  We develop a new approach to construction of Br\`{e}gman relative entropies
over nonreflexive Banach spaces, based on nonlinear mappings into reflexive
Banach spaces. We apply it to derive few families of Br\`{e}gman relative
entropies over several radially compact base normed spaces in spectral duality.
In particular, we prove generalised pythagorean theorem and norm-to-norm
continuity of the corresponding entropic projections for a family induced on
preduals of any W$^*$-algebras and of semifinite JBW-algebras using Mazur maps
into corresponding noncommutative and nonassociative $L_p$ spaces. We also
prove generalised pythagorean theorem for a family induced using Kaczmarz maps
into Orlicz spaces over semifinite W$^*$-algebras, and for a family over
generalised spin factors. Additionally, we establish Lipschitz--H\"{o}lder
continuity of the nonassociative Mazur map on positive parts of unit balls,
characterise several geometric properties of the Morse-Transue-Nakano-Luxemburg
norm on noncommutative Orlicz spaces, and introduce a new family of $L_p$
spaces over order unit spaces.
