arXiv subjects

Prahlad Vaidyanathan

Publications and source records attributed to Prahlad Vaidyanathan.

11 recordsLinked to original sources

arXiv2026

Weak Centrality of unital C(X)-algebras

This paper establishes a fibrewise characterization of weak centrality for unital $C(X)$-algebras whose defining homomorphism maps $C(X)$ onto the center: such an algebra is weakly central if and only if each of its nonzero fibres has a unique maximal ideal. This yields a corresponding characterization for arbitrary unital $C^*$-algebras through their canonical fibres over the spectra of their centers. The resulting criterion explains Vesterstrøm's AF-algebra counterexample, whose obstruction is also interpreted through its Bratteli diagram. A parallel criterion characterizes centrality by simplicity of the fibres. Applications to full group $C^*$-algebras give an alternative proof for the discrete Heisenberg group and show that the full group $C^*$-algebra of every countable non-abelian torsion-free nilpotent group is not weakly central.

math.OA↗

arXiv2026

Rokhlin Dimension and Inductive Limit Actions on AF-algebras

Given a separable, AF-algebra A and an inductive limit action on A of a finitely generated abelian group with finite Rokhlin dimension with commuting towers, we give a local description of the associated crossed product C*-algebra. In particular, when A is unital and $α\in Aut(A)$ is approximately inner and has the Rokhlin property, we conclude that $A \rtimes_α \mathbb{Z}$ is an A$\mathbb{T}$-algebra.

math.OA↗

arXiv2024

Rokhlin Dimension: Permanence Properties and Ideal Separation

We study the Rokhlin dimension for actions of residually finite groups on C*-algebras. We give a definition equivalent to the original one due to Szabo, Wu and Zacharias. We then prove a number of permanence properties and discuss actions on C_0(X)-algebras and commutative C*- algebras. Finally, we use a theorem of Sierakowski to show that, for an action with finite Rokhlin dimension, every ideal in the associated reduced crossed product C*-algebra arises from an invariant ideal of the underlying algebra.

math.OA↗

arXiv2022

$K$-Stability of A$\mathbb{T}$-Algebras

We describe a procedure to compute the rational nonstable K-groups of A$\mathbb{T}$-algebras. As an application, we show that an A$\mathbb{T}$-algebra is K-stable if and only if it has slow dimension growth.

math.OA↗

arXiv2021

Rokhlin Dimension and Equivariant Bundles

Given an action of a compact group on a complex vector bundle, there is an induced action of the group on the associated Cuntz-Pimsner algebra. We determine conditions under which this action has finite Rokhlin dimension.

math.OA↗

arXiv2021

Rational $K$-Stability of Continuous $C(X)$-Algebras

We show that the property of being rationally $K$-stable passes from the fibers of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) $K$-stable provided the underlying C*-algebra is (rationally) $K$-stable, and the action has finite Rokhlin dimension with commuting towers.

math.OA↗

arXiv2020

Homotopical Stable Ranks for Certain $C^{\ast}$-algebras Associated to Groups

We study the general and connected stable ranks for $C^{\ast}$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast}$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast}$-algebras by finite group actions with the Rokhlin property.

math.OA↗

arXiv2020

AF-algebras and rational homotopy theory

We give a procedure to compute the rational homotopy groups of the group of quasi-unitaries of an AF-algebra. As an application, we show that an AF-algebra is K-stable if and only if it is rationally K-stable.

math.OA↗

arXiv2020

K-stability of continuous C(X)-algebras

A C*-algebra is said to be K-stable if its nonstable K-groups are naturally isomorphic to the usual K-theory groups. We study continuous $C(X)$-algebras, each of whose fibers are K-stable. We show that such an algebra is itself K-stable under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension.

math.OA↗

arXiv2018

Homotopical Stable Ranks for Certain C*-algebras

We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras, and non-commutative CW-complexes.

math.OA↗

arXiv2015

Roots of Dehn twists about multicurves

A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{\C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.

math.GT↗