[Paper Review] C*-bundle dynamical systems

Rachel A. D. Martins|arXiv (Cornell University)|Feb 5, 2014

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper introduces C*-bundle dynamical systems as a framework for studying C*-subalgebras within Fell bundles, linking spatial automorphisms of C*-bundles to 1-parameter groups of diffeomorphisms on the base manifold. It constructs an embedding invariant equivalent to a groupoid 2-cocycle, presenting it in a form analogous to a partition function—offering a novel algebraic structure with potential applications to quantum gravity and non-commutative geometry.

ABSTRACT

C*-bundle dynamical systems are introduced and their rôle within the theory of C*-subalgebras and Fell bundles is investigated. A C*-bundle dynamical system involves an action of a 1-parameter group of "spatial automorphisms" of the C*-bundle together with a notion of covariance with respect to the diffeomorphisms of the base manifold, and turns out to define a class of examples of Arveson's $A$-dynamical systems. An embedding invariant for non-commutative C*-subalgebras (equivalent to a groupoid 2-cocycle by construction) emerges and has presentation analogous to a partition function towards a potential algebraic formulation of quantum gravity.

Motivation & Objective

• To develop a dynamical systems framework for C*-bundles that captures spatial automorphisms and base manifold diffeomorphisms.
• To characterize embeddings of non-commutative C*-subalgebras within larger C*-algebras using a novel invariant.
• To establish connections between C*-bundle dynamics, Fell bundles, and Arveson’s A-dynamical systems.
• To explore physical applications in quantum gravity, algebraic quantum field theory, and spectral geometry.
• To provide a bridge between principal groupoid Fell bundles and diagonal C*-subalgebra structures via dynamical systems.

Proposed method

• Introduces C*-bundle dynamical systems via a 1-parameter group of spatial automorphisms acting on a C*-bundle over a locally compact base space.
• Defines covariance with respect to diffeomorphisms of the base manifold, linking the dynamics to geometric flows.
• Constructs an embedding invariant Φ→ as a finite linear combination of projections and isometries derived from the Fell bundle structure.
• Shows that Φ→ is equivalent to a groupoid 2-cocycle ω: Gep × Gep → ⊕i=1^n ℂ, providing a cohomological characterization of the embedding.
• Uses the Hilbert space L²(E) to define a faithful conditional expectation P: B → A, with P(b) = ∫αg(b)dx when a unique probability measure exists.
• Relates the dynamics to unitary representations of the groupoid and to Kumjian’s normalizers for C*-subalgebra embeddings.

Experimental results

Research questions

• RQ1 How can C*-bundle dynamical systems be defined to incorporate both spatial automorphisms and base manifold diffeomorphisms?
• RQ2 What is the precise relationship between the embedding invariant Φ→ and the groupoid 2-cocycle ω in the context of Fell bundles?
• RQ3 In what way does the invariant Φ→ resemble a partition function, and how might this suggest an algebraic formulation of quantum gravity?
• RQ4 How do C*-bundle dynamical systems serve as an enveloping structure between a Fell bundle (E,π,G) and a Cartan pair (A,B) with A a diagonal subalgebra?
• RQ5 What are the implications of this framework for Poincaré covariance in algebraic quantum field theory and for Dirac operators in spectral triples?

Key findings

• The embedding invariant Φ→ is constructed as a sum over finite combinations of projections and isometries associated with the Fell bundle’s structure, specifically Φ→ = {∑Y px u(p_x,p_y) py | px,py ∈ Y ⊂ {e_xy, e_xy e_xy*} ≅ G_ep }
• Φ→ is shown to be equivalent to a groupoid 2-cocycle ω: G_ep × G_ep → ⊕i=1^n ℂ, providing a cohomological classification of the C*-subalgebra embedding.
• When E⁰ is a locally trivial Banach bundle, the pure state space admits a unique probability measure, ensuring the conditional expectation P: B → A is unique and given by P(b) = ∫αg(b)dx.
• In finite dimensions, the conditional expectation simplifies to P(b) = ∑i p_i b p_i, and P identifies with the restriction map C*(E) → C*(E⁰).
• The system realizes a C*-dynamical system that is equivalent to Arveson’s A-dynamical system, with the 1-parameter group of automorphisms lifting from diffeomorphisms of the base manifold.
• The construction provides a bridge between Fell bundles over essentially principal groupoids and diagonal C*-subalgebras, with A = C*(E⁰) being a C*-diagonal in B = C*(E) when the groupoid is principal.