  I review the various algebraic foundations of quantum mechanics. They have
been suggested since the birth of this theory till up to last year. They are
the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930),
von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi
(2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the
misinterpretation of Dirac foundation; 2) von Neumann conversion from the
analytic approach of Hilbert space to the algebraic approach of the rings of
operators; 3) the recent foundation of quantum mechanics upon the algebra of
perturbation Lagrangians. Moreover, historical considerations on the
go-and-stop path performed by the algebraic approach in the history of QM are
offered. The level of formalism has increased from the mere introduction of
matrices till up to group theory and C*-algebras. But there was no progress in
approaching closer the foundations of physics; therefore the problem of
discovering an algebraic formulation of QM organized as a problem-based theory
and making use of no more than constructive mathematics is open.
