arXiv subjects
Publications and source records attributed to Caleb Eckhardt.
Let $\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let $W=\left(\bigoplus_{G/\Gamma} \mathbb{Z}/2\mathbb{Z}\right) \rtimes G.$ We show that $W$ is not an MF group and that $C^*(G)$ is not a finite C*-algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.
We construct the first examples of residually finite amenable groups that are not Hilbert-Schmidt (HS) stable. We construct finitely generated, class 3 nilpotent by cyclic examples and solvable linear finitely presented examples. This also provides the first examples of amenable groups that are very flexibly HS-stable but not flexibly HS-stable and the first examples of residually finite amenable groups that are not locally HS-stable. Along the way we exhibit (necessarily not-finitely-generated) class 2 nilpotent groups $G = A\rtimes \Z$ with $A$ abelian such that the periodic points of the dual action are dense but it does not admit dense periodic measures. Finally we use the Tikuisis-White-Winter theorem to show all of the examples are not even operator-HS-stable; they admit operator norm almost homomorphisms that can not be HS-perturbed to true homomorphisms.
We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on $C_0(X)$-algebras.
We show that a C*-algebra generated by an irreducible representation of a finitely generated virtually nilpotent group satisfies the universal coefficient theorem and has real rank 0. This combines with previous joint work with Gillaspy and McKenney to show these C*-algebras are classified by their Elliott invariants. When we further assume the group is nilpotent we build explicit Cartan subalgebras that are closely related to the group and representation, although the Cartan subalgebras are generally not C*-diagonals.
We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. We give a short, elementary proof that finitely generated nilpotent groups are HS-stable. We investigate the permanence of HS-stability under central extensions by showing HS-stability is preserved by finite central quotients, but is not preserved in general. We give a characterization of HS-stability for semidirect products $G\rtimes_\gamma \mathbb{Z}$ with $G$ abelian. We use it to construct the first example of a finitely generated amenable HS-stable group which is not permutation stable. Finally, it is proved that for amenable groups flexible HS-stability is equivalent to HS-stability, and very flexible HS stability is equivalent to maximal almost periodicity. There is some overlap of our work with the very recent and very nice preprint of Levit and Vigdorovich. We detail this overlap in the introduction. Where our work overlaps it appears that we take different approaches to the proofs and we feel the two works compliment each other.
We initiate the program of extending to higher-rank graphs ($k$-graphs) the geometric classification of directed graph $C^*$-algebras, as completed in the 2016 paper of Eilers, Restorff, Ruiz, and Sorensen [ERRS16]. To be precise, we identify four "moves," or modifications, one can perform on a $k$-graph $Λ$, which leave invariant the Morita equivalence class of its $C^*$-algebra $C^*(Λ)$. These moves -- insplitting, delay, sink deletion, and reduction -- are inspired by the moves for directed graphs described by Sorensen [S\o13] and Bates-Pask [BP04]. Because of this, our perspective on $k$-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a $k$-graph and its underlying directed graph.
We show that torsion-free finitely generated nilpotent groups are characterised by their group C*-algebras and we additionally recover their nilpotency class as well as the subquotients of the upper central series. We then use a C*-bundle decomposition and apply K-theoretic methods based on noncommutative tori to prove that every torsion-free finitely generated 2-step nilpotent group can be recovered from its group C*-algebra.
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.
We use free groups to settle a couple questions about the values of the Pimsner-Popa-Voiculescu modulus of quasidiagonality for a set of operators $Ω$, denoted by qd$(Ω)$. Along the way we deduce information about the operator space structure of finite dimensional subspaces of $\mathbb{C}[\mathbb{F}_d]\subseteq C^*_{\ell^p}(\mathbb{F}_d)$ where $C^*_{\ell^p}(\mathbb{F}_d)$ is the so-called $\ell^p$-completion of $\mathbb{C}[\mathbb{F}_d].$ Roughly speaking, we use free groups and qd$(Ω)$ to put a quantitative face on the two known qualitative obstructions to quasidiagonality; absence of an amenable trace or the presence of a proper isometry. The modulus of quasidiagonality for a proper isometry is equal to 1. We show that qd$(\{λ_a,λ_b\})\in [1/2,\sqrt{3}/2]$ where $a$ and $b$ are free group generators and $λ$ is the left regular representation. In another direction, we use certain $\ell^p$ representations of free groups constructed by Pytlik and Szwarc and a recent result of Ruan and Wiersma to show that qd$(Ω)$ may be positive, yet arbitrarily close to zero when $Ω$ is a set of unitaries.
We show that C*-algebras generated by irreducible representations of finitely generated nilpotent groups satisfy the universal coefficient theorem of Rosenberg and Schochet. This result combines with previous work to show that these algebras are classifiable by their Elliott invariants within the class of unital, simple, separable, nuclear C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. We also show that these C*-algebras are central cutdowns of twisted group C*-algebras with homotopically trivial cocycles.
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.
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.
We show that every unitary representation of a solvable discrete virtually nilpotent group G is quasidiagonal. Roughly speaking, this says that every unitary representation of G approximately decomposes as a direct sum of finite dimensional approximate representations. In operator algebraic terms we show that C*(G) is strongly quasidiagonal.
In this note we address a question of Don Hadwin: "Which groups have strongly quasidiagonal C*-algebras?" In recent work we showed that all finitely generated virtually nilpotent groups have strongly quasidiagonal C*-algebras, while together with Carrión and Dadarlat we showed that most wreath products fail to have strongly quasidiagonal C*-algebras. These two results raised the question of whether or not strong quasidiagonality could characterize virtual nilpotence among finitely generated groups. The purpose of this note is to provide examples of finitely generated groups (in fact of the form $\Z^3\rtimes \Z^2$) that are not virtually nilpotent yet have strongly quasidiagonal C*-algebras. Moreover we show these examples are the "simplest" possible by proving that a group of the form $\Z^d\rtimes \Z$ is virtually nilpotent if and only if its group C*-algebra is strongly quasidiagonal.
We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong quasidiagonality for group C*-algebras, exhibiting classes of amenable groups with and without strongly quasidiagonal C*-algebras.
Let $A$ be a homogeneous C*-algebra and $ϕ$ a state on $A.$ We show that if $ϕ$ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $ϕ$-preserving maps on $A$ that tend to the identity pointwise. This combined with results of Ricard and Xu show that the reduced free product of homogeneous C*-algebras with respect to these states have the completely contractive approximation property. We also give an example of a faithful state on $M_2\otimes C[0,1]$ for which no such state-preserving approximation of the identity map exists, thus answering a question of Ricard and Xu.
The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra $\mathfrak{A}$ considered separately by F. Boca and D. Mundici. The center of $\ga$ is isomorphic to $C[0,1]$, so we first consider the action of the Gauss map on $C[0,1]$ and then extend the map to $\mathfrak{A}$ and show that the extension inherits many desirable properties.
Let $ϕ:M_n\to B(H)$ be an injective, completely positive contraction with $\Vϕ^{-1}:ϕ(M_n)\to M_n\V_{cb}\leq1+δ(ε).$ We show that if either (i) $ϕ(M_n)$ is faithful modulo the compact operators or (ii) $ϕ(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $ψ:M_n\to B(H)$ with $\Vϕ-ψ\V_{cb}<ε.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial answer to a question of Junge, Ozawa and Ruan.