Geometric Property (T) and Positive Cones of Real Algebraic Roe Algebras

We give a characterization of geometric property (T) for a coarse disjoint union of finite graphs with bounded degree using the idea of noncommutative real algebraic geometry. In the proof, we define a $\ast$-subalgebra $I_u[X]$ of real algebraic Roe algebra ${\mathbb{R}}_u[X]$ over a graph $X$ with bounded degree. Then we show that $I_u[X]$ contains the Laplacian $\mathrm{\Delta }$ as an order unit with respect to the positive cone ${\sum }^2{I}_u[X]$ which consists of sums of hermitian squares.