A Non-Spectral Dense Banach Subalgebra of the Irrational Rotation   Algebra
  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]. 
Larry B. Schweitzer