Some inverse limits of Cuntz algebras

TL;DR

The paper constructs an inverse system of Cuntz algebras whose limit is isomorphic to ${ m O}_ infty$, revealing that the $K_0$-functor is discontinuous under inverse limits in C*-algebras.

Contribution

It introduces a novel inverse system of Cuntz algebras demonstrating the discontinuity of the $K_0$-functor at the inverse limit.

Findings

Inverse system of Cuntz algebras constructed

Inverse limit is *-isomorphic to ${ m O}_ infty$

Discontinuity of the $K_0$-functor shown

Abstract

We construct a nontrivial inverse system of Cuntz algebras $\{{\cal O}_{n}:2\leq n<\infty\}$, whose inverse limit is *-isomorphic onto ${\cal O}_{\infty}$. By using this result, it is shown that the $K_{0}$-functor is discontinuous with respect to the inverse limit even if the limit is a C$^{*}$-algebra.