Noncommutative Semialgebraic Sets in Nilpotent Variables

TL;DR

This paper addresses the lifting problem in C*-algebras for noncommutative semialgebraic sets defined by nilpotent relations and norm constraints, expanding the class of sets with known liftability properties.

Contribution

It introduces new liftability results for noncommutative semialgebraic sets involving nilpotent and homogeneous relations in C*-algebras.

Findings

Liftability proven for sets with relations like x^N=0 and norm bounds

Extended the class of noncommutative semialgebraic sets with topological properties

Identified sets that are noncommutative absolute retracts

Abstract

We solve the lifting problem in C^*-algebras for many sets of relations that include the relations x_j^{N_j} = 0 on each variable. The remaining relations must be of the form \| p(x_1,...,x_n) \| \leq C for C a positive constant and p a noncommutative *-polynomial that is in some sense homogeneous. For example, we prove liftability for the set of relations x^3=0, y^4=0, z^5=0, xx^*+yy^*+zz^* \leq 1. Thus we find more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts.