Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras

In this paper, we introduce a notion of geometric Banach property (T) for metric spaces, which jointly generalizes Banach property (T) for groups and geometric property (T) for metric spaces. Our framework is achieved by Banach representations of Roe algebras of metric spaces. We show that geometric Banach property (T) is a coarse geometric invariant, and it is equivalent to the existence of the Kazhdan projections in the Banach-Roe algebras. Further, we study the implications of this property for sequences of finite Cayley graphs, establishing two key results: 1. geometric Banach property (T) of such sequences implies Banach property (T) for their limit groups; 2. while the Banach coarse fixed point property implies geometric Banach property (T), the converse fails. Additionally, we provide a geometric characterization of V. Lafforgue's strong Banach property (T) for a residually finite group in terms of geometric Banach property (T) of its box spaces.