Left ideal structure of C∗-algebras

Simultaneous Lifting From Irreducible Representations of $C^*$ -Algebras

“…Introduction. The starting point for this investigation is a result of Akemann [1,II.10] that strengthened an earlier result of Tomiyama [10, 4.2.5] concerning simultaneous lifting from irreducible representations. This states that if (π n ) n≥1 is a sequence of distinct elements in the spectrum A of a liminal C * -algebra A, if (π n ) has no cluster points and if (b n ) is a null sequence with b n ∈ π n (A), for all n, then there exists a ∈ A such that π n (a) = b n , for all n ≥ 1.…”

“…In Theorem 1, we give the following sufficient condition on (π n ) n≥1 for the existence of a ∈ A and (V n ) n≥1 satisfying (1) and (2):…”

Unbounded Disjointness Preserving Linear Functionals

“…The above proof was designed with the hope of being relatively accessible to non-specialists in C*-algebras. A more conceptual proof uses closed left ideals and their associated closed projections and the"Urysohn Lemma" of C. A. Akemann [6]. This latter method can also be used for real C*-algebras, but the conclusion has to be modified in the real case.…”

On peak phenomena for non-commutative H ∞

“…By [16,Lemma 3.6], a peaks at p and moreover (a * a) n ց p in σ(M ⋆⋆ , M ⋆ ) as n → ∞ so that p is a closed projection in the sense of Akemann [1], [2]. For any positive ψ ∈ M ⋆ one has…”