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1h

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Correspondence between real forms and real structures on complex Lie groups
@YCor: Yes, I thought the same, an involutive antilinear Lie algebra automorphism. So this is not a onetoone correspondence?... I think, at least for reductive groups (i.e. for complexifications of compact real Lie groups) this must be so... That's true? 
8h

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Correspondence between real forms and real structures on complex Lie groups
@YCor: Pardon, I corrected the text in MSE (and here also). 
9h

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Correspondence between real forms and real structures on complex Lie groups
Does this give an answer in the case, when $K$ is a compact real Lie group and $G$ is its complexification? 
May 4 
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Gérard, as far as I understand, the exponential map of $\text{Diff}({\mathbb T})$ is also not locally invertible. I think, you should accept Alexander Schmeding's answer, it sounds much more useful. 
May 4 
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
@YCor, I thought this means that there is a neighbourhood $U$ of 1 in $G$, and a neighbourhood $V$ of 0 in $L(G)$ such that $\exp(V)=U$. 
May 4 
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Gérard, the Lie algebra for ${\mathbb T}^{\mathfrak m}$ is ${\mathbb R}^{\mathfrak m}$, the ${\mathfrak m}$th power of $\mathbb R$. 
May 4 
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Gérard, some infinite dimensional locally compact groups have exponential maps that are surjective. For example, the infinite power ${\mathbb T}^{\mathfrak m}$ (where ${\mathfrak m}$ is any infinite cardinal) of the circle $\mathbb T$ has this property. 
Apr 24 
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Will it be correct to reformulate your question as follows: when is the exponential map $\exp:L(G)\to G$ (locally) surjective? 
Apr 19 
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Quantum mechanics formalism and C*algebras
I bet, $F^*$algebras will not save quantum mechanics, but stereotype ones will do: arxiv.org/abs/1303.2424 :) 
Mar 23 
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What are the applications of operator algebras to other areas?
Sébastien, the problem exists. Not only in this concrete field, but in many fields in mathematics. You were right when asking this question, this is a normal reaction of a human (contrary to robot). The discussions like this must be common in science, otherwise this activity turns into a religion (where references to "tone" replace doubts :). 
Mar 22 
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What are the applications of operator algebras to other areas?
Nice to hear this from a specialist in operator algebras. :) When at the Last Judgment we will be accused of snobbery, we will point at Sébastien Palcoux and say: "Here's a conscientious man among us!" 
Feb 7 
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Which group algebras in analysis are “true group algebras”?
And are there other constructions? Will $W^*(G)$ be initial object anywhere? 
Feb 7 
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Which group algebras in analysis are “true group algebras”?
Ah, OK! That's interesting... What is the problem in the nondiscrete case? 
Feb 7 
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Which group algebras in analysis are “true group algebras”?
So as far as I understand, for the case of discrete $G$ you and Tobias suggest two examples: 1) $C^*(G)$ will be an initial object in the category of unitary representations of $G$ in C*algebras, and 2) $\ell^1(G)$ will be an initial object in the category of representations of $G$ in Banach algebras. Is it correct? 
Feb 7 
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Which group algebras in analysis are “true group algebras”?
As to unitality, I would bear (you could appreciate this, by the way!), if there would exist some constructions of group algebras with this universality property in some categories of ``representations in nonunital algebras''. 
Feb 7 
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Which group algebras in analysis are “true group algebras”?
Yemon, yes if $L^1(G)$ or $C^*(G)$ could be characterized as initial objects, that would be interesting. I suppose, you mean initial object in some ``category of representations of $G$''? I did not understand, are $C^*(G)$ or $\ell^1(G)$ such an initial objects for discrete $G$? 
Jan 26 
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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! :) 
Jan 26 
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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. 
Jan 26 
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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. 
Jan 26 
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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 $M({\mathbb T})$). 