Mathematical Foundations of Quantum Mechanics 183The right-hand side of the above identity is calledspectral radiusofa.Ifais
not normal,a∗ais selfadjoint and thus normal in any cases. Therefore theC∗-
property of the norm||a||^2 =||a∗a||permits us to write down||a||in terms of the
spectrum ofa∗a. As the spectrum is a completely algebraic property, we conclude
that it is impossible to change the norm of aC∗-algebra preserving theC∗-algebra
property of the new norm. A unital∗-algebra admits at most oneC∗-norm.
(d) Unital C∗-algebras are very rigid structures. In particular, every ∗-
homomorphismπ:A→B(which is a pure algebraic notion) between two unital
C∗-algebras is necessarily[ 5 ]norm decreasing (||π(a)|| ≤ ||a||) thus continuous.
Its image,π(A),isaC∗-subalgebra ofB.Finallyπis injective if and ony if it
is isometric. The spectra satisfy a certain permanence property[ 5 ], with obvious
meaning of the symbols
σB(π(a)) =σπ(A)(a)⊂σA(a), ∀a∈A,where the last inclusion becomes and equality ifπis injective.
The most evidenta posteriori justification of the algebraic approach lies in its
powerfulness[ 26 ]. However there have been a host of attempts to account for
assumptions(AA1)and(AA2)and their physical meaning in full generality (see
the study of[ 32 ],[ 27 ]and[25; 29]and especially the work of I. E. Segal[ 33 ]based
on so-calledJordan algebras). Yet none seems to be definitive[ 34 ].
An evident difference with respect to the standard QM, where states are mea-
sures on the lattice of elementary propositions, is that we have now a complete
identification of the notion of state with that of expectation value. This identifi-
cation would be natural within the Hilbert space formulation, where the class of
observables includes the elementary ones, represented by orthogonal projectors,
and corresponding to “Yes-No” statements. The expectation value of such an ob-
servable coincides with the probability that the outcome of the measurement is
“Yes”. The set of all those probabilities defines, in fact, a quantum state of the
system as we know. However, the analogues of these elementary propositions gen-
erally do not belong to theC∗-algebra of observables in the algebraic formulation.
Nevertheless, this is not an insurmountable obstruction. Referring to a completely
general physical system and following[ 27 ], the most general notion of state,ω,is
the assignment of all probabilities,w(ωA)(a), that the outcome of the measurement
of the observableAisa, for all observablesAand all of valuesa. On the other
hand, it is known[ 25 ]that all experimental information on the measurement of an
observableAin the stateω– the probabilitieswω(A)(a) in particular – is recorded
in the expectation values of the polynomials ofA. Here, we should think ofp(A)
as the observable whose values are the valuesp(a) for all valuesaofA. This char-
acterization of an observable is theoretically supported by the various solutions