[Paper Review] One-Sided Projections on C*-algebras

David P. Blecher, Roger R. Smith+1 more|ArXiv.org|Mar 7, 2002

Advanced Operator Algebra Research20 references8 citations

TL;DR

This paper establishes equivalent characterizations of complete left M-projections on C*-algebras, showing they are precisely left multiplication by orthogonal projections in the multiplier algebra. The key contribution is proving that in C*-algebras, classical M-projection properties (like isometric extension to matrix amplifications) fully characterize such maps, and that preduals of von Neumann algebras admit no nontrivial complete one-sided M-ideals.

ABSTRACT

In [BEZ] the notion of a complete one-sided M-ideal for an operator space X was introduced as a generalization of Alfsen and Effros' notion of an M-ideal for a Banach space [AE72]. In particular, various equivalent formulations of complete one-sided M-projections were given. In this paper, some sharper equivalent formulations are given in the special situation that $X = \mathcal{A}$, a $C^*$-algebra (in which case the complete left M-projections are simply left multiplication on $\mathcal{A}$ by a fixed orthogonal projection in $\mathcal{A}$ or its multiplier algebra). The proof of the first equivalence makes use of a technique which is of interest in its own right--a way of ``solving'' multi-linear equations in von Neumann algebras. This technique is also applied to show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals. In addition, we show that in a $C^*$-algebra, the intersection of finitely many complete one-sided M-summands need not be a complete one-sided M-summand, unlike the classical situation.

Motivation & Objective

• To characterize complete left M-projections on C*-algebras using multiple equivalent conditions.
• To extend the classical notion of M-projections to the noncommutative operator space setting via complete isometry and contractivity.
• To show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals.
• To demonstrate that the intersection of finitely many complete one-sided M-summands in a C*-algebra need not be a complete one-sided M-summand, unlike the classical case.
• To clarify the relationship between left multipliers and complete left M-projections in C*-algebras using a novel technique for solving multilinear equations in von Neumann algebras.

Proposed method

• Established equivalence between six conditions characterizing complete left M-projections on a C*-algebra A, including isometric embedding into C₂(A) and contractivity of matrix extensions.
• Introduced a novel technique for solving multilinear operator equations in von Neumann algebras, used to prove the equivalence of conditions and to analyze M-ideals.
• Applied the solution technique to show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals.
• Used a counterexample construction involving projections on a Hilbert space to show that the intersection of two complete right M-summands need not be a complete right M-summand.
• Constructed a unital C*-algebra A ⊂ B(H) generated by projections P, Q, and rank-one projections Eₙ, where P ∧ Q ∉ A, to demonstrate failure of the classical M-summand intersection property.
• Leveraged the Paulsen system construction to embed general operator spaces into non-unital operator systems, enabling application of Werner’s earlier results on left multipliers.

Experimental results

Research questions

• RQ1 What are the complete set of equivalent conditions characterizing a complete left M-projection on a C*-algebra?
• RQ2 Can the classical characterization of M-projections via ∥x∥ = max{∥Px∥, ∥x−Px∥} be generalized to the noncommutative, completely isometric setting?
• RQ3 Do preduals of von Neumann algebras admit nontrivial complete one-sided M-ideals?
• RQ4 Does the intersection of finitely many complete one-sided M-summands in a C*-algebra remain a complete one-sided M-summand?
• RQ5 How do left multipliers on operator spaces relate to complete left M-projections in the special case of C*-algebras?

Key findings

• A complete left M-projection on a C*-algebra A is equivalent to left multiplication by an orthogonal projection e ∈ M(A), the multiplier algebra of A.
• The map x ↦ [Px; x−Px] is completely isometric from A to C₂(A) if and only if P is a complete left M-projection.
• The map [x; y] ↦ [Px; y] on C₂(A) is completely contractive if and only if P is a complete left M-projection.
• Preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals, as shown via the solution technique for multilinear equations.
• The intersection of two complete right M-summands in a C*-algebra need not be a complete right M-summand, even if the projections commute.
• There exists a unital C*-algebra A and projections P, Q ∈ A such that P ∧ Q ∉ A, and (P A) ∩ (Q A) ≠ r A for any projection r ∈ A, showing failure of the classical M-summand intersection property.