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Phase Transition on the Toeplitz Algebra of the Affine Semigroup Over the Natural Numbers
University of Wollongong Research Online Faculty of Engineering and Information Faculty of Informatics - Papers (Archive) Sciences 1-1-2010 Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Marcelo Laca University of Newcastle, Australia Iain F. Raeburn University of Wollongong, [email protected] Follow this and additional works at: https://ro.uow.edu.au/infopapers Part of the Physical Sciences and Mathematics Commons Recommended Citation Laca, Marcelo and Raeburn, Iain F.: Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers 2010, 643-688. https://ro.uow.edu.au/infopapers/1749 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Abstract We show that the group of orientation-preserving affine ansformationstr of the rational numbers is quasi- lattice ordered by its subsemigroup N⋊N×. The associated Toeplitz C∗-algebra T(N⋊N×) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(N⋊N×) in terms of generators and relations, and use this to show that the C∗-algebra QN recently introduced by Cuntz is the boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra T(N⋊N×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient QN, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (T(N⋊N×),R,σ) for all values of the inverse temperature β. -
Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
Orbits of Automorphism Groups of Fields
ORBITS OF AUTOMORPHISM GROUPS OF FIELDS KIRAN S. KEDLAYA AND BJORN POONEN Abstract. We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann \generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic. Can an infinite field K have so many automorphisms that the action of the automorphism group on the elements of K has only finitely many orbits? In Section 1, we prove that the answer is \no" (Theorem 1.1), even though the corresponding answer for division rings is \yes" (see Remark 1.2). Our proof constructs a \trace map" from the given field to a finite field, and exploits the peculiar combination of additive and multiplicative properties of this map. Section 2 attempts to prove a relative version of Theorem 1.1, by considering, for a non- trivial extension of fields k ⊂ K, the action of Aut(K=k) on K. In this situation each element of k forms an orbit, so we study only the orbits of Aut(K=k) on K − k. -
Monoid Kernels and Profinite Topologies on the Free Abelian Group
BULL. AUSTRAL. MATH. SOC. 20M35, 20E18, 20M07, 20K45 VOL. 60 (1999) [391-402] MONOID KERNELS AND PROFINITE TOPOLOGIES ON THE FREE ABELIAN GROUP BENJAMIN STEINBERG To each pseudovariety of Abelian groups residually containing the integers, there is naturally associated a profinite topology on any finite rank free Abelian group. We show in this paper that if the pseudovariety in question has a decidable membership problem, then one can effectively compute membership in the closure of a subgroup and, more generally, in the closure of a rational subset of such a free Abelian group. Several applications to monoid kernels and finite monoid theory are discussed. 1. INTRODUCTION In the early 1990's, the Rhodes' type II conjecture was positively answered by Ash [2] and independently by Ribes and Zalesskii [10]. The type II submonoid, or G-kernel, of a finite monoid is the set of all elements which relate to 1 under any relational morphism with a finite group. Equivalently, an element is of type II if 1 is in the closure of a certain rational language associated to that element in the profinite topology on a free group. The approach of Ribes and Zalesskii was based on calculating the profinite closure of a rational subset of a free group. They were also able to use this approach to calculate the Gp-kernel of a finite monoid. In both cases, it turns out the the closure of a rational subset of the free group in the appropriate profinite topology is again rational. See [7] for a survey of the type II theorem and its motivation. -
(I) Eu ¿ 0 ; Received by the Editors November 23, 1992 And, in Revised Form, April 19, 1993
proceedings of the american mathematical society Volume 123, Number 1, January 1995 TRIANGULAR UHF ALGEBRAS OVER ARBITRARYFIELDS R. L. BAKER (Communicated by Palle E. T. Jorgensen) Abstract. Let K be an arbitrary field. Let (qn) be a sequence of posi- tive integers, and let there be given a family \f¥nm\n > m} of unital K- monomorphisms *F„m. Tqm(K) —►Tq„(K) such that *¥np*¥pm= %im when- ever m < n , where Tq„(K) is the if-algebra of all q„ x q„ upper trian- gular matrices over K. A triangular UHF (TUHF) K-algebra is any K- algebra that is A'-isomorphic to an algebraic inductive limit of the form &~ = lim( Tq„(K) ; W„m). The first result of the paper is that if the embeddings ^Vnm satisfy certain natural dimensionality conditions and if £? = lim (TPn(K) ; Onm) is an arbitrary TUHF if-algebra, then 5? is A-isomorphic to ^ , only if the supernatural number, N[{pn)], of (q„) is less than or equal to the super- natural number, N[(p„)], of (pn). Thus, if the embeddings i>„m also satisfy the above dimensionality conditions, then S? is À"-isomorphic to F , only if NliPn)] = N[(qn)] ■ The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital A"-monomorphism <P: Mq{K) —>MP(K), only if q\p . The first result of the paper depends directly on the second result. 1. Introduction Let K be a fixed but arbitrary field. In this paper we investigate a certain class of A.-algebras (i.e., algebras over K) called triangular UHF (TUHF) K- algebras. -
