Differences of idempotents in $C^*$-algebras and the~quantum Hall effect

A. M. Bikchentaev

Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 75-80

Résumé

Let $\varphi$ be a trace on the unital $C^*$-algebra $\mathcal{A}$ and $\mathfrak{M}_{\varphi}$ be the ideal of the definition of the trace $\varphi$. We obtain a $C^*$ analogue of the quantum Hall effect: if $P,Q\in\mathcal{A}$ are idempotents and $P-Q\in\mathfrak{M}_{\varphi}$, then $\varphi((P-Q)^{2n+1})=\varphi (P-Q)\in \mathbb{R}$ for all $n\in\mathbb{N}$. Let the isometries $U\in\mathcal{A}$ and $A=A^*\in\mathcal{A}$ be such that $I+A$ is invertible and $U-A\in\mathfrak{M}_{\varphi}$ with $\varphi (U-A)\in \mathbb{R}$. Then $I-A,\,I-U \in\mathfrak{M}_{\varphi}$ and $\varphi (I-U)\in \mathbb{R}$. Let $n\in\mathbb{N}$, $\dim \mathcal{H}=2n+1$, the symmetry operators $U,V\in\mathcal{B}(\mathcal{H})$, and $W=U-V$. Then the operator $W$ is not a symmetry, and if $V=V^*$, then the operator $W$ is nonunitary.

Détail

Keywords: Hilbert space, linear operator, idempotent, symmetry, projection, unitary operator, trace-class operator, $C^*$-algebra, trace, quantum Hall effect.