fa.functional analysis - Odd element of L^1 group algebra of the integers

I'm a bit uncertain what is meant in the third condition: is this the supremum norm of the Gelfand/Fourier transform of $a^m$, or the norm of $a*a*dots*a$ in $ell^infty(mathbb Z)$?

In the first interpretation, it would clearly suffice to find an element satisfying the first two conditions, and then multiply it by an appropriate scalar betwen $0$ and $1$.
In the second interpretation, as Matthew says, it would suffice to find $a$ satisfying conditions 1) and 2) with the additional property that the Fourier transform of $a$ has modulus $<1$ at all but finitely many points of the unit circle.

Anyway, it turns out that we can find (trigonometric) polynomials satisfying conditions 1) and 2), which should by the previous remarks be enough. I don't know where the first examples are in the literature, but you can find explicit quadratic examples in this paper of D. J. Newman:

MR0241980 (39 #3315)
Newman, D. J.
Homomorphisms of $l_{+}$.
Amer. J. Math. 91 1969 37--46

which I personally think is a gem.