Classification of Minimal Actions of a Compact Kac Algebra with Amenable Dual

Abstract

We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II 1 . This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument.

Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III

“…Write α = id B(L 2 ( G)) ⊗ α andū = 1 ⊗ u. Then we set a unitary v : [18,Section 2]. Using the 2-cocycle relation of u and (x)…”

“…The first, the second and the fourth ones are derived in a similar way as in the proof of [18,Lemma 5.11]. Thus we only present a proof for the third one.…”

“…Set a projection e = (id ⊗ ϕ 2 )(γ 2 K 2 (e)) ∈S ∩ M ω . If we show (e ⊗ 1)γ 1 π (e ) = 0 for each π ∈ K 1 · K 1 \ {1}, then we are immediately done in view of [18,Lemma 5.7]. This is verified as follows.…”

“…We freely use the notations in [18]. For example, G = (L ∞ ( G), , ϕ) denotes a discrete Kac algebra.…”

“…Our approach is on the whole same as [18], that is, we mainly handle actions of an amenable discrete Kac algebra G instead of a compact Kac algebra G, and obtain our main result through a duality argument as [6]. More precisely, we extend given actions of G on type III factors to larger von Neumann algebras, which are the crossed products by abelian group actions.…”