fa.functional analysis - Characterisation of positive elements in l¹(Z)

Consider the Banach $^*$-algebra $ell^1(mathbb Z)$ with multiplication given by convolution and involution given by $a^*(n)=overline{a(-n)}$.

I would like to find nice necessary and sufficient conditions for an element $binell^1(mathbb Z)$ to be positive, that is, to be of the form $a^* * a$ for some $ainell^1(mathbb Z)$.

By now, I have found two necessary conditions. Namely, if $binell^1(mathbb Z)$ is positive, then

$$b(-n)=overline{b(n)}$$ and $$lvert b(n)rvertleq b(0)$$ for every $ninmathbb Z$.

Edit: As t3suji states in his comment below both conditions follow from the more general fact that $a$ is a positive-definite function.

Question: Is this condition also sufficient for positivity? If not, what to I have to add?

Good references would also be great.

Motivation: In the end I want to investigate the (failure of) the Gelfand–Naimark theorem for the above non-C*-algebra.

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