Structure of Intermediate C*-subalgebras of discrete group actions

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This thesis deals with the structure of intermediate C∗-sub-algebras B, either of the form Cλ∗(Γ)⊆B⊆A⋊rΓ or of the type C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ. We begin by investigating the ideal structure of intermediate C∗-sub-algebras B of the form Cλ∗(Γ)⊆B⊆A⋊rΓ for commutative unital Γ-simple Γ-C∗-algebras A. In particular, we show that if Γ is a C∗-simple group, then every such intermediate C∗-sub-algebra B is simple. Continuing our perusal, we find examples of inclusions Cλ∗(Γ)⊆A⋊rΓ for which every intermediate C∗-sub-algebra B of the form Cλ∗(Γ)⊆B⊆A⋊rΓ is a crossed product. We show that for a large class of actions Γ↷A of C∗-simple groups Γ on unital C∗-algebras A, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate C∗-sub-algebra B, Cλ∗(Γ)⊆B⊆A⋊rΓ, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic C∗-simple group Γ on a von Neumann algebra M with separable predual, every intermediate vNa N, L(Γ)⊆N⊆M⋊Γ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate C∗-sub-algebras B of the form C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ for an inclusion of unital Γ-simple Γ-C∗-algebras C(Y)⊂C(X). We introduce a notion of generalized Powers' averaging and show that it is equivalent to the simplicity of the crossed product C(X)⋊rΓ. As an application, we show that every intermediate C∗-sub-algebras B, C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ is simple whenever C(Y)⋊rΓ is simple.

Crossed products, C*-algebras
Portions of this document appear in: Amrutam, Tattwamasi, and Mehrdad Kalantar. "On simplicity of intermediate-algebras." Ergodic Theory and Dynamical Systems 40, no. 12 (2020): 3181-3187; and in: Amrutam, Tattwamasi. "On Intermediate Subalgebras of C*-simple Group Actions." International Mathematics Research Notices 2021, no. 21 (2021): 16193-16204; and in: Amrutam, Tattwamasi, and Dan Ursu. "A generalized Powers averaging property for commutative crossed products." Transactions of the American Mathematical Society 375, no. 03 (2022): 2237-2254.