2025-10-22
Jonathan Taylor
We define essential Cartan pairs of $C^*$-algebras generalising the definition of Renault [Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. (2008), no. 61, 29–63] and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwaśniewski and Meyer [Essential crossed products by inverse semigroup actions: Simplicity and pure infiniteness, Doc. Math. 26 (2021), 271–335]. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.
2025-10-22
Nguyen Duc Trung, Le Xuan Truong, Tan Duc Do, Nguyen Ngoc Trong
We derive an interior estimate up to second orders in Hardy spaces for solutions to the parabolic problem $$ \begin {cases} u_t - \sum _{i, j=1}^n a_{i j} \partial ^2_{ij} u = f & \text {in $\Omega _T$}, \\ u \in h^p(0,T; h^{1,p}(\Omega )) \cap h^{1,p}_{loc }(0,T ; h^{2,p}_{loc }(\Omega )) \end {cases} $$ within an appropriate framework. In the course of proof, we also establish the boundedness results of parabolic singular integrals and their commutators on Hardy spaces which are of independent interest.
2025-10-22
Frédéric Chapoton
Starting from the data of an arbor, which is a rooted tree with vertices decorated by disjoint sets, we introduce a lattice polytope and a partial order on its lattice points. We give recursive formulas for various classical invariants of these polytopes and posets, using the tree structure. For linear arbors, we propose a conjecture exchanging the Ehrhart polynomial of the polytope with the Zeta polynomial of the poset for the reverse arbor. The general motivation comes from a transmutation operator acting on $M$-triangles, which should link the posets considered here with some kind of generalized noncrossing partitions and generalized associahedra. We give some evidence for this relationship in several cases, including notably some polytopes, namely halohedra and Hochschild polytopes.
2025-10-22
Mathematica Scandinavica
Cover vol 131-3