Crossed product splitting of intermediate operator algebras via 2-cocycles

TL;DR

The paper addresses the problem of describing intermediate algebras in inclusions $B\subset A\rtimes_{\mathrm{r}}\Gamma$ arising from $\Gamma$-C$^*$-algebras. It develops a splitting mechanism: after tensoring with $\mathcal{O}_2$, every intermediate algebra $B\subset C\subset A\rtimes_{\mathrm{r}}\Gamma$ splits canonically as $\mathcal{O}_2\otimes C=(\mathcal{O}_2\otimes (C\cap A))\rtimes_{\mathrm{r},\gamma,\mathfrak{w}}\Lambda$ for a subgroup $\Lambda<\Gamma$, a subalgebra $D=C\cap A$, and a unitary-perturbed cocycle action $(\gamma,\mathfrak{w})$ on $\mathcal{O}_2\otimes A$. The work shows why the $2$-cocycle $\mathfrak{w}$ and the tensor component $\mathcal{O}_2$ are generally necessary (with K$_0$ obstructions supporting minimality of $\mathcal{O}_2$), extends the decomposition to twisted/non-unital settings, and provides both a von Neumann algebra analogue and a Galois-type theorem for compact-by-discrete group actions. It also constructs hard-to-untwist cocycles on $\mathcal{O}_2$ to illustrate limits of the splitting, and demonstrates a concrete lattice isomorphism between the Galois data and intermediate algebras. Overall, the results give a canonical, structure-theoretic description of intermediate algebras in a broad dynamical context, with significant implications for crossed product theory and Galois correspondences in operator algebras.

Abstract

We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} Γ$ arising from inclusions $B \subset A$ of $Γ$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of Rørdam, and is centrally $Γ$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} Γ$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, γ, \mathfrak{w}} Λ\] for $D:= C \cap A$, some $Λ<Γ$, and a subsystem $(γ, \mathfrak{w})$ of a unitary perturbed cocycle action $Λ\curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} Γ\] for actions of compact-by-discrete groups $K \rtimes Γ$ on simple C*-algebras. Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice. We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: For the precise statements and the proofs, see Section \ref{['section:O2']}
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3: Central freeness of inclusions for automorphisms and actions
• Lemma 3.1