What do we get from an euclidian affine structure ?

Question

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 ?

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 http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

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 http://aimsciences.org/journals/pdfsnews.jsp?paperID=2458&mode=full 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.