# C*-algebra positive element invertibility criteria in terms of $L_1$-norms equivalence

## TL;DR

This paper establishes a criterion for invertibility of positive elements in unital C*-algebras based on the equivalence of their associated $L_1$-norms to the algebra's norm.

## Contribution

It provides a new characterization of invertibility for positive elements using $L_1$-norms in C*-algebras, linking norm equivalence to invertibility.

## Key findings

- $L_1$-norms are equivalent to the C*-algebra norm if and only if the positive element is invertible.
- The criterion offers a norm-based condition for invertibility in unital C*-algebras.
- The result bridges norm equivalence and algebraic invertibility for positive elements.

## Abstract

We prove that the $L_1$-norms associated with a positive element $a$ of a unital C*-algebra are equivalent to the norm of C*-algebra if and only if $a$ is invertible.