Non-commutativity and many-valuedness: The topological representation of the spectrum of C∗-algebras

In the past there have been made various attempts to define the spectrum of a non-commutative C∗-algebra. But all these definitions have certain drawbacks — e.g. C.J. Mulvey's definition does not coincide with the standard definition of the spectrum in the commutative case. The aim of this paper is to give an alternative definition of the spectrum which does not suffer under this deficit — i.e. coincides with the standard situation in the commutative setting. For this purpose we recall some properties of balanced and bisymmetric quantales, introduce a definition of the spectrum of a C∗-algebra working for the general case and develop subsequently its topological representation.