The Realization Problem for some wild monoids and the Atiyah Problem

Abstract:

The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid $\mathcal M$ obtained by successive refinements of the identity $x_0+y_0=x_0+z_0$. This monoid is known to be realizable by the algebra $A= K[\mathcal F]$ of the monogenic free inverse monoid $\mathcal F$, for any choice of field $K$, but $A$ is not an exchange ring. We show that, for any uncountable field $K$, $\mathcal M$ is not realizable by a regular $K$-algebra, but that a suitable universal localization $\Sigma ^{-1}A$ of $A$ provides an exchange, non-regular, $K$-algebra realizing $\mathcal M$. For any countable field $F$, we show that a skew version of the above construction gives a regular $F$-algebra realizing $\mathcal M$. Finally, we develop some connections with the Atiyah Problem for the lamplighter group. We prove that the algebra $A$ can be naturally seen as a $*$-subalgebra of the group algebra $kG$ over the lamplighter group $G= \mathbb {Z}_2\wr \mathbb {Z}$, for any subfield $k$ of $\mathbb {C}$ closed under conjugation, and we determine the structure of the $*$-regular closure of $A$ in $\mathcal U(G)$. Using this, we show that the subgroup of $\mathbb R$ generated by the von Neumann dimensions of matrices over $kG$ contains $\mathbb {Q}$.