J. Kiefer proved a peripherally related convexity result in 1959, which is a corollary of an operator Schwarz inequality proved by E.H.Lieb and M.B.Ruskai. However, these results are comparatively simple, and the proofs do not use the results of Lieb's 1973 paper on convex and concave trace functionals. It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai. The extension from a Hilbert space setting to a von Neumann algebra setting, where states are not given by density matrices, was done by Narnhofer and Thirring .