Author

# Diego López-Álvarez

Other affiliations: Autonomous University of Madrid

Bio: Diego López-Álvarez is an academic researcher from Spanish National Research Council. The author has contributed to research in topics: Group algebra & Division algebra. The author has an hindex of 3, co-authored 5 publications receiving 21 citations. Previous affiliations of Diego López-Álvarez include Autonomous University of Madrid.

##### Papers

More filters

••

TL;DR: In this paper, it was shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups and that one-relator groups satisfy both conjectures.

Abstract: It is shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

9 citations

•

TL;DR: In this article, it was shown that the strong Atiyah conjecture holds for locally indicable groups over a field of characteristic zero and that the group algebra of a locally indepth group over zero has a Hughes-free epic division algebra and that it is embedded in a division algebra.

Abstract: It is shown that the strong Atiyah conjecture holds for locally indicable groups. In particular, this implies that one-relator groups satisfy the strong Atiyah conjecture. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups.
As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

8 citations

••

TL;DR: It was shown in this article that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups, which implies that one-relator groups satisfy both conjectures.

Abstract: It is shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups.
As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

5 citations

•

TL;DR: In this paper, a homological criterion for a ring to be a pseudo-Sylvester domain was proposed, that is, to admit a non-commutative field of fractions over which all stably full matrices become invertible.

Abstract: Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a non-commutative field of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with free-by-cyclic and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell-Jones conjecture for poly-free groups and extends previous results of Linnell-Luck and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.

2 citations

•

TL;DR: In this article, the space of Sylvester rank functions on certain families of rings, including Dedekind domains, simple left noetherian rings and skew Laurent polynomial rings, has been studied.

Abstract: Given a ring $R$, the notion of Sylvester rank function was conceived within the context of Cohn's classification theory of epic division $R$-rings.
In this paper we study and describe the space of Sylvester rank functions on certain families of rings, including Dedekind domains, simple left noetherian rings and skew Laurent polynomial rings $\mathcal{D}[t^{\pm 1};\tau]$ for any division ring $\mathcal{D}$ and any automorphism $\tau$ of $\mathcal{D}$.

##### Cited by

More filters

••

393 citations

••

01 Jan 2006

TL;DR: In this paper, the authors present a list of special notation for preiminaries on modules over firs and semi-firs, including principal ideal domains, centralizers, subalgebras and Skew fields of fractions.

Abstract: Preface Note to the reader Terminology, notations and conventions used List of special notation 0. Preliminaries on modules 1. Principal ideal domains 2. Firs, semifirs and the weak algorithm 3. Factorization 4. 2-firs with a distributive factor lattice 5. Modules over firs and semifirs 6. Centralizers and subalgebras 7. Skew fields of fractions Appendix Bibliography and author index Subject index.

11 citations

••

TL;DR: In this paper, it was shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups and that one-relator groups satisfy both conjectures.

Abstract: It is shown that the strong Atiyah conjecture and the Luck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

9 citations

••

TL;DR: In this paper, a general construction of an approximating sequence of -subalgebras which are embeddable into an algebraic crossed product induced by a homeomorphism on the Cantor set is presented, where is an arbitrary field with involution and denotes the -algebra of locally constant -valued functions on.

Abstract: In this paper we consider the algebraic crossed product induced by a homeomorphism on the Cantor set , where is an arbitrary field with involution and denotes the -algebra of locally constant -valued functions on . We investigate the possible Sylvester matrix rank functions that one can construct on by means of full ergodic -invariant probability measures on . To do so, we present a general construction of an approximating sequence of -subalgebras which are embeddable into a (possibly infinite) product of matrix algebras over . This enables us to obtain a specific embedding of the whole -algebra into , the well-known von Neumann continuous factor over , thus obtaining a Sylvester matrix rank function on by restricting the unique one defined on . This process gives a way to obtain a Sylvester matrix rank function on , unique with respect to a certain compatibility property concerning the measure , namely that the rank of a characteristic function of a clopen subset must equal the measure of .

7 citations

••

TL;DR: The existence of Hughes-free division E ∗G-ring DE∗G for an arbitrary locally indicable group G is still an open question as discussed by the authors, even if G is amenable or G is bi-orderable.

Abstract: Let E∗G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group.

5 citations