The calculus of one-sided 𝑀-ideals and multipliers in operator spaces

Abstract

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 =...

Operator spaces which are one-sided M-ideals in their bidual

“…Hence by Proposition 3.3, T = S * * , for some complete surjective isometry on X. So, 2P = T + Id X * * = (S + Id X ) * * , and since by [BZ2,Section 5.3…”

“…Then by [BEZ,Theorem 5.1], P ∈ Ball(M ℓ (X (4) )) and Q ∈ Ball(M r (X (4) )), which implies that P Q = QP . Hence by [BZ2,Theorem 5.30 (ii)], J ∩X is a right M -ideal in X * * . But J ∩X ⊂ X, so by [BZ2,Theorem 5.3], J ∩ X is a right M -ideal in X.…”

“…Hence by [BZ2,Theorem 5.30 (ii)], J ∩X is a right M -ideal in X * * . But J ∩X ⊂ X, so by [BZ2,Theorem 5.3], J ∩ X is a right M -ideal in X. Hence by (i), J ∩ X is a right M -summand.…”

“…1) X is completely M -embedded if and only if X is a complete M -ideal (in the sense of [ER1]) in its bidual (see e.g. [BEZ,Lemma 3.1] and [BZ2,Chapter 7]). …”

“…Let X be a right M-summand in X * * . Since X * * is a dual operator space, X is w * -closed, see [BZ2,p.8]. So X = XProof.…”