Bruhat decompositions of operator algebras
We introduce and study a notion of decomposition of a C$^*$-algebra over a Coxeter system based on Tits' definition of a $W$-distance. When such a decomposition comes with suitable conditional expectations, we build an associated Fock Hilbert module and reduced C$^*$-algebra $A^r$. This unifies constructions of Voiculescu [Voi85] and Caspers--Fima [CF17] and provides a noncommutative analogue of the situation of a discrete group $G$ acting on a building, in which case $A^r \cong C^*_r(G)$. We construct covariance C$^*$-algebras $\mathscr{C}(i)\supset A^r$ as a noncommutative analogue of the crossed product $C(\Omega)\rtimes_r G \supset C^*_r(G) $ where $\Omega$ is Caprace and L\'ecureux's minimal combinatorial compactification [CL11] of the locally finite building. Following the approach of Hasegawa [Has17] and Klisse [Kli25], we prove a universal property for the covariance algebras. This structural result yields different approximation properties, some of which are new even for the group case.