[Paper Review] Homogeneity of the pure state space for the separable nuclear C*-algebras

Akitaka Kishimoto, Shôichirô Sakai|arXiv (Cornell University)|Apr 27, 2001

Advanced Operator Algebra Research9 references3 citations

TL;DR

This paper establishes that the pure state space of any separable nuclear C*-algebra is homogeneous under the action of the group of asymptotically inner automorphisms. Using techniques from nuclearity theory and Kadison's transitivity, it proves that pure states with equivalent GNS representations can be connected via continuous paths of unitary conjugations, extending known results to all separable nuclear C*-algebras and showing strong transitivity for finite families of pure states with identical kernels.

ABSTRACT

We prove that the pure state space is homogeneous under the action of the group of asymptotically inner automorphisms for all the separable simple nuclear C*-algebras. If simplicity is not assumed for the C*-algebras, the set of pure states whose GNS representations are faithful is homogeneous for the above action.

Motivation & Objective

• To establish homogeneity of the pure state space under the action of asymptotically inner automorphisms for all separable nuclear C*-algebras.
• To extend known transitivity results—previously valid for specific classes of C*-algebras—to the full class of separable nuclear C*-algebras.
• To prove that the group of asymptotically inner automorphisms acts strongly transitively on pure states with identical kernels.
• To provide a new characterization of nuclearity linked to amenability via unitary conjugation of finite families of pure states.
• To generalize Lemma 3.1 to strengthen the transitivity result to finite sequences of mutually disjoint pure states with identical kernels.

Proposed method

• Leverages the characterization of nuclear C*-algebras via nets of completely positive contractions factoring through finite-dimensional C*-algebras, as established by Choi and Effros.
• Applies a technical lemma (Lemma 3.1) showing that nuclear C*-algebras admit unitary approximations in finite-dimensional quotients with controlled error.
• Uses the polar decomposition and unitary perturbation techniques to construct paths of unitaries that implement automorphisms approximating state conjugation.
• Employs Kadison’s transitivity theorem to mimic one pure state as a vector state in the GNS representation of another, enabling approximation of states via unitary conjugation.
• Constructs finite ε-nets in unitary groups of finite-dimensional C*-algebras and uses bijection arguments to match elements under unitary actions.
• Generalizes the core technical lemma to handle finite families of mutually disjoint irreducible representations, enabling strong transitivity for multiple pure states.

Experimental results

Research questions

• RQ1 Can the homogeneity of the pure state space under asymptotically inner automorphisms be extended from specific classes of C*-algebras to all separable nuclear C*-algebras?
• RQ2 Is the action of the group of asymptotically inner automorphisms on the pure state space strongly transitive for finite families of pure states with identical kernels?
• RQ3 Can the technical core of the proof—based on unitary approximation in finite-dimensional quotients—be generalized to handle multiple irreducible representations simultaneously?
• RQ4 Does the existence of such transitive actions provide a new characterization of nuclearity in terms of amenability and unitary conjugation?
• RQ5 Can the condition that a finite-rank projection lies in the commutant of the representation be relaxed without losing the approximation result?

Key findings

• The pure state space of any separable simple nuclear C*-algebra is homogeneous under the action of the group of asymptotically inner automorphisms.
• For general separable nuclear C*-algebras, the set of pure states whose GNS representations are faithful is homogeneous under the same action.
• The group of asymptotically inner automorphisms acts strongly transitively on finite families of pure states with identical kernels.
• A technical lemma (Lemma 3.1) establishes that nuclear C*-algebras admit unitary approximations in finite-dimensional quotients with error bounds depending on the finite set and tolerance.
• The proof constructs continuous paths of unitaries that implement automorphisms approximating state conjugation to within any given ε on any finite set.
• The main result is generalized to show that for any finite sequence of pure states with identical kernels, there exists an asymptotically inner automorphism mapping one sequence to the other.