a compact quantum group on a factor may allow non-trivial relative commutant of the fixed point subalgebra. We give a probabilistic description of the relative commutant in terms of a non-commutative Markov operator. In particular, we show that the following three objects can be naturally identified in the case of the quantum group SU q ð2Þ: (1) the relative commutant of the fixed point algebra under the action, (2) the space of harmonic elements for some non-commutative Markov operator on the dual quantum group of SU q ð2Þ; and (3) the weak closure L 1 ðT=SU q ð2ÞÞ of one of the Podles quantum spheres. In view of the ordinary Poisson boundary theory of random walks on discrete groups, it shows that symbolically the quantum homogeneous space T=SU q ð2Þ may be regarded as the ''Poisson boundary'' of a non-commutative random walk on the dual object of SU q ð2Þ: An analogy of the Poisson integral formula is also given. # 2002 Elsevier Science (USA)