Weak∗ continuous states on Banach algebras

Abstract

We prove that if a unital Banach algebra A is the dual of a Banach space A then the set of normal states is weak * dense in the set of all states on A. Further, normal states linearly span A .

Real positivity and approximate identities in Banach algebras

“…That is, ϕ(p 2 − p 1 ) ≥ 0 for all ϕ ∈ S(A 1 ). By [40] this is also true if ϕ ∈ S((A 1 ) * * ), and hence if ϕ ∈ S(∆). Therefore p 1 ≤ p 2 in ∆, so that indeed p 1 ≤ p 2 in the usual ordering of projections in A * * .…”

“…For the other, by the observation above the Proposition, we can assume that f : A → C is C-linear and real positive. If A is unital then the result follows from the proof of [40,Theorem 2.2]. Otherwise by Proposition 3.2 (4) applied to the inclusion A ⊂ A 1 we see that the condition in Corollary 2.8 (iii) holds.…”

“…Indeed the F A * * case of this is obvious. By [40], r (A 1 ) * * is weak* closed, hence so is r A * * = A * * ∩ r (A 1 ) * * .…”

“…ǫ + ǫ, as can be easily seen from the well known fact that the numerical range of a is contained in the right half plane iff 1 − ta ≤ 1 + t 2 a 2 for all t > 0 (see e.g. [40,Lemma 2.1]).…”

Generalization of 𝐶*-algebra methods via real positivity for operator and Banach algebras

“…Clearly (5) implies (2). That (2) implies (1) follows by applying a state ϕ to see |1 − tϕ(x)| ≤ 1 + Kt 2 , which forces Re ϕ(x) ≥ 0) (see [63,Lemma 2.1]). Given (4) with t replaced by 1 t , we have 1…”