Noncommutative semialgebraic sets in nilpotent variables (Q713037)

Lifting problems involving norms and star-polynomials are fundamental in \(C^*\)-algebras. For references, see [\textit{S. Eilers, T. A. Loring} and \textit{G. K. Pedersen}, Adv. Math. 147, No. 1, 74--109 (1999; Zbl 1014.46030)]; [\textit{T. A. Loring}, Lifting solutions to perturbing problems in \(C^*\)-algebras. Providence, RI: American Mathematical Society (1997; Zbl 1155.46310)]; [\textit{C. L. Olsen} and \textit{G. K. Pedersen}, Math. Scand. 64, No. 1, 63--86 (1989; Zbl 0668.46029)]. In the present paper, the authors show how to lift a nilpotent element and its norm conditions and so the liftability of the set of relations \[ \|x^j\|\leq C_j,\;\;j=1,\dots,n, \] even if \(C_n=0\), is shown. In the particular case where the quotient is the Calkin algebra and the lifting is to \(\mathbb{B}(\mathbb{H})\), the proof uses different methods; see [the authors, J. Funct. Anal. 262, No. 2, 719--731 (2012; Zbl 1243.46048)]. The paper uses many technical results from the previous paper by the authors [Trans. Am. Math. Soc. 364, No. 2, 721--744 (2012; Zbl 1243.46047)]. As for potential applications in noncommutative real algebraic geometry, the liftability for the set of relations \[ x^3=0,\;y^4=0,\;z^5=0,\;\;xx^*+yy^*+zz^*\leq 1, \] is proved. Thus, more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts are found.