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Author: David Signorielli Penneys
Publisher:
ISBN:
Category :
Languages : en
Pages : 145
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Book Description
This dissertation consists of three self-contained papers from my graduate work at UC Berkeley. The chapters increase in complexity from the annular Temperley-Lieb category to strongly Markov inclusions of finite von Neumann algebras to infinite index II_1-subfactors. In Chapter 2, we discuss how two copies of the cyclic category generate the annular Temperley-Lieb category. In the process, we give a presentation of the annular Temperley-Lieb category via generators and relations, and we see the cyclic category evolve from the simplicial and semi-simplicial categories. Chapter 3 is joint work with Vaughan F.R. Jones. First, we define a canonical planar *-algebra associated to a strongly Markov inclusion of finite von Neumann algebras (the notion of such an inclusion is defined within). Second, we show for an inclusion of finite dimensional C^*-algebras with the Markov trace, the canonical planar algebra is isomorphic to the graph planar algebra of the Bratteli diagram of the inclusion. We use this fact to show that a subfactor planar algebra embeds into the graph planar algebra of its principal graph. In Chapter 4, we expand upon Burns' work on rotations for infinite index II_1-subfactors. We start with a II_1-factor bimodule, and we construct a tower of centralizer algebras and a sequence of central L^2-vectors. In the finite index setting, the centralizer algebras and central L^2-vectors agree, but in the infinite index setting, these spaces can differ dramatically. We develop planar calculi for both sequences which are compatible. Interestingly, we obtain planar structure without Jones' basic construction or the resulting Jones projections! We also generalize Burns work on extremality and the existence of rotations to the bimodule setting, and we recover his main theorem. Along the way, we prove some results about relative tensor products of extended positive cones, and we give an example of an infinite index subfactor with finite dimensional higher relative commutants.
Author: David Signorielli Penneys
Publisher:
ISBN:
Category :
Languages : en
Pages : 145
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Book Description
This dissertation consists of three self-contained papers from my graduate work at UC Berkeley. The chapters increase in complexity from the annular Temperley-Lieb category to strongly Markov inclusions of finite von Neumann algebras to infinite index II_1-subfactors. In Chapter 2, we discuss how two copies of the cyclic category generate the annular Temperley-Lieb category. In the process, we give a presentation of the annular Temperley-Lieb category via generators and relations, and we see the cyclic category evolve from the simplicial and semi-simplicial categories. Chapter 3 is joint work with Vaughan F.R. Jones. First, we define a canonical planar *-algebra associated to a strongly Markov inclusion of finite von Neumann algebras (the notion of such an inclusion is defined within). Second, we show for an inclusion of finite dimensional C^*-algebras with the Markov trace, the canonical planar algebra is isomorphic to the graph planar algebra of the Bratteli diagram of the inclusion. We use this fact to show that a subfactor planar algebra embeds into the graph planar algebra of its principal graph. In Chapter 4, we expand upon Burns' work on rotations for infinite index II_1-subfactors. We start with a II_1-factor bimodule, and we construct a tower of centralizer algebras and a sequence of central L^2-vectors. In the finite index setting, the centralizer algebras and central L^2-vectors agree, but in the infinite index setting, these spaces can differ dramatically. We develop planar calculi for both sequences which are compatible. Interestingly, we obtain planar structure without Jones' basic construction or the resulting Jones projections! We also generalize Burns work on extremality and the existence of rotations to the bimodule setting, and we recover his main theorem. Along the way, we prove some results about relative tensor products of extended positive cones, and we give an example of an infinite index subfactor with finite dimensional higher relative commutants.
Author: Institut des hautes études scientifiques (Paris, France)
Publisher: American Mathematical Soc.
ISBN: 0821852035
Category : Mathematics
Languages : en
Pages : 695
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Book Description
The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.
Author: Vaughan F. R. Jones
Publisher: Cambridge University Press
ISBN: 0521584205
Category : Mathematics
Languages : en
Pages : 178
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Book Description
Subfactors have been a subject of considerable research activity for about 15 years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. A new pictorial approach to subfactors is presented in a late ch apter.
Author:
Publisher: World Scientific
ISBN:
Category :
Languages : en
Pages : 1001
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Author: American Mathematical Society
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 182
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Author: Source Wikipedia
Publisher: University-Press.org
ISBN: 9781230512310
Category :
Languages : en
Pages : 28
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Book Description
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 26. Chapters: Abelian von Neumann algebra, Affiliated operator, Baer ring, Central carrier, Commutation theorem, Connes embedding problem, Continuous geometry, Crossed product, Direct integral, Dixmier trace, Finite dimensional von Neumann algebra, Hyperfinite type II factor, Kaplansky density theorem, Octacube (mathematics), Schroder-Bernstein theorems for operator algebras, Sherman-Takeda theorem, Subfactor, Temperley-Lieb algebra, Tomita-Takesaki theory, Ultrastrong topology, Ultraweak topology, Von Neumann bicommutant theorem.
Author: Matti Pitkanen
Publisher: Bentham Science Publishers
ISBN: 1681081792
Category : Science
Languages : en
Pages : 1235
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Book Description
Topological geometrodynamics (TGD) is a modification of the theory of general relativity inspired by the problems related to the definition of inertial and gravitational energies in the earlier hypotheses. TGD is also a generalization of super string models. TGD brings forth an elegant theoretical projection of reality and builds upon the work by renowned scientists (Wheeler, Feynman, Penrose, Einstein, Josephson to name a few). In TGD, Physical space-time planes are visualized as four-dimensional surfaces in a certain 8-dimensional space (H). The choice of H is fixed by symmetries of standard model and leads to a geometric mapping of known classical fields and elementary particle numbers. TGD differs from Einstein’s geometrodynamics in the way space-time planes or ‘sheets’ are lumped together. Extending the theory based on fusing number concepts implies a further generalisation of the space-time concept allowing the identification of space-time correlates of cognition and intentionality. Additionally, zero energy ontology forces an extension of quantum measurement theory to a theory of consciousness and a hierarchy of phases is identified. Dark matter is thus predicted with far reaching implications for the understanding of consciousness and living systems. Therefore, it sets a solid foundation for modeling our universe in geometric terms. Topological Geometrodynamics: An Overview explains basic and advanced concepts about TGD. The book covers introductory information and classical TGD concepts before delving into twistor-space theory, particle physics, infinite-dimensional spinor geometry, generalized number theory, Planck constants, and the applications of TGD theory in research. The book is a valuable guide to TDG theory for researchers and advanced graduates in theoretical physics and cosmology.
Author: Frederick M. Goodman
Publisher: Springer
ISBN:
Category : Mathematics
Languages : en
Pages : 312
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Book Description
A recent paper on subfactors of von Neumann factors has stimulated much research in von Neumann algebras. It was discovered soon after the appearance of this paper that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links. Recent efforts to understand the fundamental nature of the new link invariants has led to connections with invariant theory, statistical mechanics and quantum theory. In turn, the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had a great impact on the study of subfactors. Our subject is certain algebraic and von Neumann algebraic topics closely related to the original paper. However, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expository material.
Author: Narjess Afzaly
Publisher: American Mathematical Society
ISBN: 1470447126
Category : Mathematics
Languages : en
Pages : 94
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Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 872
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