Profinite Surface Groups and the Congruence Kernel Of
Pro¯nite surface groups and the congruence kernel of arithmetic lattices in SL2(R) P. A. Zalesskii* Abstract Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion of top the fundamental group ¼1 (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, caracteristic, accessible) subgroup of ¡ is isomorphic to a normal (respectively, caracteristic, accessible) subgroup of a free pro¯nite group. We use this description to give a complete solution of the congruence subgroup problem for aritmetic lattices in SL2(R). Introduction Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. As a topological space X is a compact oriented 2-manifold and is simply a sphere with g handles top added. The group ¦ = ¼1 (X) is called a surface group and has 2g generators ai; bi (i = 1; : : : ; g) subject to one relation [a1; b1][a2; b2] ¢ ¢ ¢ [ag; bg] = 1. The subgroup structure of ¦ is well known. Namely, if H is a subgroup of ¦ of ¯nite index n, then H is again a surface group of genus n(g ¡ 1) + 1. If n is in¯nite, then H is free. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion top of the fundamental group ¼1 (X) in the topological sense (see Exp. 10, p. 272 in [SGA-1]). -
Model-Complete Theories of Pseudo-Algebraically Closed Fields
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Annals of Mathematical Logic 17 (1979) 205-226 © North-Holland Publishing Company MODEL-COMPLETE THEORIES OF PSEUDO-ALGEBRAICALLY CLOSED FIELDS William H. WHEELER* Department of Mathematics, Indiana University, Bloomington, IN 47405, U. S.A. Received 14 June 1978; revised version received 4 January 1979 The model-complete, complete theories of pseudo-algebraically closed fields are characterized in this paper. For example, the theory of algebraically closed fields of a specified characteristic is a model-complete, complete theory of pseudo-algebraically closed fields. The characterization is based upon algebraic properties of the theories' associated number fields and is the first step towards a classification of all the model-complete, complete theories of fields. A field F is pseudo-algebraically closed if whenever I is a prime ideal in a F[xt, ] F[xl F is in polynomial ring ..., x, = and algebraically closed the quotient field of F[x]/I, then there is a homorphism from F[x]/I into F which is the identity on F. The field F can be pseudo-algebraically closed but not perfect; indeed, the non-perfect case is one of the interesting aspects of this paper. Heretofore, this concept has been considered only for a perfect field F, in which case it is equivalent to each nonvoid, absolutely irreducible F-variety's having an F-rational point. The perfect, pseudo-algebraically closed fields have been promi- nent in recent metamathematical investigations of fields [1,2,3,11,12,13,14, 15,28]. -
A Note on the Non-Existence of Prime Models of Theories of Pseudo-Finite
A note on the non-existence of prime models of theories of pseudo-finite fields Zo´eChatzidakis∗ DMA (UMR 8553), Ecole Normale Sup´erieure CNRS, PSL Research University September 23, 2020 Abstract We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we generalise this result to κ-prime models, for κ a regular uncountable cardinal or . ℵε Introduction In this short note, we show that prime models of the theory of pseudo-finite fields do not exist. More precisely, we consider the following theory T (A): F is a pseudo-finite field, A a relatively algebraically closed subfield of F, and T (A) is the theory of the field F in the language of rings augmented by constant symbols for the elements of A. Our first result is: Theorem 2.5. Let T (A) be as above. If A is not pseudo-finite, then T (A) has no prime model. The proof is done by constructing 2|A|+ℵ0 non-isomorphic models of T (A), of transcendence degree 1 over A (Proposition 2.4). Next we address the question of existence of κ-prime models of T (A), where κ is a regular uncountable cardinal or ε. We assume GCH, and again show non-existence of κ-prime models arXiv:2004.05593v2 [math.LO] 22 Sep 2020 ℵ of T (A), unless A < κ or A is κ-saturated when κ 1, and when the transcendence degree of A is infinite when| | κ = (Theorem 3.4). -
One Dimensional Groups Definable in the P-Adic Numbers
