When Dirac first formalizedquantum theory, he posited that the ultimate state of reality was a vector belongingto 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 endowedwith the added structure of an algebra) on which a conjugation is an involution extending the conjugation on thescalar field (using the same postfixed star "*" notation):
