Abstract
Coherent superposability of pure states is one characteristic feature of traditional quantum mechanics. It is one of those points, which forces the departure from classical physics and the use of Hilbert space quantum mechanics. If one want to use the algebraic approach to quantum mechanics, two problems arise. The first one is how to formulate superposability of states if there is no Hilbert space, and the second one is how to handle mixed states, which are the only states relevant to certain thermodynamic representations.
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