# An Infinite C*-algebra with a Dense, Stably Finite *-Subalgebra

## TL;DR

This paper constructs a specific algebraic structure that is stably finite at the algebraic level but becomes infinite upon C*-completion, revealing nuanced differences between algebraic and topological properties.

## Contribution

It introduces a unital pre-C*-algebra with a dense, stably finite *-subalgebra that becomes infinite after C*-completion, highlighting a novel example of divergence between algebraic and topological properties.

## Key findings

- The pre-C*-algebra is stably finite.
- Its C*-completion contains a non-unitary isometry.
- The resulting C*-algebra is infinite.

## Abstract

We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a non-unitary isometry, and so it is infinite.