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Trigonometric polynomials on noncompact and nonabelian groups
Yes! I indeed need a reference, so if somebody could write something about this, it would be great! :) 
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What are the reasons for considering rings without identity?
$M(\mathbb T)$ is a good illustration: if you want to have a Banach algebra, then you get $M(\mathbb T)$, and this is neither a Hopf algebra, nor a group algebra. But if you allow your algebras to be just stereotype, the very same space of measures $M(\mathbb T)$, with the same multiplication (but with another topology) turns into a stereotype Hopf algebra ${\mathcal C}^\star(\mathbb T)$ which is a group algebra. 
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Trigonometric polynomials on noncompact and nonabelian groups
Yemon, could you, please, make a little paper from this (for example, in arxiv), so that I could cite it? A separate request: if ${\tt Trig}(G)$ is not an algebra even for Moore groups, then this should be mentioned there, because this is an important proposition for me. 
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What are the reasons for considering rings without identity?
Yemon, I don't understand you. Does anybody consider $\ell^1({\mathbb T})$? The stereotype theory suggests several natural group algebras, in particular, the algebra of measures ${\mathcal C}^\star(G)$, the algebra of distributions ${\mathcal E}^\star(G)$, etc., and they are all Hopf algebras in contrast to $L^1(G)$. The full measure algebra $M({\mathbb T})$ is unital, but it is not a Hopf algebra and not a group algebra, since it does not generate equivalence between representations (of $\mathbb T$ and of $M({\mathbb T})$). 
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What are the reasons for considering rings without identity?
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What are the reasons for considering rings without identity?
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What are the reasons for considering rings without identity?
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answered  What are the reasons for considering rings without identity? 
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accepted  Is a matrix element of a norm continuous representation always a trigonometric polynomial? 
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Trigonometric polynomials on noncompact and nonabelian groups
OK. They will be my heroes as well. :) 
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accepted  Trigonometric polynomials on noncompact and nonabelian groups 
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Trigonometric polynomials on noncompact and nonabelian groups
Yemon, for a long time I am dreaming of getting acquaintance with you. :) 
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Is a matrix element of a norm continuous representation always a trigonometric polynomial?
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asked  Is a matrix element of a norm continuous representation always a trigonometric polynomial? 
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Trigonometric polynomials on noncompact and nonabelian groups
Александр Игоревич, хочу с вами подружиться. :) If you don't mind, I'll continue this discussion by email. :) 
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Trigonometric polynomials on noncompact and nonabelian groups
Is this true for Moore groups (where every irreducible unitary representation is finite dimensional)? 
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Trigonometric polynomials on noncompact and nonabelian groups
Хм... Большое спасибо! :) 
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Trigonometric polynomials on noncompact and nonabelian groups
Are there at least some natural classes of groups, where this is true? 
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asked  Trigonometric polynomials on noncompact and nonabelian groups 
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Are normcontinuous representations smooth?
It's strange, that there is no direct reference. 