On truncated $t$-free Fock spaces: spectrum of position operators and shift-invariant states

TL;DR

This paper investigates the ergodic properties of shift actions on truncated and full t-free C*-algebras, characterizes invariant states, and determines the spectrum of position operators on truncated t-free Fock spaces.

Contribution

It provides a detailed analysis of ergodic behavior, invariant states, and spectral properties in truncated t-free Fock spaces, extending understanding of their operator algebraic structure.

Findings

Shift is uniquely ergodic with respect to the fixed-point algebra.

Invariant states form a (m+1)-dimensional Choquet simplex for m-truncated cases.

Spectrum of position operators on m-truncated t-free Fock space is explicitly determined.

Abstract

The ergodic properties of the shift on both full and $m$-truncated $t$-free $C^*$-algebras are analyzed. In particular, the shift is shown to be uniquely ergodic with respect to the fixed-point algebra. In addition, for every $m\geq 1$, the invariant states of the shift acting on the $m$-truncated $t$-free $C^*$-algebra are shown to yield a $m+1$-dimensional Choquet simplex, which collapses to a segment in the full case. Finally, the spectrum of the position operators on the $m$-truncated $t$-free Fock space is also determined.