Pure states as a dual object for 𝐶*-algebras

A Gelfand–Naimark theorem for C*-algebras

“…Shultz [3] based on the results in F.W. Shultz [37], where the Hilbert space H was assumed to have a larger dimension than ours, such that every unitary equivalence of cyclic representations in H can be implemented by a unitary operator on H, and also the continuity of the equivariant functions was restricted to the strong* topology. We shall show that the theorem can be released from these constraints as follows.…”

“…[16,17] for the history of this theorem). Since then, there have been various attempts to generalize this beautiful representation theorem for non-commutative C*-algebras, i.e., to reconstruct the original algebra as a certain function space on some "generalized spectrum", or the duality theory in the broad sense, for which we can refer to e.g., [2], [3], [4], [11], [14], [19], [29], [32], [37], [39]. Among them, we cite the following notable directions which motivated our present work.…”

“…II.4.5]), and the duality arguments on the pure states for C*-algebras by F.W. Shultz [37] in the context of the Alfsen-Shultz theory, which was further developed by C.A. Akemann and F.W.…”

“…Then we can consider the CP-affine extension of continuous equivariant functions on the CP-extreme boundary, and as applications of this argument, we shall give a simple characterization of perfect C*-algebras (Proposition 3.2), the notion of which was initially introduced by F.W. Shultz [37] together with its Stone-Weierstrass property. Moreover, we can generalize the notion of semi-perfect C*-algebras to non-separable C*-algebras and prove its StoneWeierstrass property (Theorem 3.5), which was initially defined and proved by C.J.K.…”

On perfect 𝐶*-algebras

“…Then Ac is defined to be the set of those elements 6 G zA** such that b, b*b and bb* are continuous on P{A) U {0}. In fact, Ac is a C*-subalgebra of zA** [17,2]. The algebra A is said to be perfect if Ac -zA.…”