Odd element of L^1 group algebra of the integers

Question

Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a_n)\in\ell^1(\mathbb Z)$ such that:

• $\|a\|>1$
• a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach algebra for the convolution product)
• we have also that $\|a^m\|_\infty \rightarrow 0$.

I'm sure a clever use of the Fourier transform would work. For example, the third condition is ensured if, letting $f\in C(\mathbb T)$ be the Fourier transform of $a$, we have that $|f|<1$ almost everywhere. The 2nd condition implies that $|f|\leq1$, but of course this isn't if and only if.

Answer 1

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.