More Like this

1. Fell’s absorption principle for semigroup operator algebras

   https://doi.org/10.4171/JNCG/566  

   Katsoulis, Elias G (April 2025, Journal of Noncommutative Geometry)

   Cuntz, J (Ed.)

   Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica-covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper, we explain how to extend Fell’s absorption principle to an arbitrary submonoidPof a groupGby using an enhanced version of the left regular representation. Li’s semigroup\mathrm{C}^{*}-algebra\mathrm{C}^{*}_{s}(P)and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for\mathrm{C}^{*}_{s}(P)but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell’s absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouâtre and Dor-On on the residual finite dimensionality of certain\mathrm{C}^{*}-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a\mathrm{C}^{*}-algebra which is co-universal for equivariant, Li-covariant representations of a submonoidPof a groupG. 

   more » « less
2. DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

   https://doi.org/10.4171/QT/216  

   Jones, Corey (October 2024, Quantum Topology)

   For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry. 

   more » « less
3. Reflective centers of module categories and quantum K-matrices

   https://doi.org/10.1017/fms.2025.10055  

   Laugwitz, Robert; Walton, Chelsea; Yakimov, Milen (January 2025, Forum of Mathematics, Sigma)

   Abstract Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category$${\mathcal {C}}$$and$${\mathcal {C}}$$-module category$${\mathcal {M}}$$, we introduce a version of the Drinfeld center$${\mathcal {Z}}({\mathcal {C}})$$of$${\mathcal {C}}$$adapted for$${\mathcal {M}}$$; we refer to this category as thereflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$of$${\mathcal {M}}$$. Just like$${\mathcal {Z}}({\mathcal {C}})$$is a canonical braided monoidal category attached to$${\mathcal {C}}$$, we show that$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$is a canonical braided module category attached to$${\mathcal {M}}$$; its properties are investigated in detail. Our second goal pertains to when$${\mathcal {C}}$$is the category of modules over a quasitriangular Hopf algebraH, and$${\mathcal {M}}$$is the category of modules over anH-comodule algebraA. We show that the reflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$here is equivalent to a category of modules over an explicit algebra, denoted by$$R_H(A)$$, which we call thereflective algebraofA. This result is akin to$${\mathcal {Z}}({\mathcal {C}})$$being represented by the Drinfeld double$${\operatorname {Drin}}(H)$$ofH. We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangularH-comodule algebras, and we examine their corresponding quantumK-matrices; this yields solutions to the qRE. We also establish that the reflective algebra$$R_H(\mathbb {k})$$is an initial object in the category of quasitriangularH-comodule algebras, where$$\mathbb {k}$$is the ground field. The case whenHis the Drinfeld double of a finite group is illustrated. 

   more » « less
4. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

   more » « less
5. Local topological order and boundary algebras

   https://doi.org/10.1017/fms.2025.16  

   Jones, Corey; Naaijkens, Pieter; Penneys, David; Wallick, Daniel; Izumi, Masaki (January 2025, Forum of Mathematics, Sigma)

   Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states. 

   more » « less