Intercalation events visualized in single microcrystals of graphite.

The electrochemical intercalation of layered materials, particularly graphite, is fundamental to the operation of rechargeable energy-storage devices such as the lithium-ion battery and the carbon-enhanced lead-acid battery. Intercalation is thought to proceed in discrete stages, where each stage represents a specific structure and stoichiometry of the intercalant relative to the host. However, the three-dimensional structures of the stages between unintercalated and fully intercalated are not known, and the dynamics of the transitions between stages are not understood. Using optical and scanning transmission electron microscopy, we video the intercalation of single microcrystals of graphite in concentrated sulfuric acid. Here we find that intercalation charge transfer proceeds through highly variable current pulses that, although directly associated with structural changes, do not match the expectations of the classical theories. Evidently random nanoscopic defects dominate the dynamics of intercalation

The ideal structure of measure algebras and asymptotic properties of group representations

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closedness and finite generation, proving that in many cases $M(G)$ has no finitely-generated maximal left ideals. We also show that the measure algebra of the 2D Euclidean rigid motion group has a weak*-closed maximal left ideal that is not generated by a projection, and investigate whether or not it has any weak*-closed left ideals which are not finitely-generated.Comment: 21 pages. Comments welcom

Finitely-generated left ideals in Banach algebras on groups and semigroups

Let G be a locally compact group. We prove that the augmentation ideal in L1(G) is (algebraically) finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason

The Radical of the Bidual of a Beurling Algebra

We prove that the bidual of a Beurling algebra on Z , considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that rad(ℓ1(⊕∞i=1Z)'') contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight ω on Z such that the bidual of ℓ1(Z,ω) contains a radical element which is not nilpotent

Banach algebras on groups and semigroups

This thesis concerns the theory of Banach algebras, particularly those coming from abstract harmonic analysis. The focus for much of the thesis is the theory of the ideals of these algebras. In the final chapter we use semigroup algebras to solve an open probelm in the theory of C*-algebras. Throughout the thesis we are interested in the interplay between abstract algebra and analysis. Chapters 2, 4, and 5 are closely based upon the articles [88], [89], and [56], respectively. In Chapter 2 we study (algebraic) finite-generation of closed left ideals in Banach algebras. Let G be a locally compact group. We prove that the augmentation ideal in L 1 pGq is finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason. In Chapter 3 we turn our attention to topological finite-generation of closed left ideals in Banach algebras. We define a Banach algebra to be topologically left Noetherian if every closed left ideal is topologically finitely-generated, and we seek infinitedimensional examples of such algebras. We show that, given a compact group G, the group algebra L 1 pGq is topologically left Noetherian if and only if G is metrisable. For a Banach space E satisying a certain condition we show that the Banach algebra of approximable operators ApEq is topologically left Noetherian if and only if E 1 is separable, whereas it is topologically right Noetherian if and only if E is separable. We also define what it means for a dual Banach algebra to be weak*-topologically left Noetherian, and give examples which satisfy and fail this condition. Along the way, we give classifications of the weak*-closed left ideals in MpGq, for G a compact group, and in BpEq, for E a reflexive Banach space with AP. Chapter 4 looks at the Jacobson radical of the bidual of a Banach algebra. We prove that the bidual of a Beurling algebra on Z, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that rad p` 1 p‘8 i“1Zq 2 q contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight ω on Z such that the bidual of ` 1 pZ, ωq contains a radical element which is not nilpotent. In Chapter 5 we move away from the theory of ideals and consider a question about the notion of finiteness in C*-alegebras. We construct a unital pre-C*-algebra A0 which is stably finite, in the sense that every left invertible square matrix over A0 is right invertible, while the C*-completion of A0 contains a non-unitary isometry, and so it is infinite. This answers a question of Choi. The construction is based on semigroup algebras

Publisher Correction: Intercalation events visualized in single microcrystals of graphite.

The Peer Review File associated with this Article was updated shortly after publication to redact confidential comments to the editor

High-Dispersion Optical Spectra of Nearby Stars Younger Than The Sun

We present high-dispersion (R~16,000) optical (3900-8700 A) spectra of 390 stars obtained with the Palomar 60 inch telescope. The majority of stars observed are part of the Spitzer Legacy Science Program "The Formation and Evolution of Planetary Systems." Through detailed analysis we determine stellar properties for this sample, including radial and rotational velocities, Li I 6708 and Ha equivalent widths, the chromospheric activity index R'_HK, and temperature- and gravity-sensitive line ratios. Several spectroscopic binaries are also identified. From our tabulations, we illustrate basic age- and rotation-related correlations among measured indices. One novel result is that Ca II chromospheric emission appears to saturate at vsini values above ~30 km/s, similar to the well established saturation of X-rays that originate in the spatially separate coronal regions.Comment: 1 electronic table; published in the Astronomical Journa

An infinite C*-algebra with a dense, stably finite *-subalgebra

We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a nonunitary isometry, and so it is infinite