Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems

Abstract

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.

arXiv commentMinor (non-Math) modification of v3, to appear in Commun. Math. Phys., 24 pages

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Cited by in corpus (18)

• On characterizations of amenable C*-dynamical systems and new examples
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Equivariant ${\mathcal{O}}_2$-absorption theorem for exact groups
• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Non-amenable tight squeezes by Kirchberg algebras
• Charmenability of higher rank arithmetic groups
• Noncommutative ergodic theory of higher rank lattices
• Subalgebras, subgroups, and singularity
• Maximal Haagerup subalgebras in $L({\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z}))$
• The approximation property and exactness of locally compact groups
• C*-irreducibility for reduced twisted group C*-algebras
• On simplicity of intermediate C*-algebras
• On pathological properties of fixed point algebras in Kirchberg algebras
• The noncommutative factor theorem for lattices in product groups
• Discrete Inclusions of C*-algebras
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra
• Amenable actions on ill-behaved simple C*-algebras
• Crossed product splitting of intermediate operator algebras via 2-cocycles