The theory of M -ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M -ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A-B-bimodule for C * -algebras A and B, then the module operations on X are automatically weak * continuous. One sided L-projections are introduced, and analogues of various results from the classical theory are proved. An assortment of examples is considered.

The theory of one-sided M -ideals and multipliers of operator spaces is simultaneously a generalization of classical M -ideals, ideals in operator algebras, and aspects of the theory of Hilbert C * -modules and their maps. Here we give a systematic exposition of this theory; a reference tool for 'noncommutative functional analysts' who may encounter a one-sided M -ideal or multiplier in their work.Since ran(|T | 1/n ) ⊂ ran(|T |), we have ran(W |T | 1/n ) ⊂ ran(T ), for all n ∈ N. Thus ran(W W ⋆ ) ⊂ ran(T ) wk* .Now suppose that T ⋆ x = 0. Then |T |W ⋆ x = 0, which implies that |T | 1/n W ⋆ x = 0 for all n ∈ N. Therefore, W W ⋆ x = 0. Conversely, suppose that W W ⋆ x = 0. Then W ⋆ x = 0. Thus, T ⋆ x = |T |W ⋆ x = 0. Hence, ker(W W ⋆ ) = ker(T ⋆ ). By Proposition 3.2.1 again, ran(W ) = ran(T ) wk* and ker(W ⋆ ) = ker(T ⋆ ).Corollary 3.5.7. Let J be a linear subspace of a dual operator space X. Then the following are equivalent: (i) J is a right M -summand of X.(ii) J = ran(T ) wk* for some T ∈ A ℓ (X).(iii) J = ker(T ) for some T ∈ A ℓ (X).Corollary 3.5.8. Let X be an operator space and T ∈ A ℓ (X). Then ran(T ) is a right M -ideal of X.Proof. By basic functional analysis we have ran(T ) ⊥⊥ = ker(T * ) ⊥ = ( ⊥ ran(T * * )) ⊥ = ran(T * * ) wk* .We shall see in Section 5.3 that T * * ∈ A ℓ (X * * ). Thus the result follows from Corollary 3.5.7.Dual to Theorem 3.5.6 we have the following result:Corollary 3.5.9. Let X be an operator space and T ∈ C r (X). Then T = W |T |, where W ∈ C r (X) is a partial isometry such that ran(W ⋆ ) = ran(|T |), ran(W ) = ran(T ), ker(W ) = ker(|T |), and ker(W ⋆ ) = ker(T ⋆ ).In particular, X = ran(T ) ⊕ ℓL ker(T ⋆ ).The proof is essentially the same as that of Theorem 3.5.6.Corollary 3.5.10. Let J be a linear subspace of an operator space X. Then the following are equivalent:One question remains here: Is the intersection of two right M -ideals of X * again a right M -ideal of X * ? We doubt it, but no counter-example comes to mind. 5.5. Algebraic Direct Sum. We turn to another item in the classical M -ideal 'calculus'. Namely, if X is a Banach space, and J and K are M -ideals of X such that X = J ⊕ K (internal algebraic direct sum), then J and K are in fact complementary M -summands, i.e. X = J ⊕ ∞ K (see [Beh1], Proposition 2.8). Furthermore, an M -ideal J of a Banach space X is an M -summand of X if and only if there exists an M -ideal K of X such that X = J ⊕ K (same reference). The corresponding statements are not true for operator spaces and one-sided M -ideals. Example 5.5.1. Let A = C[0, 1] and I = {h ∈ C[0, 1] : h(0) = 0}. Define X = A ⊕ I (external direct sum). Then X is a left Hilbert C * -module over A. Set J = {(h, h) : h ∈ I} and K = {(f, 0) : f ∈ A}.Then J and K are closed left submodules (i.e. left M -ideals) of X such that X = J ⊕K (internal direct sum). However, J is not orthogonally complemented (i.e. not a left M -summand). Indeed, if (f, g), (h, h) = 0 for all h ∈ I, then (f + g)h = 0 for all h ∈ I. This yields f + g = 0 on (0, 1], which in turn yields that f + g =...

Given an inclusion D ⊆ C of unital C * -algebras (with common unit), a unital completely positive linear map Φ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C, D) of all pseudoexpectations is a convex set, and when D is abelian, we prove a Krein-Milman type theorem showing that PsExp(C, D) can be recovered from its set of extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D.In general, PsExp(C, D) is not a singleton. However there are large and natural classes of inclusions (e.g., when D is a regular MASA in C) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For example, we show that when D ⊆ C ⊆ B(H) are von Neumann algebras, uniqueness of the pseudo-expectation implies that D ′ ∩ C is the center of D; moreover, when H is separable and D is abelian, we are able to characterize which von Neumann algebra inclusions have the unique pseudo-expectation property.For general inclusions of C * -algebras with D abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property.As applications, we show that if an inclusion D ⊆ C has a unique pseudo-expectation Φ which is also faithful, then the C * -envelope of any operator space X with D ⊆ X ⊆ C is the C * -subalgebra of C generated by X; we also show that for many interesting classes of C * -inclusions, having a faithful unique pseudo-expectation implies that D norms C, although this is not true in general. We provide a number of examples to illustrate the theory, and conclude with several unresolved questions.2010 Mathematics Subject Classification. Primary: 46L05, 46L07, 46L10 Secondary: 46M10.

Let G be a discrete group acting on a unital C * -algebra A by * -automorphisms. In this note, we show that the inclusion A ⊆ A ⋊rG has the pure extension property (so that every pure state on A extends uniquely to a pure state on A ⋊rG) if and only if G acts freely on A, the spectrum of A. The same characterization holds for the inclusion A ⊆ A ⋊G. This generalizes what was already known for A abelian.2010 Mathematics Subject Classification. Primary: 47L65, 46L55, 46L30 Secondary: 46L07.

The one-sided multipliers of an operator space X are a key to ''latent operator algebraic structure'' in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals.