Strengthening of Banach property (T) and applications

Property (T) was introduced by Kazhdan in 1967 as a way to establish compact generation of groups and since then was found useful for many other applications such as group cohomology and expander graphs. I will introduce a new notion of property (T) in Banach spaces, present a criterion for the fulfillment of this property and discuss its applications to the construction of Banach expanders and to group fix point properties in Banach spaces.