One dimensional groups definable in the p-adic numbers Juan Pablo Acosta L´opez November 27, 2018 Abstract A complete list of one dimensional groups definable in the p-adic num- bers is given, up to a finite index subgroup and a quotient by a finite subgroup. 1 Introduction The main result of the following is Proposition 8.1. There we give a complete list of all one dimensional groups definable in the p-adics, up to finite index and quotient by finite kernel. This is similar to the list of all groups definable in a real closed field which are connected and one dimensional obtained in [7]. The starting point is the following result. Proposition 1.1. Suppose T is a complete first order theory in a language extending the language of rings, such that T extends the theory of fields. Suppose T is algebraically bounded and dependent. Let G be a definable group. If G arXiv:1811.09854v1 [math.LO] 24 Nov 2018 is definably amenable then there is an algebraic group H and a type-definable subgroup T G of bounded index and a type-definable group morphism T H with finite kernel.⊂ → This is found in Theorem 2.19 of [8]. We just note that algebraic boundedness is seen in greater generality than needed in [5], and it can also be extracted from the proofs in cell decomposition [4]. That one dimensional groups are definably amenable comes from the following result. 1 Proposition 1.2. Suppose G is a definable group in Qp which is one dimen- sional. -
F-Isocrystals on the Line
F -isocrystals on the line Richard Crew The University of Florida December 16, 2012 I V is a complete discrete valuation with perfect residue field k of characteristic p and fraction field K. Let π be a uniformizer of V and e the absolute ramification index, so that (πe ) = (p). I A is a smooth V-algebra of finite type, and X = Spec(A). n+1 I A1 is the p-adic completion of A, and An = A/π , so that A = lim A . 1 −n n I We then set Xn = Spec(An) and X1 = Spf(A1). th I φ : A1 ! A1 is a ring homomorphism lifting the q -power Frobenius of A0. We denote by the restriction of φ to V, and by σ : K ! K its extension to K. I If X has relative dimension d over V, t1;:::; td will usually denote local parameters at an (unspecified) point of X , so 1 that ΩX =V has dt1;:::; dtd at that point. Same for local parameters on the completion X1. The Setup We fix the following notation: f I p > 0 is a prime, and q = p . I A is a smooth V-algebra of finite type, and X = Spec(A). n+1 I A1 is the p-adic completion of A, and An = A/π , so that A = lim A . 1 −n n I We then set Xn = Spec(An) and X1 = Spf(A1). th I φ : A1 ! A1 is a ring homomorphism lifting the q -power Frobenius of A0. We denote by the restriction of φ to V, and by σ : K ! K its extension to K. -
MATHEMATICS Algebra, Geometry, Combinatorics
MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson November 17, 2014 ii Contents Preface v Introduction vii 1 The nature of mathematics 1 1.1 The scope of mathematics . .1 1.2 Pure versus applied mathematics . .3 1.3 The antiquity of mathematics . .4 1.4 The modernity of mathematics . .6 1.5 The legacy of the Greeks . .8 1.6 The legacy of the Romans . .8 1.7 What they didn't tell you in school . .9 1.8 Further reading and links . 10 2 Proofs 13 2.1 How do we know what we think is true is true? . 14 2.2 Three fundamental assumptions of logic . 16 2.3 Examples of proofs . 17 2.3.1 Proof 1 . 17 2.3.2 Proof 2 . 20 2.3.3 Proof 3 . 22 2.3.4 Proof 4 . 23 2.3.5 Proof 5 . 25 2.4 Axioms . 31 2.5 Mathematics and the real world . 35 2.6 Proving something false . 35 2.7 Key points . 36 2.8 Mathematical creativity . 37 i ii CONTENTS 2.9 Set theory: the language of mathematics . 37 2.10 Proof by induction . 46 3 High-school algebra revisited 51 3.1 The rules of the game . 51 3.1.1 The axioms . 51 3.1.2 Indices . 57 3.1.3 Sigma notation . 60 3.1.4 Infinite sums . 62 3.2 Solving quadratic equations . 64 3.3 *Order . 70 3.4 *The real numbers . 71 4 Number theory 75 4.1 The remainder theorem . 75 4.2 Greatest common divisors . -
Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs
http://dx.doi.org/10.1090/surv/124 Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs Volume 124 Valuations, Orderings, and Milnor K-Theory Ido Efrat American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 12J10, 12J15; Secondary 12E30, 12J20, 19F99. For additional information and updates on this book, visit www.ams.org/bookpages/surv-124 Library of Congress Cataloging-in-Publication Data Efrat, Ido, 1963- Valuations, orderings, and Milnor if-theory / Ido Efrat. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 124) Includes bibliographical references and index. ISBN 0-8218-4041-X (alk. paper) 1. Valuation theory. 2. Ordered fields. 3. X-theory I. Title. II. Mathematical surveys and monographs ; no. 124. QA247.E3835 2006 515/.78-dc22 2005057091 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society.