Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction

Abstract

We consider how the outputs of the Kadison transitivity theorem and Gelfand-Naimark-Segal construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation $(\mathcal{H}, π)$ of a $C^*$-algebra $\mathfrak{A}$ and $n \in \mathbb{N}$, there exists a continuous function $A:X \rightarrow \mathfrak{A}$ such that $π(A(\mathbf{x}, \mathbf{y}))x_i = y_i$ for all $i \in \{1, \ldots, n\}$, where $X$ is the set of pairs of $n$-tuples $(\mathbf{x}, \mathbf{y}) \in \mathcal{H}^n \times \mathcal{H}^n$ such that the components of $\mathbf{x}$ are linearly independent. Versions of this result where $A$ maps into the self-adjoint or unitary elements of $\mathfrak{A}$ are also presented. Regarding the Gelfand-Naimark-Segal construction, we prove that given a topological $C^*$-algebra fiber bundle $p:\mathfrak{A} \rightarrow Y$, one may construct a topological fiber bundle $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ whose fiber over $y \in Y$ is the space of pure states of $\mathfrak{A}_y$ (with the norm topology), as well as bundles $\mathscr{H} \rightarrow \mathscr{P}(\mathfrak{A})$ and $\mathscr{N} \rightarrow \mathscr{P}(\mathfrak{A})$ whose fibers $\mathscr{H}_ω$ and $\mathscr{N}_ω$ over $ω\in \mathscr{P}(\mathfrak{A})$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $ω$. When $p:\mathfrak{A} \rightarrow Y$ is a smooth fiber bundle, we show that $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ and $\mathscr{H}\rightarrow \mathscr{P}(\mathfrak{A})$ are also smooth fiber bundles; this involves proving that the group of $*$-automorphisms of a $C^*$-algebra is a Banach-Lie group. In service of these results, we review the geometry of the topology and pure state space. A simple non-interacting quantum spin system is provided as an example.