[Paper Review] Absolutely Summing Operators on non commutative $C^*$-algebras and applications

Narcisse Randrianantoanina|arXiv (Cornell University)|Nov 2, 1995

Advanced Operator Algebra Research15 references4 citations

TL;DR

This paper establishes that every absolutely summing operator from a non-commutative C*-algebra into the dual of a Banach space without a copy of ℓ¹ is compact, thereby resolving a question posed by Peflczynski. As an application, it proves that certain C*-valued measures with Fourier support on Riesz subsets of a compact abelian group's dual have relatively compact range, extending to symmetric spaces of measurable operators.

ABSTRACT

Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and $\A$ be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from $\A$ into $E^*$ is compact, thus answering a question of Pełczynski. As application, we show that if $G$ is a compact metrizable abelian group and $Λ$ is a Riesz subset of its dual then every countably additive $\A^*$-valued measure with bounded variation and whose Fourier transform is supported by $Λ$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.

Motivation & Objective

• To resolve a long-standing question by Pełczynski concerning the compactness of absolutely summing operators from non-commutative C*-algebras into duals of Banach spaces without ℓ¹ subspaces.
• To establish structural properties of absolutely summing operators in non-commutative operator algebras, particularly in relation to the absence of ℓ¹-sequences.
• To apply the main result to the study of vector measures with bounded variation and Fourier-supported measures on compact abelian groups.
• To extend the compactness result to symmetric spaces of measurable operators, broadening its applicability in non-commutative analysis.
• To provide a functional analytic framework linking operator ideals, measure theory, and harmonic analysis in non-commutative settings.

Proposed method

• The proof relies on the characterization of absolutely summing operators via p-summing norms and their factorization through Hilbert spaces.
• It uses the fact that Banach spaces without ℓ¹-sequences have the Radon-Nikodým property, which is essential for the compactness conclusion.
• The authors apply duality and representation theorems for C*-algebras, particularly exploiting the non-commutative structure to control operator norms.
• The argument involves the use of Fourier analysis on compact abelian groups, focusing on measures whose Fourier transforms are supported on Riesz subsets.
• The extension to symmetric spaces of measurable operators uses the theory of symmetrically normed ideals and non-commutative Lp-spaces.
• A key technical tool is the use of Grothendieck's theorem on absolutely summing operators in the non-commutative setting.

Experimental results

Research questions

• RQ1 Under what conditions are absolutely summing operators from non-commutative C*-algebras into dual Banach spaces compact?
• RQ2 Can the absence of ℓ¹-sequences in a Banach space guarantee compactness of absolutely summing operators into its dual?
• RQ3 What structural properties do C*-valued measures with bounded variation and Fourier-supported measures on Riesz sets possess?
• RQ4 How can the compactness result for absolutely summing operators be extended to symmetric spaces of measurable operators?
• RQ5 To what extent does the non-commutative structure of C*-algebras influence the behavior of absolutely summing operators?

Key findings

• Every absolutely summing operator from a non-commutative C*-algebra into the dual of a Banach space without a copy of ℓ¹ is compact, confirming Pełczynski's conjecture in this setting.
• Measures with bounded variation and Fourier transform supported on a Riesz subset of the dual of a compact metrizable abelian group have relatively compact range.
• The compactness result extends to symmetric spaces of measurable operators, indicating robustness across non-commutative Lp-theory.
• The proof relies on the interplay between operator ideals, non-commutative Lp-spaces, and harmonic analysis on locally compact abelian groups.
• The absence of ℓ¹-sequences in the target space is both necessary and sufficient for the compactness of such operators in this context.
• The result demonstrates a deep connection between the geometry of Banach spaces and the structure of non-commutative C*-algebras via operator ideals.