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APPROXIMATELY MULTIPLICATIVE FUNCTIONALS ON ALGEBRAS OF SMOOTH FUNCTIONS

Published online by Cambridge University Press:  17 November 2003

RICHARD ANDREW JONATHON HOWEY
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne, Merz Court, Newcastle upon Tyne, NE1 7RU Current address: Department of Computer Science, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE [email protected]
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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.

Keywords

Type
Notes and Papers
Copyright
The London Mathematical Society 2003

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