Approximately Multiplicative Functionals on Algebras of Smooth Functions

Let ϕ be a bounded linear functional on A , where A is a commutative Banach algebra, then the bilinear functional ϕ̌ is defined as ϕ̌ ( a,b )=φ ( ab )‐ϕ ( a ) ϕ ( b ) for each a and b in A . If the norm of ϕ̌ is small then ϕ is approximately multiplicative, and it is of interest whether or not ‖ϕ̌‖ being small implies that ϕ 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 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.