Irreducible subfactors derived from Popa's construction for non-tracial state

For an inclusion of the form ℂ⊆Mn(ℂ) , where M n (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, Nλ(Q)⊆Mλ(Q) of index ∑1λi . Mλ(Q) is identified with a subfactor inside the centralizer algebra of the canonical free product state on Q ⋆ M N (ℂ). Its structure is described by “infinite” semicircular elements as in {xc[32]}. The irreducible subfactor inclusions obtained by this method are similar to the first irreducible subfactor inclusions, of index in [{xc4},∞) constructed in {xc[24]}, starting with the Jones’ subfactors inclusion Rs⊆R , s gt; 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 Mλ(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 Mλ(L(FN)) is Morita equivalent to  L (F∞). This method gives also another approach to find, as also recently proved in {xc[36]}, irreducible subfactors of  L (F∞) for index values bigger than 4.