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Approximately Multiplicative Functionals on Algebras of Smooth Functions
Author(s) -
Howey Richard Andrew Jonathon
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703004551
Subject(s) - multiplicative function , mathematics , commutative property , lipschitz continuity , pure mathematics , norm (philosophy) , bounded function , banach algebra , order (exchange) , algebra over a field , discrete mathematics , banach space , mathematical analysis , finance , political science , law , economics
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.