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Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar product $\langle \cdot , \cdot \rangle$ (namely an euclidian affine space). My first thought was "Nice, I can equip $\mathcal{E}$ with a structure of metric space and explore the meaning of the associated convergence". But is there moreover some generic advantage to have an euclidian structure ?

One may formulate alternatively : "What informations you get for $\mathcal{E}$ knowing that there exists a notion of orthogonality on ?"

This is a very broad question, but this is motivated by a question I posted few months ago : What do we actually know about logarithmic energy ?

I guess I'm looking for an answer pointing some Theorems characterizing euclidian affine spaces between metric affine spaces, but also for your first reaction, or direction for investigation, after such a discovery.

Maybe it is of interest to precise that $\mathcal{E}$ is in fact a subspace of signed measures on $\mathbb{C}$ having total mass equals to $1$, and thus is an infinite dimensional real space.

EDIT : I precise my question,

Is there some geometric theory developed for infinite dimensional real pre-Hilbert spaces (which are not complete) ?

I emphasis that the elements of the space I have in mind are measures, and the scalar product has the form $$ \langle \mu, \nu \rangle = \iint K(x,y)d\mu(x)d\mu(y) $$ with a kernel $K : \mathbb{C}^2\rightarrow \mathbb{R}$ given by $$ K(x,y)=\log|x-y|. $$ Maybe there also exists some references for general kernels which are less singular ?

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look for an orthogonal basis and do "fourier series" ? –  Pierre Oct 22 '11 at 17:43
My only quibble with this question is that I do not consider the inner product to be part of an "affine structure". The affine flat structure of $R^n$ corresponds to the action of the affine group, which is generated by linear transformations and translations. This is really just part of the vector space structure of $R^n$ and does not use the inner product. If you introduce the inner product and restrict to the group of rigid motions (generated by rotations and translations), then the resulting geometric structure is usually called the Euclidean structure. –  Deane Yang Jun 4 '12 at 9:06
@Deane Yang : I agree, that's why I speak about "Euclidean affine structure" (with a typo, I just realize, sorry for the gallicism...). Maybe I should call it "Euclidean structure", but I preferred to emphases that we do not have a norm, but just a metric and the notion of angles. –  Adrien Hardy Jun 4 '12 at 10:42

1 Answer 1

Affine Hilbert spaces have been of relatively great importance recently both in operator algebras and geometric group theory, through the study of affine isometric actions on Hilbert spaces. Let me give two examples:

1) For $\sigma$-compact, locally compact groups $G$, Kazhdan's property (T) is equivalent to the fact that every affine isometric action of $G$ on a Hilbert space, is conjugate via a translation to a linear isometric action, i.e. a unitary representation. This is the Delorme-Guichardet theorem, see e.g. the book

Note that there are groups with property (T) which admit non-trivial (i.e. fixed point-free) affine isometric actions on $L^p$ for $p$ large; this is the case of the simple Lie group $Sp(n,1)\;(n\geq 2)$.

2) A locally compact group has the Haagerup property, or is a-(T)-menable (a pun due to Gromov), if it admits a proper affine isometric action on a Hilbert space. Here it is important to allow infinite-dimensional Hilbert spaces, since in finite dimension we would only get the crystallographic groups (by Bieberbach's theorem). Allowing infinite-dimensional spaces we get a huge class of groups containing free groups, amenable groups, Coxeter groups, etc... A remarkable result by Higson and Kasparov is that a-(T)-menable groups satisfy the strongest form of the Baum-Connes conjecture, see The proof involves the construction of a $C^*$-algebra associated with a Hilbert space, which captures the geometry of closed affine subspaces and their orthogonality.

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Thx for your answer! Question : Is the character "Hilbert" important ? (In my setting, $\mathcal{E}$ is unfortunately not complete). I understand that in your examples the main character is the group $G$, for which you investigate its properties by letting it acting as isometries on different Hilbert spaces. But my question is more about the (pre-)Hilbert space itself ! I also understand from your answer the importance of "knowing the isometries associated to my scalar product". NB : I've attended your mini-course in Leuven, it was very nice. –  Adrien Hardy Oct 22 '11 at 23:09
@ Adrien: My excuse is that, in your own words, your question is a very broad one! I agree that, from my point of view, you study a group by letting it act isometrically on various metric spaces, so you may as well assume that these spaces are complete. In Hilbert spaces, this provides you with useful tools like projection onto closed subsets. When you say "characterizing euclidean affine spaces between metric affine spaces", I presume you are after something else than the parallelogram identity? –  Alain Valette Oct 23 '11 at 10:13
@ Alain : You're right, I shouldn't have used the word "characterized", but "what is the typical things you can do in euclidian affine spaces you can't a priori do in non euclidian one's". You again provide an element of answer by mentioning projections over closed convex spaces. –  Adrien Hardy Oct 24 '11 at 11:23

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