The mathematical formulation of Quantum Mechanics in terms of complex Hilbert
space is derived for finite dimensions, starting from a general definition of
"physical experiment" and from five simple Postulates concerning "experimental
accessibility and simplicity". For the infinite dimensional case, on the other
hand, a C*-algebra representation of physical transformations is derived,
starting from just four of the five Postulates via a Gelfand-Naimark-Segal
(GNS) construction. The present paper simplifies and sharpens the previous
derivation in version 1. The main ingredient of the axiomatization is the
postulated existence of "faithful states" that allows one to calibrate the
experimental apparatus. Such notion is at the basis of the operational
definitions of the scalar product and of the "transposed" of a physical
transformation. What is new in the present paper with respect to
quant-ph/0603011 is the operational deduction of an involution corresponding to
the "complex-conjugation" for effects, whose extension to transformations
allows to define the "adjoint" of a transformation when the extension is
composition-preserving.Comment: New improvements have been made. Work presented at the conference
"Foundations of Probability and Physics-4, Quantum Theory: Reconsideration of
Foundations-3" held on 4-9 June at the International Centre for Mathematical
Modelling in Physics, Engineering and Cognitive Sciences, Vaxjo University,
Sweden. Also contains an errata to "How to Derive the Hilbert-Space
Formulation of Quantum Mechanics From Purely Operational Axioms",
quant-ph/060301
