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 $b\in\ell^1(\mathbb Z)$ to be positive, that is, to be of the form $a^\* \* a$ for some $a\in\ell^1(\mathbb Z)$.


By now, I have found two necessary conditions. Namely, if $b\in\ell^1(\mathbb Z)$ is positive, then $$b(-n)=\overline{b(n)}$$ and $$\lvert b(n)\rvert\leq b(0)$$ for every $n\in\mathbb Z$.


**Edit:** As t3suji states in his comment below both conditions follow from the more general fact that $a$ is a [positive-definite function](http://en.wikipedia.org/wiki/Positive_definite_function).



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



Characterisation of positive elements in l¹(Z)
Although I am not sure that answering this question will help all that much with your original motivating question/problem, I may as well post a link to Bochner's theorem (see [these remarks on Wikipedia](http://en.wikipedia.org/wiki/Positive-definite_function)).


The passage I have in minds says:



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> Positive-definiteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite is a necessary condition ... to be the Fourier transform of a function $g$ on the real line with $g(y) \geq 0$.
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> The converse result is Bochner's theorem, stating that a continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.
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