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Only three suitable division rings exist: the real numbers, the complex numbers and the quaternions. The fist two are contained in the last one. Thus the most elaborate separable Hilbert space is a quaternionic Hilbert space. “Division algebras and quantum theory” by John Baez. http://arxiv.org/abs/1101.5690

Also most scientist do not notice what separable stands for. It means that eigenspaces of operators can only contain a countable number of eigenvalues. For example operators whose eigenspaces contain all rational numbers may exist, but operators whose eigenspaces contain all (or a closed set of) real numbers can only exist in a non-separable Hilbert space, such as a Gelfand triple. By the way, each infinite dimensional separable Hilbert space owns a Gelfand triple.

Another fact that hardly anyone knows is that quaternionic number systems, coherent sets of quaternionic numbers and continuous quaternionic functions exits in 16 versions that only differ in their discrete symmetry sets. This is due to the four dimensions of quaternions. For example quaternionic number systems exist in left handed and right handed versions.

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