SYLVESTER MATRIX RANK FUNCTIONS ON CROSSED PRODUCTS AND THE ATIYAH PROBLEM

Abstract

This thesis is primarily concerned with the famous Atiyah problem, which asks about the possible_x000D_ l2-Betti numbers of discrete countable groups G. This question motivated a series of research papers_x000D_ in which where formulated (and proved in some relevant cases) statements stronger than the Atiyah_x000D_ original question, which asked whether there were countable discrete groups G with irrational l2-Betti_x000D_ numbers. Recently, the original Atiyah question has been solved, and some authors (including Austin_x000D_ and Grabowski) found examples of groups with irrational l2-Betti numbers. The lamplighter group is_x000D_ one of these groups._x000D_ The thesis presents an algebraic approach to the Atiyah problem by considering the -regular_x000D_ closure of the group algebra inside the algebra U(G) of (possibly unbounded) a liated operators of_x000D_ the group von Neumann algebra of G. By working on the lamplighter group, and following ideas of_x000D_ Ara and Goodearl, a sequence of approximating -subalgebras of the group algebra is constructed,_x000D_ giving a way of embedding the group algebra inside the well-known von Neumann continuous factor_x000D_ M. This allows us to construct a Sylvester matrix rank function on the group algebra, which in_x000D_ this particular case coincides with the rank function it inherits from U(G). By observing that the_x000D_ lamplighter group algebra can be realized as a crossed product algebra arising from a dynamical_x000D_ system, and using ideas of Putnam, it is shown in Chapter 2 that the above construction can be_x000D_ generalized to general crossed product algebras of a Cantor space by a homeomorphism, thus giving_x000D_ an explicit way of constructing Sylvester matrix rank functions on such crossed product algebras. The_x000D_ uniqueness of these rank functions is also studied. This rank function gives a notion of dimension, so_x000D_ allows us to de ne l2-Betti numbers in this more general setting in such a way that they coincide with_x000D_ the classical notion of l2-Betti numbers in the situation of the motivating example of the lamplighter_x000D_ group algebra. By following ideas of Grabowski and applying the previous techniques, we have been_x000D_ able to nd a whole family of irrational l2-Betti numbers arising from the lamplighter group algebra,_x000D_ and in fact we completely determined the possible l2-Betti numbers that can arise from the so-called_x000D_ odometer algebras, which are also a special case of crossed product algebras. This is done in Chapter_x000D_ 3._x000D_ Chapter 4 concerns about the rank completions of ultramatricial k-algebras, being k an arbitrary_x000D_ eld. A generalization of a result of von Neumann and Halperin is given, establishing an interesting_x000D_ analogy with the structure of the hyper nite II1 factor in the theory of von Neumann algebras._x000D_ Analogous results are obtained in the case of D-rings, and rings with involution *._x000D_ We also present a possible analytical approach to attack the Atiyah problem in Chapter 5, through_x000D_ studying the KMS states over the Toeplitz algebra of a group(oid) G acting on a graph E.

Date of Award	5 Dec 2018
Original language	English
Supervisor	Pere Ara Bertran (Director)