APPROXIMATELY MULTIPLICATIVE FUNCTIONALS ON ALGEBRAS OF SMOOTH FUNCTIONS

Abstract

Let $\phi$ be a bounded linear functional on $A$, where $A$ is a commutative Banach algebra, then the bilinear functional $\skew5\check{\phi}$ is defined as $\skew5\check{\phi} (a,b)\,{=}\,\phi (ab)-\phi (a) \phi (b)$ for each $a$ and $b$ in $A$. If the norm of $\skew5\check{\phi}$ is small then $\phi$ is approximately multiplicative, and it is of interest whether or not $\|\skew5\check{\phi}\|$ being small implies that $\phi$ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then $A$ is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that ${\text{C}}^N {[0,1]}^M$ (the complex valued functions defined on ${[0,1]}^M$ with all $N$th order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.