arXiv subjects
Publications and source records attributed to Pradyut Karmakar.
Let $G$ be a finitely generated virtually abelian group and $[σ]\in H^2(G;\mathbb{T})$ such that $σ(x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^*$-algebra $C^*(G,σ)$ is equal to the rank of a finite index abelian subgroup of $G$. We also show that $\mbox{dim}_{\text{nuc}}(C^*(\mathbb{Z}^r,σ))=r$ if and only if $σ$ is type I.
Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.
We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the construction of a unitary tensor functor from a category of $C(G)$-modules to the category of endomorphisms of $M$. This functor maps a $C(G)$-module, realized as the space of square-integrable functions on a measure space, to a continuous family of endomorphisms of $M$. The resulting structure is a continuous category of endomorphisms, which provides a new framework for studying the interplay between subfactor theory and the representation theory of continuous groups.
We introduce a notion of the Rapid Decay Property (RDP) for Fell bundles over locally compact Hausdorff étale groupoids, extending earlier rapid decay theories for étale groupoids and twists. Our approach yields analytic control on convolution norms and leads to the existence of dense Schwartz-type $*$-subalgebras of the reduced cross-sectional $C^*$-algebra $C_r^*(E)$. As an application, we obtain approximation results showing that, under suitable hypotheses, sections of $C_r^*(E)$ with support contained in an open subset $U\subseteq G$ can be approximated in the reduced norm by compactly supported sections supported inside $U$. In this sense, the Rapid Decay Property provides an analytic mechanism leading to a form of localizability for Fell bundles. We also investigate the relationship between RDP, polynomial growth, and dynamical systems. We show that Fell bundles over groupoids with polynomial growth naturally satisfy the RDP. Furthermore, for a transformation groupoid $G=Γ\ltimes_θX$ associated with a partial action, we prove that RDP for a Fell bundle over $G$ is equivalent to RDP for a naturally associated Fell bundle over the discrete group $Γ$. Finally, we apply these tools to Deaconu-Renault groupoids. By realizing them as partial crossed products of free groups, we show that the presence of persistent branching forces exponential growth, completely obstructing the RDP. This provides a striking illustration of a system where the acting group has RDP, but the associated groupoid fails to inherit it, fully clarifying the boundary between the group and groupoid theories.
We introduce a notion of the Fejér property for topological étale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal étale second countable groupoid satisfying the Fejér property, every closed $C_0(\mathcal{G}^0)$-bimodule $M\subset C_r^*(\mathcal{G})$ is of the form $\overline{C_c(U)}^r$ for some open set $U$. Moreover, we get a Galois correspondence in the sense that every intermediate $C^*$-algebra $\mathcal{B}$ with $C_0(\mathcal{G}^0)\subseteq \mathcal{B}\subseteq C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\leq \mathcal{G}$.
Let $Σ\rightarrow G$ be a twist over a locally compact Hausdorff étale groupoid $G$. Given $f$ in the reduced C$^*$-algebra $C_r^*(Σ;G)$ with open support $U \subseteq G$ we ask when $f$ lies in the closure of the compactly supported sections on $U$. Suppose $G$ satisfies the rapid decay property with respect to a length function $L$. We give a positive answer to our question in two instances: when $L$ is conditionally negative-definite, and when $L$ is the square-root of a locally negative type function on $G$.
Let $L\subseteq \mathbb{R}^{n}$ an even lattice and $T_{L}=\mathbb{R}^{n}/L$ the associated torus. Associated with $L$ we construct $T_{L}$--kernel on a hyperfinite factor type $\mathcal{A}_{L}$, i.e. a monomorphism $T_{L}\to\mathsf{Out}(\mathcal{A}_{L})$, and compute Sutherland's obstruction class in $H^{3}_{\mathrm{Borel}}(T_{L},\mathbb{T})\cong H^{4}(BT_{L} ,\mathbb{Z})$, which is an invariant of the $T_{L}$--kernel and an obstruction to the existence of a twisted crossed product by $T_{L}$. As a Corollary, we obtain that for any $n$-torus $T$ any class in $H^{3}_{\mathrm{Borel}}(T,\mathbb{T})$ arises as an obstruction for a $T$-kernel on the hyperfinite type III${}_{1}$ factor $R$. The construction is an analogue of the construction of Vaughan Jones for finite groups on the hyperfinite type II${}_{1}$ factor but is also motivated by and has applications to conformal nets. Namely, there is an associated local extension $\mathcal{A}_{L}\supseteq \mathcal{A}_{\mathbb{R}^n}$ of conformal nets and the $T_{L}$--kernel corresponds to a family of $T_{L^\ast}$--twisted sectors representations whose anomaly (obstruction) can be identified with the inner product on $L$ seen as a class in $H^{4}(BT_{L},\mathbb{Z})\cong \operatorname{Sym}^{2}(L,\mathbb{Z})$.