Closed Projections and Peak Interpolation for Operator Algebras

Abstract

The closed one-sided ideals of a C * -algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C * -algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B * * which also lies in A ⊥⊥ . Although this seems quite natural, the proof requires a set of new techniques which may may be viewed as a noncommutative version of the subject of peak interpolation from the theory of function spaces. Thus, the right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction, we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces.

Open partial isometries and positivity in operator spaces

“…[12,Chapter 8] and references therein). For example, it has strong relations with the recent study of peak projections and peak tripotents [27,10,9]. In any case, our paper, like its predecessor, in some sense "marries" the notion of positivity of Hilbert space operators to ideas from the basic structure theory of JBW * -triples.…”

On the quest for positivity in operator algebras

“…The main unsolved question in the thesis of D.M. Hay [5] was whether this might be true; the result puts the theory of "noncommutative peak sets" on a much firmer foundation. A key observation in [2] was that the "noncommutative Glicksberg theorem" would follow if every algebra A as in Theorem 1.1 had a b.a.i.…”

On peak phenomena for non-commutative H ∞

“…More precisely, what we want to prove here is that for any non-zero singular ϕ ∈ M * in the sense of Takesaki [29] one can find a "peak" projection p for A in the sense of Hay [16] such that p dominates the (right) support projection of ϕ but is smaller than the central support projection z s ∈ M ⋆⋆ of the singular part M ⋆ ⊖ M ⋆ . This is not exactly same as Amar and Lederer's result, but is enough for usual applications (even in classical theory for H ∞ (D)).…”

“…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…”

“…[28,Theorem 9.8]). Thus, letting ξ : [16,Lemma 3.7] the new contraction a := (1 + b)/2 satisfies that a n converges to a certain projection p ∈ M ⋆⋆ in σ(M ⋆⋆ , M ⋆ ) as n → ∞, and p 0 ≤ p so that |ϕ|, p = |ϕ|(1). If a vector ξ ∈ H satisfies aξ 2,τ = ξ 2,τ , then 2 ξ 2,τ = ξ + bξ 2,τ ≤ ξ 2,τ + bξ 2,τ ≤ 2 ξ 2,τ , which implies bξ 2,τ = ξ 2,τ and ξ + bξ 2,τ = ξ 2,τ + bξ 2,τ .…”