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.

Abstract. This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from 'noncommutative functional analysis', more specifically the new field of operator spaces. In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.

Let p be a polynomial in the non-commuting variables (a, x) = (a 1 , . . . , a ga , x 1 , . . . , x gx ). If p is convex in the variables x, then p has degree two in x and moreover, p has the formwhere L has degree at most one in x and Λ is a (column) vector which is linear in x, so that Λ T Λ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the x and a variables separately are presented.

Abstract. We compute the monoid V[L K (E)] of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno, and Pardo in which they computed the monoid V[L K (E)] of a Leavitt path algebra over a countable row-finite directed graph.

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