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Paul McKenney

Publications and source records attributed to Paul McKenney.

11 recordsLinked to original sources
arXiv2025

Density of monochromatic infinite subgraphs II

In 1967, Gerencsér and Gyárfás proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erd\H os regarding the Ramsey numbers of trees, graphs with bounded maximum degree, and graphs with bounded degeneracy. In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd\H os and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, it was finally shown that $d={(12+\sqrt{8})}/{17}$. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.

math.CO↗
arXiv2021

Forcing axioms and coronas of $C^*$-algebras

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable $C^*$-algebras with the metric approximation property and an increasing approximate identity of projections.

math.LO↗
arXiv2018

Density of monochromatic infinite subgraphs

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of $G$ we can find with respect to some measure -- for instance, the density (or upper density) of the vertex set of $G$ in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of $K_\mathbb{N}$, there exists a monochromatic path whose vertex set has upper density at least $2/3$, but it is not possible to do better than $8/9$. They also showed that for some sequence $ε_n\to 0$, there exists a monochromatic path $P$ such that for infinitely many $n$, the set $\{1,2,...,n\}$ contains the first $(\frac{1}{3+\sqrt{3}}-ε_n)n$ vertices of $P$, but it is not possible to do better than $2n/3$. We improve both results, in the former case achieving an upper density at least $3/4$ and in the latter case obtaining a tight bound of $2/3$. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.

math.CO↗
arXiv2018

Finite decomposition rank for virtually nilpotent groups

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $G$ is bounded by $2\cdot h(G)!-1$, where $h(G)$ is the Hirsch length of $G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.

math.OA↗
arXiv2016

Ulam stability for some classes of C*-algebras

We prove some stability results for certain classes of C*-algebras. We prove that whenever $A$ is a finite-dimensional C*-algebra, $B$ is a C*-algebra and $ϕ\colon A\to B$ is approximately a $^*$-homomorphism then there is an actual $^*$-homomorphism close to $ϕ$ by a factor depending only on how far is $ϕ$ from being a $^*$-homomorphism and not on $A$ or $B$.

math.OA↗
arXiv2015

Automorphisms of $\mathscr{P}(λ)/\mathscr{I}_κ$

We study conditions on automorphisms of Boolean algebras of the form $P(λ)/I_κ$ (where $λ$ is an uncountable cardinal and $I_κ$ is the ideal of sets of cardinality less than $κ$) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every cardinality-preserving automorphism of $P(2^κ)/I_{κ^+}$ which is trivial on all sets of cardinality $κ^+$ is trivial, and that $MA_{\aleph_1}$ implies that every automorphism of $P(\mathbb{R})/Fin$ is trivial on a cocountable set.

math.LO↗
arXiv2015

Classification of C*-algebras generated by representations of the unitriangular group $UT(4,\mathbb{Z})$

It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group $UT(4,\mathbb{Z})$ we calculate the ordered K-theory of each C*-algebra generated by a faithful irreducible representation of $UT(4,\mathbb{Z})$ and see that they are all simple A$\mathbb{T}$ algebras. We also point out that there are many simple non A$\mathbb{T}$ algebras generated by irreducible representations of nilpotent groups.

math.OA↗
arXiv2015

Finitely generated nilpotent group C*-algebras have finite nuclear dimension

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, $A$ satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Elliott invariant and is approximately subhomogeneous. We give a large class of irreducible representations of nilpotent groups (of arbitrarily large nilpotency class) that satisfy the universal coefficient theorem and therefore are classifiable and approximately subhomogeneous.

math.OA↗
arXiv2013

Reduced products of UHF algebras under forcing axioms

If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic's Axiom and Martin's Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.

math.LO↗
arXiv2013

Some Calkin algebras have outer automorphisms

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, consistently, there are many outer automorphisms.

math.LO↗