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This paper presents some elements of a new approach to construction ofBr\`{e}gman relative entropies over nonreflexive Banach spaces, based onnonlinear mappings into reflexive Banach spaces. We apply it to derive a newfamily of Br\`{e}gman relative entropies over preduals of any W$^*$-algebrasand of semifinite JBW-algebras, induced using the Mazur maps into correspondingnoncommutative and nonassociative $L_p$ spaces. We prove generalisedpythagorean theorem and norm-to-norm continuity of the corresponding entropicprojections, as well as H\"{o}lder continuity of the nonassociative Mazur mapon positive parts of unit balls. We also discuss the possibility of extensionof these results to base normed spaces in spectral duality, pointing to an openproblem of construction of $L_p$ spaces over the corresponding order unitspaces.

New approach to quantum and nonassociative Br\`{e}gman relative entropies