Spectra of the Γ-invariant of uniform modules

For a ring R, denote by SpecΓ(κ,R) the κ-spectrum of the Γ-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that SpecΓ(aleph, Hebrew1,R) is full for a suitable von Neumann regular algebra R, but the techniques do not extend to cardinals κ>aleph, Hebrew1. By a direct construction, we prove that for any field F and any regular uncountable cardinal κ there is an F-algebra R such that SpecΓ(κ,R) is full. We also derive some consequences for the Γ-invariant of strongly dense lattices of two-sided ideals, and for the complexity of Ziegler spectra of infinite-dimensional algebras.