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On the Weak Contractibility of the Space of Pure States

Daniel D. Spiegel, Markus J. Pflaum

Abstract

We prove that the space $\mathscr{P}(\mathfrak{A})$ of pure states of a nonelementary, simple, separable, real rank zero $C^*$-algebra $\mathfrak{A}$ has trivial homotopy groups of all orders when $\mathscr{P}(\mathfrak{A})$ is equipped with the weak* topology. The convex-valued and finite-dimensional selection theorems of Michael are used to deform a family of pure states via the action of a homotopy of unitaries so that the entire family evaluates to one on a given projection $P \in \mathfrak{A}$. Then, the excision theorem of Akemann, Anderson, and Pedersen is used to iterate this deformation for a sequence of projections in $\mathfrak{A}$ excising a base point of the family of pure states, thereby contracting the family to the base point. Finally, we compare our weak contractibility result to the spaces of pure states of commutative $C^*$-algebras and rational rotation algebras, and compute the homotopy groups of the latter in terms of the homotopy groups of spheres.

On the Weak Contractibility of the Space of Pure States

Abstract

We prove that the space of pure states of a nonelementary, simple, separable, real rank zero -algebra has trivial homotopy groups of all orders when is equipped with the weak* topology. The convex-valued and finite-dimensional selection theorems of Michael are used to deform a family of pure states via the action of a homotopy of unitaries so that the entire family evaluates to one on a given projection . Then, the excision theorem of Akemann, Anderson, and Pedersen is used to iterate this deformation for a sequence of projections in excising a base point of the family of pure states, thereby contracting the family to the base point. Finally, we compare our weak contractibility result to the spaces of pure states of commutative -algebras and rational rotation algebras, and compute the homotopy groups of the latter in terms of the homotopy groups of spheres.
Paper Structure (7 sections, 29 theorems, 179 equations)

This paper contains 7 sections, 29 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Michael's Theory of Selections
  3. Continuous Families of Unitaries
  4. Proof of the Equi-LCn Condition
  5. Applying the Finite-Dimensional Selection Theorem
  6. Comparison with the Cases of Commutative and Rotation Algebras
  7. Contractibility of the Infinite-Dimensional Unit Sphere

Key Result

Theorem 2.2

If $X$ is paracompact Hausdorff, $Y$ is a (real or complex) Banach space, and $\phi:X \rightarrow \mathcal{P}(Y)\setminus \qty{\varnothing}$ is a lower semicontinuous carrier such that $\phi(x)$ is closed and convex for all $x \in X$, then there exists a selection for $\phi$.

Theorems & Definitions (57)

  • Definition 2.1
  • Theorem 2.2: MichaelSelection
  • Definition 2.3
  • Theorem 2.4: MichaelSelectionII
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 47 more