Quasidiagonality of nuclear C*-algebras

We prove that faithful traces on separable and nuclear C*- algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras

Experimental and computational results on the interaction of carbon dioxide laser radiation with a dense plasma

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Groupoid normalisers of tensor products: infinite von Neumann algebras

The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$establishing the formula$$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$and normalisers$v_i\in\mathcal{N}_{M_i}(B_i)$). We obtain a positive result when one of the$M_i$is finite or both of the$B_i$are infinite. For the remaining case, we characterise the II$_1$factors$B_1$for which such factorisations always occur (for all$M_1, B_2$and$M_2$) as those with a trivial fundamental group.Comment: 22 page

A new genus and species of sabretooth, Oriensmilus liupanensis (Barbourofelinae, Nimravidae, Carnivora), from the middle Miocene of China suggests barbourofelines are nimravids, not felids

Since the early 2000s, a revival of a felid relationship for barbourofeline sabretooths has become popular due to recent discoveries of fragmentary fossils from Africa. According to this view, barbourofelines trace their common ancestor with felids through shared similarities in dental morphology going back to the early Miocene of Africa and Europe. However, whether or not such an idea is represented in the basicranial morphology, a conservative area of high importance in family-level relationships, is yet to be tested. A nearly complete skull of Oriensmilus liupanensis gen. and sp. nov. from the middle Miocene Tongxin Basin of northern China represents the most primitive known barbourofeline with an intact basicranial region, affording an opportunity to re-examine the relationship of felids and nimravines. We also present an update on East Asian records of barbourofelines. The new skull of Oriensmilus possesses a suite of characters shared with nimravines, such as the lack of an ossified (entotympanic) bullar floor, absence of an intrabullar septum, lack of a ventral promontorial process of the petrosal, presence of a small rostral entotympanic on the dorsal side of the caudal entotympanic, and a distinct caudal entry of the internal carotid artery and nerve that pierces the caudal entotympanic at the junction of the ossified and unossified caudal entotympanics. The absence of an ossified bullar floor in O. liupanensis and its presence in those from the middle Miocene of Sansan, France thus help to bracket the transition of this character, which must have happened in the early part of the middle Miocene. Spatial relationships between bullar construction and the middle ear configuration of the carotid artery in Oriensmilus strongly resemble those in nimravines but are distinctly different from felids and other basal feliforms. Despite the attractive notion that early barbourofelines arose from a Miocene ancestor that also gave rise to felids, the basicranial evidence argues against this view. http://zoobank.org/urn:http://lsid:zoobank.org:pub:2DE98DBC-4D02-4E18-9788-0B0D8587E73F

Z-stability and finite dimensional tracial boundaries

We show that a simple separable unital nuclear nonelementary C∗-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies Z-stability for these algebras

Preduals of semigroup algebras

For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup $S$, the convolution algebra $\ell^1(S)$ also carries a coproduct. In this paper we examine preduals for $\ell^1(S)$ making both the product and the coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that $\ell^1(S)$ has a unique such predual. Such $S$ include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on $\ell^1(S)$ when $S$ is either $\mathbb Z_+\times\mathbb Z$ or $(\mathbb N,\cdot)$.Comment: 17 pages, LaTe

The Radial Masa in a Free Group Factor is Maximal Injective

The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.Comment: 25 Pages, Typos corrected and exposition improve

Abstract classification theorems for amenable C*-algebras

Operator algebras are subalgebras of the bounded operators on a Hilbert space. They divide into two classes: C∗-algebras and von Neumann algebras according to whether they are required to be closed in the norm or weak-operator topology, respectively. In the 1970s Alain Connes identified the appropriate notion of amenabilty for von Neumann algebras, and used it to obtain a deep internal finite-dimensional approximation structure for these algebras. This structure is exactly what is needed for classification, and one of many consequences of Connes’ theorem is the uniqueness of amenable II 1 factors, and later a complete classification of all simple amenable von Neumann algebras acting on separable Hilbert spaces. The Elliott classification programme aims for comparable structure and classification results for C∗-algebras using operator K-theory and traces. The definitive unital classification theorem was obtained in 2015. This is a combination of the Kirchberg–Phillips theorem and the large scale activity in the stably finite case by numerous researchers over the previous 15–20 years. It classifies unital simple separable amenable C∗-algebras satisfying two extra hypotheses: a universal coefficient theorem which computes KK-theory in terms of K-theory and a regularity hypothesis excluding exotic high-dimensional behaviour. Today the regularity hypothesis can be described in terms of tensor products (Z-stability). These hypotheses are abstract, and there are deep tools for verifying the universal coefficient theorem and Z-stability in examples. This article describes the unital classification theorem, its history and context, together with the new abstract approach to this result developed in collaboration with Carrión, Gabe, Schafhauser, and Tikuisis. This method makes a direct connection to the von Neumann algebraic results, and does not need to obtain any kind of approximation structure inside C∗-algebras en route to classification. The companion survey (White, in preparation) focuses on the role of the Z-stability hypothesis, and the associated work on “regularity.

On spectral triples on crossed products arising from equicontinuous actions

The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.Comment: 22 pages (v4 corrects a mistake in the discussion of the equicontinuity condition and modifies the terminology used). The paper will appear in Mathematica Scandinavic