Let

$\U0001d504$ be a commutative Banach algebra with dual

${\U0001d504}^{*}$. For

$\phi \in {A}^{*}$, define

$\stackrel{\u02d8}{\phi}$(a,b)

$=\phi \left(ab\right)-\phi \left(a\right)\phi \left(b\right)$, and call

$\phi \delta $-multiplicative iff

$\parallel \stackrel{\u02d8}{\phi}\parallel \le \delta $.

$\U0001d504$ is an algebra in which approximately multiplicative functionals are near multiplicative (AMNM) if for each

$\u03f5>0$, there is

$\delta >0$ such that

$inf\{\parallel \phi -\psi \parallel :\psi $ is a

$character\}<\u03f5$ whenever

$\phi $ in

${\U0001d504}^{*}$ is

$\delta $-multiplicative. The author studies these entities and shows that AMNM algebras include the well-known examples (finite dimensional,

${C}_{0}\left(X\right)$,

${L}^{1}\left(G\right)$,

${\ell}^{1}\left(\mathbb{Z}\right)$, disc algebra), but not all. A result of Gleason about multiplicativeness of functions with range contained in the spectrum is studied in a more general context.