[Paper Review] Inner coactions, Fell bundles, and abstract uniqueness theorems

S. Kaliszewski, Nadia S. Larsen+1 more|arXiv (Cornell University)|Aug 25, 2011

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper establishes abstract gauge-invariant uniqueness theorems for C*-algebras associated to Fell bundles and product systems of C*-correspondences using the framework of inner coactions and maximal/normal coactions. By proving that a C*-algebra homomorphism is injective if and only if its restriction to the unit fiber is injective and it intertwines coactions, the authors generalize classical uniqueness results and characterize injectivity in Nica's Toeplitz algebra under a finite lower bound condition for quasi-lattice ordered groups.

ABSTRACT

We prove gauge-invariant uniqueness theorems with respect to maximal and normal coactions for $C^*$-algebras associated to product systems of $C^*$-correspondences. Our techniques of proof are developed in the abstract context of Fell bundles. We employ inner coactions to prove an essential-inner uniqueness theorem for Fell bundles. As application, we characterise injectivity of homomorphisms on Nica's Toeplitz algebra $\Tt(G, P)$ of a quasi-lattice ordered group $(G, P)$ in the presence of a finite non-trivial set of lower bounds for all non-trivial elements in $P$.

Motivation & Objective

• To develop a general framework for gauge-invariant uniqueness theorems in the context of Fell bundles and coactions.
• To characterize injectivity of homomorphisms on Nica's Toeplitz algebra T(G,P) for quasi-lattice ordered groups (G,P) with finite lower bounds.
• To establish abstract uniqueness theorems for C*-algebras with maximal and normal coactions using the theory of inner coactions.
• To apply these results to Cuntz-Nica-Pimsner algebras and their Toeplitz-like extensions, refining known uniqueness properties.

Proposed method

• Employ the theory of Fell bundles to abstractly characterize C*-algebras associated with product systems of C*-correspondences.
• Use inner coactions to prove an essential-inner uniqueness theorem for Fell bundles, linking coaction structure to injectivity.
• Apply the normalization and maximalization functors in the category of coactions to reduce abstract uniqueness to coaction properties.
• Prove that a surjective homomorphism π:A→B is injective if and only if π|Aₑ is injective and π intertwines coactions δ and ε, depending on whether δ is maximal or ε is normal.
• Use the fact that the canonical coaction on T(G,P) is normal when (G,P) is amenable, and extend this to non-amenable cases via finite lower bound conditions.
• Leverage the equivalence of maximal coactions and the faithfulness of the normalization functor to derive uniqueness criteria.

Experimental results

Research questions

• RQ1 Under what conditions is a homomorphism from a C*-algebra with a maximal coaction to another C*-algebra injective?
• RQ2 When is a homomorphism from a C*-algebra with a normal coaction injective, given the intertwining of coactions?
• RQ3 How can the injectivity of representations on Nica's Toeplitz algebra T(G,P) be characterized for quasi-lattice ordered groups with finite lower bounds?
• RQ4 What is the role of inner coactions in establishing uniqueness theorems for Fell bundles and their associated C*-algebras?
• RQ5 How do the abstract uniqueness theorems for maximal and normal coactions refine known results for Cuntz-Nica-Pimsner algebras?

Key findings

• A surjective homomorphism π:A→B between C*-algebras is injective if and only if π|Aₑ is injective and there exists a coaction ε on B such that π is δ–ε equivariant, provided δ is maximal.
• A surjective homomorphism π:A→B is injective if and only if π|Aₑ is injective and there exists a coaction δ on A such that π is δ–ε equivariant, provided ε is normal.
• The canonical coaction on the Toeplitz algebra T(G,P) is normal when (G,P) is amenable, and this property extends to non-amenable cases under a finite lower bound condition.
• For a quasi-lattice ordered group (G,P) with a finite non-trivial set F of lower bounds for all non-trivial elements in P, injectivity of homomorphisms on T(G,P) is characterized by the injectivity of the restriction to the unit fiber and coaction intertwining.
• The Cuntz-Nica-Pimsner algebra NO_X admits a gauge-invariant uniqueness property if and only if its canonical coaction is normal, which is equivalent to the existence of a normal coaction on the target algebra.
• The abstract uniqueness theorems for maximal and normal coactions are proven via the faithfulness of the normalization and maximalization functors, establishing that a morphism is an isomorphism if and only if its normalization is the identity.