Irreducible Subfactors Derived from Popa’s Construction for Non-tracial States

For an inclusion of the form C ⊆ Mn(C), where Mn(C) is endowed with a state with diagonal weights λ = (λ1, ..., λn), we use Popa’s construction, for non-tracial states, to obtain an irreducible inclusion of II1 factors, N(Q) ⊆ M(Q) of index ∑ 1 λi . M(Q) is identified with a subfactor inside the centralizer algebra of the canonical free product state on Q ⋆MN (C). Its structure is described by “infinite” semicircular elements as in [Ra3]. The irreducible subfactor inclusions obtained by this method are similar to the first irreducible subfactor inclusions, of index in [4,∞) constructed in [Po1], starting with the Jones’ subfactors inclusion R ⊆ R, s > 4. In the present paper, since the inclusion we start with has a simpler structure, it is easier to control the algebra structure of the subfactor inclusions. If the weights correspond to a unitary, finite dimensional representation of a Woronowicz’s compact quantum group G, then the factor M(Q) is contained in the fixed point algebra of an action of the quantum group on Q ⋆MN (C) , with equality if G is SUq(N), (or SOq(3) when N = 2). By Takesaki duality, the factor M(L(FN )) is Morita equivalent to L(F∞). This method gives also another approach to find, as also recently proved in [ShUe], irreducible subfactors of L(F∞) for index values bigger than 4. 0. Introduction and definitions In this paper we consider the structure of subfactors obtained from Popa’s construction, for non-tracial states, for the inclusion MN(C) ⊆ MN (C)⊗MN (C). We fix a diagonal matrix with non-zero weights λ1, ...λN . The states on the two algebras are respectively tr(D·) and tr(D⊗D·). This is then the Jones’ iterated basic 1