# Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras

## TL;DR

This paper characterizes closed ideals and Lie ideals of minimal tensor products of certain C*-algebras, linking their structure to properties of the component algebras and the underlying space.

## Contribution

It provides a characterization of closed ideals in $C_0(X,A)$ for algebras with finitely many ideals and describes the structure of Lie ideals in these tensor products.

## Key findings

- Closed ideals of $C_0(X,A)$ are characterized via ideals of $A$ and subspaces of $X$.
- Closed ideals of $C_0(X) ensor^{ ext{min}} A$ are finite sums of product ideals.
- The centre-quotient property of $C_0(X,A)$ is equivalent to that of $A$.

## Abstract

For a locally compact Hausdorff space $X$ and a $C^*$-algebra $A$ with only finitely many closed ideals, we discuss a characterization of closed ideals of $C_0(X,A) $ in terms of closed ideals of $A$ and certain (compatible) closed subspaces of $X$. We further use this result to prove that a closed ideal of $C_0(X) \otimes^{\min} A$ is a finite sum of product ideals. We also establish that for a unital $C^*$-algebra $A$, $C_0(X,A)$ has centre-quotient property if and only if $A$ has centre-quotient property. As an application, we characterize the closed Lie ideals of $C_0(X,A)$ and identify all closed Lie ideals of $ C_0(X) \otimes^{\min} B(H) $, $H$ being a separable Hilbert space.