Abstract. The context poset of Abelian $C^\ast $-subalgebras of a given $C^\ast $-algebra is an operator theoretic invariant of growing interest. We review recent results describing order isomorphisms between context posets in terms of Jordan type maps (linear or not) between important types of operator algebras. We discuss the important role of the generalized Gleason theorem on linearity of maps preserving linear combinations of commuting elements for studying symmetries of context posets. Related results on maps multiplicative with respect to commuting elements are investigated.

DOI: 10.14232/actasm-017-582-8

AMS Subject Classification (1991): 46L40

Keyword(s): context posets, piecewise Jordan maps, Mackey-Gleason problem

Received December 9, 2017, and in revised form January 8, 2018. (Registered under 82/2017.)