The trace formula in Banach spaces

A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ℓ2 sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.