Decomposable approximations and coloured isomorphisms for C*-algebras

In this thesis we introduce nuclear dimension and compare it with a stronger form of the completely positive approximation property. We show that the approximations forming this stronger characterisation of the completely positive approximation property witness finite nuclear dimension if and only if the underlying C*-algebra is approximately finite dimensional. We also extend this result to nuclear dimension at most omega. We review interactions between separably acting injective von Neumann algebras and separable nuclear C*-algebras. In particular, we discuss aspects of Connes' work and how some of his strategies have been used by C^*-algebraist to estimate the nuclear dimension of certain classes of C*-algebras. We introduce a notion of coloured isomorphisms between separable unital C*-algebras. Under these coloured isomorphisms ideal lattices, trace spaces, commutativity, nuclearity, finite nuclear dimension and weakly pure infiniteness are preserved. We show that these coloured isomorphisms induce isomorphisms on the classes of finite dimensional and commutative C*-algebras. We prove that any pair of Kirchberg algebras are 2-coloured isomorphic and any pair of separable, simple, unital, finite, nuclear and Z-stable C*-algebras with unique trace which satisfy the UCT are also 2-coloured isomorphic.

2-categories and cyclic homology

The topic of this thesis is the application of distributive laws between comonads to the theory of cyclic homology. The work herein is based on the three papers 'Cyclic homology arising from adjunctions', 'Factorisations of distributive laws', and 'Hochschild homology, lax codescent,and duplicial structure', to which the current author has contributed. Explicitly, our main aims are: 1) To study how the cyclic homology of associative algebras and of Hopf algebras in the original sense of Connes and Moscovici arises from a distributive law, and to clarify the role of different notions of bimonad in this generalisation. 2) To extend the procedure of twisting the cyclic homology of a unital associative algebra to any duplicial object defined by a distributive law. 3) To study the universality of Bohm and Stefan’s approach to constructing duplicial objects, which we do in terms of a 2-categorical generalisation of Hochschild (co)homology. 4) To characterise those categories whose nerve admits a duplicial structure.

Bande dessinée on the periphery

This thesis examines how Brittany and Corsica are represented in the medium of bande dessinée. Both are peripheral French regions with cultural identities markedly different from that of the overarching French norm, and both have been historically subject to ridicule from the political and cultural centre. By comparing a fair selection of bandes dessinées which are either set in Brittany or Corsica or feature characters from the relevant regions, this thesis sets out to discover whether representations of Brittany and Corsica differ according to the origin of the creators of the bandes dessinées and, if so, how. To facilitate this analysis, the bandes dessinées included for study have been classified as either external representations (published by mainstream bande dessinée publishers and/or the work of creators originating from outside the two regions) or internal representations (published by local Breton or Corsican companies and/or the work of local creators). It transpires that there are clear differences between mainstream and local bande dessinée authors and illustrators with regard to their portrayal of the local culture of both ‘outlying’ regions. External representations rely on broad stereotypes and received ideas, while internal representations draw on local folklore, regional history and regional identity to create works with more local relevance. In some cases internal representations are or were clearly aimed at a local market, while others aim both at local readers and at the wider bande dessinée market. Those aimed at a wider readership have an additional function, namely that of promoting their regional cultures in French culture generally and offering an alternative to the stereotypical representations presented by larger publishers of bandes dessinées. Brittany and Corsica are examined separately, each taking up roughly half of the thesis. Each half has the same general structure, beginning with discussion of how historical events have shaped perceptions of Brittany and Corsica in French popular consciousness, followed by analysis of the respective external representations and lastly internal representations. There are also two case studies of representations of Corsica in wider visual culture. Owing to its widespread appeal, its adaptability and its capacity to reflect popular opinion in different sectors of society, the medium of bande dessinée offers a potentially rich field for the investigation of social and cultural attitudes and prejudices. It is hoped that this thesis points the way to further research on the topic.

Learning to construct our identities over the life course: a study with lesbian, gay, bisexual and transgender adults in Scotland

To date, adult educational research has had a limited focus on lesbian, gay, bisexual and transgendered (LGBT) adults and the learning processes in which they engage across the life course. Adopting a biographical and life history methodology, this study aimed to critically explore the potentially distinctive nature and impact of how, when and where LGBT adults learn to construct their identities over their lives. In-depth, semi-structured interviews, dialogue and discussion with LGBT individuals and groups provided rich narratives that reflect shifting, diverse and multiple ways of identifying and living as LGBT. Participants engage in learning in unique ways that play a significant role in the construction and expression of such identities, that in turn influence how, when and where learning happens. Framed largely by complex heteronormative forces, learning can have a negative, distortive impact that deeply troubles any balanced, positive sense of being LGBT, leading to self- censoring, alienation and in some cases, hopelessness. However, learning is also more positively experiential, critically reflective, inventive and queer in nature. This can transform how participants understand their sexual identities and the lifewide spaces in which they learn, engendering agency and resilience. Intersectional perspectives reveal learning that participants struggle with, but can reconcile the disjuncture between evolving LGBT and other myriad identities as parents, Christians, teachers, nurses, academics, activists and retirees. The study’s main contributions lie in three areas. A focus on LGBT experience can contribute to the creation of new opportunities to develop intergenerational learning processes. The study also extends the possibilities for greater criticality in older adult education theory, research and practice, based on the continued, rich learning in which participants engage post-work and in later life. Combined with this, there is scope to further explore the nature of ‘life-deep learning’ for other societal groups, brought by combined religious, moral, ideological and social learning that guides action, beliefs, values, and expression of identity. The LGBT adults in this study demonstrate engagement in distinct forms of life-deep learning to navigate social and moral opprobrium. From this they gain hope, self-respect, empathy with others, and deeper self-knowledge.

Connected Hopf algebras of finite Gelfand-Kirillov dimension

Following the seminal work of Zhuang, connected Hopf algebras of finite GK-dimension over algebraically closed fields of characteristic zero have been the subject of several recent papers. This thesis is concerned with continuing this line of research and promoting connected Hopf algebras as a natural, intricate and interesting class of algebras. We begin by discussing the theory of connected Hopf algebras which are either commutative or cocommutative, and then proceed to review the modern theory of arbitrary connected Hopf algebras of finite GK-dimension initiated by Zhuang. We next focus on the (left) coideal subalgebras of connected Hopf algebras of finite GK-dimension. They are shown to be deformations of commutative polynomial algebras. A number of homological properties follow immediately from this fact. Further properties are described, examples are considered and invariants are constructed. A connected Hopf algebra is said to be "primitively thick" if the difference between its GK-dimension and the vector-space dimension of its primitive space is precisely one . Building on the results of Wang, Zhang and Zhuang,, we describe a method of constructing such a Hopf algebra, and as a result obtain a host of new examples of such objects. Moreover, we prove that such a Hopf algebra can never be isomorphic to the enveloping algebra of a semisimple Lie algebra, nor can a semisimple Lie algebra appear as its primitive space. It has been asked in the literature whether connected Hopf algebras of finite GK-dimension are always isomorphic as algebras to enveloping algebras of Lie algebras. We provide a negative answer to this question by constructing a counterexample of GK-dimension 5. Substantial progress was made in determining the order of the antipode of a finite dimensional pointed Hopf algebra by Taft and Wilson in the 1970s. Our final main result is to show that the proof of their result can be generalised to give an analogous result for arbitrary pointed Hopf algebras.