© 2018, Hebrew University of Jerusalem. We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.
Alpay, D., Luna-Elizarrarás, M. E., Shapiro, M., & Struppa, D. (2018). Gleason’s problem, rational functions and spaces of left-regular functions: The split-quaternion setting. Israel Journal of Mathematics, 319-349. https://doi.org/10.1007/s11856-018-1696-y