Abstract We define operator-valued Schur and Herz–Schur multipliers in terms of module actions, show that the standard properties these follow from well-known facts about actions duality theory for group actions. These results are applied to study an abelian acting on its Pontryagin dual: it is shown a natural subset can be identified with classical direct product dual group.

We will extend the topological Gromov-Hausdorff stability [ 2 ] from homeomorphisms to finitely generated actions. prove that if an action is expansive and has shadowing property, then it topologically GH-stable. From this we derive examples of GH-stable actions discrete Heisenberg group on tori. Finally, GH-stability invariant under isometric conjugacy.

We show that every action of a smooth algebraic group on variety admits normal projective model. Along the way, we present new proofs some basic results transformation groups, including Weil’s regularization theorem.

We study the derived critical locus of a function $f:[X/G]\to \mathbb{A}_{\mathbb{K}}^1$ on quotient stack smooth affine scheme $X$ by action group $G$. It is shown that $\mathrm{dCrit}(f) \simeq [Z/G]$ for $Z$, whose dg-algebra functions described explicitly. Our results generalize classical BV formalism in finite dimensions from Lie algebra to actions.

We show that an effective action of the one-dimensional torus $${\mathbb G}_m$$ on a normal affine algebraic variety X can be extended to semi-direct product G}_m\rightthreetimes {\mathbb G}_a$$ with same general orbit closures if and only there is divisor D consists -fixed points. This result applied study orbits automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ X.

We consider expansive group actions on a compact metric space containing special fixed point denoted by [Formula: see text], and endomorphisms of such systems whose forward trajectories are attracted toward text]. Such called asymptotically nilpotent, we study the conditions in which they that is, map entire to text] finite number iterations. show for large class discrete groups, this property ...

Abstract Let $X$ be a measure space with measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem understanding structure measurable tilings $F \odot A = X$ by tile $A \subset translated finite set G$ shifts, thus translates $f A$, \in F$ partition up to null sets. Adapting arguments from previous literature, we establish “dilation lemma” that asserts...