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Apurva Seth

Publications and source records attributed to Apurva Seth.

10 recordsLinked to original sources
arXiv2026

Pureness of Certain Crossed Product C*-Algebras

We establish comparison and divisibility properties for crossed product C*-algebras arising from automorphisms of algebras C (X, D) which lie over minimal homeomorphisms, from actions of compact groups which have finite Rokhlin dimension with commuting towers, and from actions of compact groups which have the restricted tracial Rokhlin property with comparison. We deduce that these crossed products we consider are pure, and conclude they have stable rank one, and in certain cases have real rank zero. We give examples in which these properties do not follow from previous results, in the case of C (X, D) due to the lack of Z-stability of D, the underlying topological spaces not being finite dimensional, or both.

math.OA↗
arXiv2026

Continuous functions over a pure C*-algebra

Let $X$ be a compact metric space, and let $A$ be a pure $\mathrm{C}^*$-algebra. We show that $C(X,A)$ is pure whenever $A$ is simple; or every quotient of $A$ is stably finite (e.g., $A$ has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such $A$ with any ASH-algebra is pure.

math.OA↗
arXiv2025

Tensorial Permanence of $K$-Stability for Diagonal AH-Algebras

We study $K$-stability for tensor products of diagonal AH-algebras with arbitrary C*-algebras. Our main result provides a characterization of $K$-stability: for a diagonal AH-algebra $A = \varinjlim (A_i, \varphi_i)$, $A \otimes B$ is $K$-stable for every C*-algebra $B$ if and only if the sizes of the matrix blocks in the inductive system grow without bound. As applications, we show that non-$\mathcal{Z}$-stable Villadsen algebras of the first kind are $K$-stable when tensored with any C*-algebra. Moreover, any simple, unital, infinite-dimensional diagonal AH-algebra automatically satisfies this growth condition, and therefore its tensor product with arbitrary C*-algebras is always $K$-stable.

math.OA↗
arXiv2024

Powers averaging for actions on $C(X)$-algebras

Given a unital $C(X)$-algebra $A$ discrete group $Γ$ and an action $α: Γ\to \text{aut}(A)$ which leaves $C(X)$ invariant and such that $C(X)\rtimes_{α,r} Γ$ is simple, and a $2$-cocycle $ω$, we obtain a bijective correspondence between maximal $Γ$-invariant ideals of $A$ and maximal ideals in $A\rtimes_{α,ω,r} Γ$. In particular, $A\rtimes_{α,ω,r} Γ$ is simple if and only if $A$ has no $Γ$-invariant ideals.

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arXiv2023

Intermediate crossed product $C^*$-algebras

Let $B$ be a separable $C^*$-algebra, let $Γ$ be a discrete countable group, let $α: Γ\to \text{Aut}(B)$ be an action, and let $A$ be an invariant subalgebra. We find certain freeness conditions which guarantee that any intermediate $C^*$-algebra $A \rtimes_{α,r} Γ\subseteq C \subseteq B \rtimes_{α,r} Γ$ is a crossed product of an intermediate invariant subalgebra $A \subseteq C_0 \subseteq B$ by $Γ$. Those are used to generalize related results by Suzuki.

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.

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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

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.

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