A Non-Spectral Dense Banach Subalgebra of the Irrational Rotation Algebra

TL;DR

This paper constructs a dense Banach subalgebra of the irrational rotation algebra that is not spectral, answering a previously open question in operator algebra theory.

Contribution

It provides a specific example of a dense, simple Banach subalgebra of the irrational rotation algebra that is not spectral, challenging existing assumptions.

Findings

Constructed a dense, simple, unital Banach subalgebra of the irrational rotation algebra.

Showed this subalgebra is not a spectral subalgebra of the larger algebra.

Answered an open question posed by T.W. Palmer.

Abstract

We give an example of a dense, simple, unital Banach subalgebra $A$ of the irrational rotation C*-algebra $B$, such that $A$ is not a spectral subalgebra of $B$. This answers a question posed in T.W. Palmer's paper [1].