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

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

Normalizers of Irreducible Subfactors

We consider normalizers of an irreducible inclusion $N\subseteq M$ of $\mathrm{II}_1$ factors. In the infinite index setting an inclusion $uNu^*\subseteq N$ can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these normalizers of $N$ in $M$ to projections in the basic construction and show that every trace one projection in the relative commutant $N'\cap$ is of the form $u^*e_Nu$ for some unitary $u\in M$ with $uNu^*\subseteq N$. This enables us to identify the normalizers and the algebras they generate in several situations. In particular each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of irreducible subfactors arising from subgroup--group inclusions $H\subseteq G$. Here the normalizers are the normalizing group elements modulo a unitary from $L(H)$. We are also able to identify the finite trace $L(H)$-bimodules in $\ell^2(G)$ as double cosets which are also finite unions of left cosets.Comment: 33 Page

Gravity darkening and brightening in binaries

We apply a von Zeipel gravity darkening model to corotating binaries to obtain a simple, analytical expression for the emergent radiative flux from a tidally distorted primary orbiting a point-mass secondary. We adopt a simple Roche model to determine the envelope structure of the primary, assumed massive and centrally condensed, and use the results to calculate the flux. As for single rotating stars, gravity darkening reduces the flux along the stellar equator of the primary, but, unlike for rotating stars, we find that gravity brightening enhances the flux in a region around the stellar poles. We identify a critical limiting separation beyond which hydrostatic equilibrium no longer is possible, whereby the flux vanishes at the point on the stellar equator of the primary facing the companion. For equal-mass binaries, the total luminosity is reduced by about 13 % when this limiting separation is reached.Comment: 7 pages, 5 figures, matches version published in Astrophysical Journa

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

The Cuntz semigroup and stability of close C*-algebras

We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several applications: we are able to prove that the property of stability is preserved by close C*-algebras provided that one algebra has stable rank one; close C*-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of C*-algebras. We also examine C*-algebras which have a positive answer to Kadison's Similarity Problem, as these algebras are completely close whenever they are close. A sample consequence is that sufficiently close C*-algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang-Su algebra tensorially.Comment: 26 pages; typos fixe

Conceptions of teaching with integrity online in higher education: a case in the field of engineering

The viewpoints of academic teaching staff take centre stage in the analysis of the changing conceptions of what it means to act with integrity when teaching online. To teach with integrity in contemporary online-supported environments in higher education is not necessarily to teach the same as if one would in teaching regularly face-to-face in the classroom. The paper argues that to teach with integrity online is to teach differently. With integrity both enhanced and in some respects diminished in teaching online, the apparent contradiction can only be resolved through developing conceptions of what teaching with integrity means in the contemporary world of higher education. Implications are drawn in the context of teaching extended and wholly online units in the field of engineering.<br /