Perfect 𝐶*-algebras

On perfect 𝐶*-algebras

“…This is easy to see directly [5,6] or may be obtained from the main theorem of [14] which states that for a unital simple C*-algebra the Dixmier property is implied by the existence of at most one tracial state. Coupling this fact with (1) above we see that for x G Ac (2) cön{uxu*\u unitary in zA} n Cz ^ 0.…”

“…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.…”

A Gelfand–Naimark theorem for C*-algebras

“…Akemann and F.W. 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.…”

“…[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.…”

“…Akemann and F.W. Shultz [3] using equivariant functions on irreducible representations (or the closure of irreducible representations) based on Takesaki's duality theorem (cf. [11], [39]).…”