When Dirac first formalized
quantum theory,
he posited that the ultimate state of reality was a vector belonging
to an abstract ad hoc Hilbert space called the space of
kets (or, equivalently, the space of the
bra covectors).
However, a ket isn't directly accessible. All we can do is apply to it an
operator associated to an observable physical quantity.
Doing so transforms the ket into an eigenvector of that operator,
whose associated eigenvalue is construed to be the result of a measurement
(it's always a real quantity if we only use hermitian operators,
henceforth called observables, for short).
The original motivation was to understand how
hermitian operators (quantum observables)
act on a system composed of several subsytems (call that entanglement if you must).
By definition, a C* algebra (pronounced "C star") is a Banach algebra
(i.e., a Banach space endowed
with the added structure of an algebra)
on which a conjugation is an involution extending the conjugation on the
scalar field (using the same postfixed star "*" notation):
