More Like this

1. Full quantum crossed products, invariant measures, and type-I lifting

   Alexandru Chirvasitu (October 2022, Munster journal of mathematics)

   We show that for a closed embedding H ≤ G of locally compact quantum groups (LCQGs) with G/H admitting an invariant probability measure, a unitary G-representation is type-I if its restriction to H is. On a related note, we also prove that if an action G ⟳ A of an LCQG on a unital C∗ -algebra admits an invariant state then the full group algebra of G embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). We also prove a few other results on crossed products of LCQG actions, some of which seem to be folklore; among them are (a) the fact that two mutually dual quantum-group morphisms produce isomorphic full crossed products, and (b) the fact that full and reduced crossed products by dual-coamenable LCQGs are isomorphic. 

   more » « less
2. Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

   https://doi.org/10.1093/imrn/rnad060  

   Brannan, Michael; Elzinga, Floris; Harris, Samuel J; Yamashita, Makoto (March 2023, International Mathematics Research Notices)

   Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups. 

   more » « less
3. Valuations, completions, and hyperbolic actions of metabelian groups

   https://doi.org/10.1112/jlms.12916  

   Abbott, Carolyn R; Balasubramanya, Sahana; Rasmussen, Alexander J (June 2024, Journal of the London Mathematical Society)

   Abstract Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations. 

   more » « less
4. C*-Framework for Higher-Order Bulk-Boundary Correspondences

   https://doi.org/10.1007/s00220-025-05415-1  

   Polo_Ojito, Danilo; Prodan, Emil; Stoiber, Tom (October 2025, Communications in Mathematical Physics)

   A typical crystal is a finite piece of a material which may be invariant under some point symmetry group. If it is a so-called intrinsic higher-order topological insulator or superconductor, then it displays boundary modes at hinges or corners protected by the crystalline symmetry and the bulk topology. We explain the mechanism behind such phenomena using operator K-theory. Specifically, we derive a groupoid C ∗ -algebra that (1) encodes the dynamics of the electrons in the infinite size limit of a crystal; (2) remembers the boundary conditions at the crystal’s boundaries, and (3) admits a natural action by the point symmetries of the atomic lattice. The filtrations of the groupoid’s unit space by closed subsets that are invariant under the groupoid and point group actions supply equivariant cofiltrations of the groupoid C ∗ -algebra. We show that specific derivations of the induced spectral sequences in twisted equivariant K-theories enumerate all non-trivial higher-order bulk-boundary correspondences. 

   more » « less
5. Fukaya categories of surfaces, spherical objects and mapping class groups

   https://doi.org/10.1017/fms.2021.21  

   Auroux, Denis; Smith, Ivan (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $$2$$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $$1$$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included. 

   more » « less