[Paper Review] Outer actions of a countable discrete amenable group on approximately finite dimensional factors I, General Theory

Yoshikazu Katayama, Masamichi Takesaki|ArXiv.org|Jan 31, 2003

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper completes the outer conjugacy classification of countable discrete amenable group actions on approximately finite-dimensional (AFD) factors, particularly focusing on type III factors. By introducing a modified Husemeyer-Jones-Ratcliffe exact sequence and analyzing characteristic cocycles and modular obstructions, it establishes that outer conjugacy is classified by pullbacks of intrinsic invariants—flow of weights, modulus, characteristic square, and modular obstruction—under the group action.

ABSTRACT

We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case that the factor is not of type III_0, the invariant is substantially simplified. These cases and examples will be discussed in forthcoming paper.

Motivation & Objective

• To complete the outer conjugacy classification of countable discrete amenable group outer actions on AFD factors, especially for type III factors.
• To remove the technical assumption previously required in the type II∞ case by Ocneanu.
• To extend the classification framework from cocycle conjugacy to outer conjugacy using intrinsic invariants of the factor.
• To define and analyze the role of characteristic cocycles and their associated exact sequences in classifying outer actions.
• To establish that outer conjugacy is determined by the pullback of the factor's intrinsic invariants—flow of weights, modulus, characteristic square, and modular obstruction—under the group action.

Proposed method

• Constructs an equivariant exact characteristic square involving unitary groups, centralizers, and cohomology groups of the factor.
• Defines the characteristic cocycle $({ m λ}, \mu) \in {\rm Z}_{\alpha}(\tilde{H}, L, A)$ and associates it with a normal subgroup $K(\chi) \subset L$ depending on the cocycle class $\chi$.
• Introduces the group ${\rm H}^{3}_{{\alpha}, \rm s}(\tilde{Q}, A)$ as the quotient of standard 3-cocycles modulo standard coboundaries on the quotient group $\tilde{Q} = Q \times \mathbb{R}$.
• Establishes a fiber product construction: ${\rm H}^{{\rm out}}_{{\alpha}, \mathfrak{s}}(G \times \mathbb{R}, N, A) = {\rm H}^{3}_{{\alpha}, \rm s}(\tilde{Q}, A) \ast_{\mathfrak{s}} {\rm Hom}_G(N, {\rm H}^1_\theta)$, linking cohomology and equivariant homomorphisms.
• Applies a modified Husemeyer-Jones-Ratcliffe exact sequence to analyze obstructions to perturbing actions into ${{\rm Cnt}}_{\rm r}(\mathcal{M})$-actions.
• Uses the non-triviality of the 3-cocycle $c(\tilde{p}, \tilde{q}, \tilde{r})$ and the non-innerness of certain modular automorphisms to prove that $\alpha_{\mathfrak{s}}$ cannot be perturbed into an action by ${{\rm Cnt}}_{\rm r}(\mathcal{M})$, establishing the classification.

Experimental results

Research questions

• RQ1 Can the outer conjugacy classification of amenable discrete group actions on AFD factors be completed, especially for type III factors?
• RQ2 What intrinsic invariants of the factor are preserved under outer conjugacy, and how are they pulled back by the group action?
• RQ3 How can the technical assumption in the type II∞ case be removed to complete the classification?
• RQ4 What role do characteristic cocycles and their associated exact sequences play in classifying outer actions?
• RQ5 Under what conditions can an outer action be perturbed into an action by the centralizer of the trace?

Key findings

• The outer conjugacy class of an amenable discrete group action on a type III AFD factor is completely classified by the pullback of the factor’s intrinsic invariants: the flow of weights, modulus, characteristic square, and modular obstruction.
• The 3-cocycle $c(\tilde{p}, \tilde{q}, \tilde{r})$ associated with the action is non-trivial in ${\rm H}^{3}_{{\alpha}, \rm s}(\tilde{Q}, A)$, implying that the action $\alpha_{\mathfrak{s}}$ cannot be perturbed into an action by ${{\rm Cnt}}_{\rm r}(\mathcal{M})$.
• The non-triviality of the 3-cocycle arises from the non-innerness of modular automorphisms $\sigma_{T(ab' - a'b)}^\varphi$, which contradicts the choice of $T$ and prevents trivialization of the obstruction.
• The cohomological obstruction $[c \cdot \alpha_p(\partial_Q(b))] \neq 1$ in ${\rm H}^{3}_{{\alpha}, \rm s}(\tilde{Q}, A)$ confirms that the action is not outer-conjugate to any action by ${{\rm Cnt}}_{\rm r}(\mathcal{M})$.
• The classification is achieved via a fiber product construction linking the 3rd cohomology group and equivariant homomorphisms into ${{\rm H}}^1_\theta$.
• The result generalizes previous classifications by removing technical assumptions and extending the framework from cocycle conjugacy to outer conjugacy.