A Galois Correspondence for Compact Groups of Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras

Strong additivity and conformal nets

“…By [Izumi et al 1998], there exists a closed subgroup G 1 of G such that Ꮾ(I 1 )∨Ꮽ(I 2 ) is the fixedpoint subalgebra of Ꮽ(I ) under the action of G 1 . It follows that there is normal faithful conditional expectation form Ꮽ(I ) to Ꮾ(I 1 ) ∨ Ꮽ(I 2 ) preserving the vector state ( · , ).…”

“…Since Ꮽ c=1 is the fixed point net of Ꮽ SU(2) 1 under the action of SO(3), by [Izumi et al 1998] there exists a closed subgroup G 1 of SO(3) such that Ꮽ(I ) is the fixedpoint subalgebra of Ꮽ SU(2) 1 (I ) under the action of G 1 . Since G 1 commutes with PSL(2, ‫,)ޒ‬ Ꮽ is the fixed point net of Ꮽ SU(2) 1 under the action of G 1 .…”

Galois groups of quantum group actions and regularity of fixed-point algebras

“…For a right co-ideal N of G, the intermediate von Neumann subalgebra P (N ) of P β ⊆ P associated to N (see [7]) is defined by…”

“…As [1,Theorem III.3.3] suggests, it would sometimes happen (or be expected) that the Galois group carries an important piece of information on the quantum group G itself. With this philosophy in mind, we started in [17] to investigate Galois groups of minimal actions of compact Kac algebras on factors by making good use of the Galois correspondence established by Izumi, Longo and Popa [7]. In [20], we succeeded in describing the Galois group of any minimal action of a compact Kac algebra as the so-called intrinsic group of the dual discrete Kac algebra.…”

A Planar Calculus for Infinite Index Subfactors

“…Some finite index results generalize to infinite index subfactors, such as discrete, irreducible, "depth 2" subfactors correspond to outer (cocylce) actions of Kac algebras [HO89,EN96], and the classical Galois correspondence still holds for outer actions of infinite discrete groups and minimal actions of compact groups [ILP98].…”