arXiv subjects
Publications and source records attributed to Miles Gould.
In this note, we show that $C^*(G)$ fails the lifting property (LP) for every discrete group $G$ containing an uncountable abelian subgroup. Consequently, for every uncountable discrete abelian group $G$, the nuclear algebra $C^*(G)$ has the local lifting property (LLP) but fails the LP. This shows that the LP and the LLP do not coincide for full discrete group $C^*$-algebras.
The purpose of this note is to address the gap in the stable rank one/purely infinite dichotomy of selfless $C^*$-probability spaces. In particular, we show that nonfaithful selfless $C^*$-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless $C^*$-algebra either has stable rank one or is purely infinite. Notably, this shows that every selfless $C^*$-algebra is pure. To accomplish this, we show that infinite reduced free products of $C^*$-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having resolved the dichotomy, we improve existing permanence properties of selfless $C^*$-probability spaces, make progress on a conjecture of Choda and Dykema, and produce several new isomorphisms arising from reduced free products.
In this note, we promote an infinite Kadison transitivity theorem on massive $C^*$-algebras, including the Calkin algebra. This transitivity stems from the analog of countable degree-1 saturation on pure states which is inherited from these algebras via excision. We show this saturation to be equivalent to several order-theoretic properties on the quantum filter associated to the state, in particular the property of being a quantum P-point. While we show their existence is independent from ZFC, under basic set theoretic assumptions, we produce a plethora of these states. Finally, we find an irreducible representation of the Calkin algebra which fails infinite transitivity.
In 1946, Garrett Birkhoff proved that the $n\times n$ doubly stochastic matrices comprise the convex hull of the $n\times n$ permutation matrices, which in turn make up the extreme points of this polytope. He proposed his problem 111, which asks whether there exists a topology on infinite matrices for which this applies to the closed convex hull of the $\mathbb{N}\times\mathbb{N}$ permutation matrices. As Isbell showed in 1955, this equality is not achieved in the line-sum norm. In this paper, we use the domain of operator theory, and its many topologies, to improve on his negative result by showing that Birkhoff's problem is not solved in any of these topologies. In Kendall's 1960 paper on this problem, he gave an answer to the affirmative, as well as a topology for which closed convex hull comprises the doubly substochastic matrices. We also show that Kendall's secondary theorem also applies for all the locally convex Hausdorff topologies finer than than Kendall's (namely that of entry-wise convergence) which make the continuous dual of the matrix space no larger than the predual of the von Neumann algebra containing them. We then show that this is a theoretical upper limit topologies with this closure property. We also discuss the exposed points of this hull for these several topologies. Moreover, we show that, in these topologies, the closed affine hull of these permutation matrices comprise all operators with real-entry matrix coefficients.
Given a symmetric operad $P$, and a signature (or generating sequence) $Φ$ for $P$, we define a notion of the "categorification" (or "weakening") of $P$ with respect to $Φ$. When $P$ is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoidal categories. We then show that this categorification is independent (up to equivalence) of the choice of signature.
It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak P-category for any strongly regular (operadic) theory P, and show that every weak P-category is equivalent via P-functors and P-transformations to a strict P-category. This strictification functor is then shown to have an interesting universal property.