Sylvester matrix rank functions on crossed products and the atiyah problem
- Autores: Joan Claramunt Carós
- Directores de la Tesis: Pedro Ara Bertran (dir. tes.)
- Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2018
- Idioma: español
- Tribunal Calificador de la Tesis: Andrei Jaikin-Zapirain (presid.), Francisco Perera Domenech (secret.), Lukasz Grabowski (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universidad Autónoma de Barcelona
- Materias:
- Enlaces
- Tesis en acceso abierto en: TDX
- Resumen
This thesis is primarily concerned with the famous Atiyah problem, which asks about the possible l2-Betti numbers of discrete countable groups G. This question motivated a series of research papers in which where formulated (and proved in some relevant cases) statements stronger than the Atiyah original question, which asked whether there were countable discrete groups G with irrational l2-Betti numbers. Recently, the original Atiyah question has been solved, and some authors (including Austin and Grabowski) found examples of groups with irrational l2-Betti numbers. The lamplighter group is one of these groups.
The thesis presents an algebraic approach to the Atiyah problem by considering the *-regular closure of the group algebra inside the algebra U(G) of (possibly unbounded) affiliated operators of the group von Neumann algebra of G. By working on the lamplighter group, and following ideas of Ara and Goodearl, a sequence of approximating *-subalgebras of the group algebra is constructed, giving a way of embedding the group algebra inside the well-known von Neumann continuous factor M. This allows us to construct a Sylvester matrix rank function on the group algebra, which in this particular case coincides with the rank function it inherits from U(G). By observing that the lamplighter group algebra can be realized as a crossed product algebra arising from a dynamical system, and using ideas of Putnam, it is shown in Chapter 2 that the above construction can be generalized to general crossed product algebras of a Cantor space by a homeomorphism, thus giving an explicit way of constructing Sylvester matrix rank functions on such crossed product algebras. The uniqueness of these rank functions is also studied. This rank function gives a notion of dimension, so allows us to define l2-Betti numbers in this more general setting in such a way that they coincide with the classical notion of l2-Betti numbers in the situation of the motivating example of the lamplighter group algebra. By following ideas of Grabowski and applying the previous techniques, we have been able to find a whole family of irrational l2-Betti numbers arising from the lamplighter group algebra, and in fact we completely determined the possible l2-Betti numbers that can arise from the so-called odometer algebras, which are also a special case of crossed product algebras. This is done in Chapter 3.
Chapter 4 concerns about the rank completions of ultramatricial K-algebras, being K an arbitrary field. A generalization of a result of von Neumann and Halperin is given, establishing an interesting analogy with the structure of the hyperfinite II1 factor in the theory of von Neumann algebras. Analogous results are obtained in the case of D-rings, and rings with involution *.
We also present a possible analytical approach to attack the Atiyah problem in Chapter 5, through studying the KMS states over the Toeplitz algebra of a group(oid) G acting on a graph E.
