Extensions, Restrictions, and Representations of States on C ∗ - Algebras

On factor states of 𝐶*-algebras and their extensions

“…Further, if B is abelian, then every pure state of B extends uniquely to a pure state of M if and only if B = Cp 1 ⊕ Cp 2 ⊕ · · · ⊕ Cp n ⊕ · · ·, where each p n is a minimal projection in M and the sum is a c 0 -sum and may be finite. This latter result complements the results of Kadison and Singer [14], Anderson [2], and Archbold, Bunce and Gregson [7]. Other extension result also leads us to study a class of C * -algebras whose states are σ-convex sums of factor states.…”

“…[9; p.104] and see also [1]). Using this interesting result, we obtain some new information concerning the unique extension of (factor) states of C * -algebras which complements the existing results in [2,4,7,14,15]. Extensions of factor states have been investigated by many authors (e.g., [3,6,17,20,21,23,25]).…”

“…We have the following immediate corollary which has been obtained by Anderson [2] under the assumption that A is unital and with a different method. (i) Every ϕ ∈ P (B) ∪ {0} extends uniquely to a ϕ ∈ P (A) ∪ {0}; (ii) There is a contractive projection Q : A → B such that ϕ = ϕ • Q for every ϕ ∈ P (B) ∪ {0} with extension ϕ on A.…”

“…In the following, we will show various necessary conditions for the unique extension of factor states. If B is abelian, the question of unique factor state extension is equivalent to that of unique pure state extension which has been considered in [2,4,7,14]; nevertheless, we are able to describe B with such unique extension explicitly if it is separable and if A is a von Neumann algebra. We begin with a simple but useful lemma.…”

Dynamical systems and extensions of states on 𝐶*-algebras

“…Furthermore, it turns out that the notion of dynamical systems is intimately related to the problem of extension of pure states of R. V. Kadison and I. M. Singer (cf. [10] and also [1,14]): let A be a C*-algebra, B be a sub-C*-algebra of A. Following J. Anderson's definition (cf.…”