Peripheral Poisson Boundary

TL;DR

This paper extends the concept of non-commutative Poisson boundaries by including peripheral eigenvectors, revealing a $C^*$-algebra structure and invariance properties in quantum dynamics.

Contribution

It introduces the peripheral Poisson boundary, incorporating the point spectrum on the unit circle, and provides a dilation-theoretic formula for its product.

Findings

Peripheral Poisson boundary has a $C^*$-algebra structure.

The boundary remains invariant under discrete quantum dynamics.

Dilation theory yields a simple formula for the new product.

Abstract

It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a $C^*$-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.