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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. -
Verification of Dade's Conjecture for Janko Group J3
JOURNAL OF ALGEBRA 187, 579]619Ž. 1997 ARTICLE NO. JA966836 Verification of Dade's Conjecture for Janko Group J3 Sonja KotlicaU Department of Mathematics, Uni¨ersity of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 View metadata, citation and similar papersCommunicated at core.ac.uk by Walter Feit brought to you by CORE provided by Elsevier - Publisher Connector Received July 5, 1996 Inwx 3 Dade made a conjecture expressing the number kBŽ.,d of characters of a given defect d in a given p-block B of a finite group G in terms of the corresponding numbers kbŽ.,d for blocks b of certain p-local subgroups of G. Several different forms of this conjecture are given inwx 5 . Dade claims that the most complicated form of this conjecture, called the ``Inductive Conjecture 5.8'' inwx 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive conjecture for all covering groups of the third Janko group J3 Žin the notation of the Atlaswx 1. This is one step in the inductive proof of the conjecture for all finite groups. Certain properties of J33simplify our task. The Schur Multiplier of J is cyclic of order 3Ž seewx 1, p. 82. Hence, there are just two covering groups of J33, namely J itself and a central extension 3 ? J33of J by a cyclic group Z of order 3. We treat these two covering groups separately. The outer automorphism group OutŽ.J33of J is cyclic of order 2 Ž seewx 1, p. -
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. -
Symmetric Generation and Existence of J3:2, the Automorphism Group of the Third Janko Group
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 304 (2006) 256–270 www.elsevier.com/locate/jalgebra Symmetric generation and existence of J3:2, the automorphism group of the third Janko group J.D. Bradley, R.T. Curtis ∗ School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2005 Available online 28 February 2006 Communicated by Jan Saxl Abstract We make use of the primitive permutation action of degree 120 of the automorphism group of L2(16), the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J3:2, to obtain the order of J3 and to prove its simplicity. A presentation for J3:2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J3:2. © 2006 Elsevier Inc. All rights reserved. 1. Introduction The group J3 was predicted by Janko in a paper in 1968 [10]. He considered a finite simple group with a single class of involutions in which the centraliser of an involution 1+4 was isomorphic to 2 :A5. The following year Higman and McKay, using an observation of Thompson that J3 must contain a subgroup isomorphic to L2(16):2, constructed the group on a computer [9]. -
(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]). -
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. -
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. -
P-Adic L-Functions and Modular Forms
p-adic L-functions and modular forms Alexei PANCHISHKIN Institut Fourier, Université Grenoble-1 B.P.74, 38402 St.Martin d'Hères, FRANCE Lectures for the International Summer School and Conference p-adic Analysis and its Applications August 31th to September 18th, 2009 (Trieste - ITALY) September, 2009 Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 1 / 1 Abstract 1) Congruences and p-adic numbers. Hensel's lemma. The Tate eld 2) Continuous and p-adic analytic functions. Mahler's criterion. Newton polygons Zeroes of analytic functions. The Weierstrass preparation theorem and its generalizations. 3) Distributions, measures, and the abstract Kummer congruences. The Kubota and Leopoldt p-adic L-functions and the Iwasawa algebra 4) Modular forms and L-functions. Congruences and families of modular forms. 5) Method of canonical projection of modular distributions. Examples of construction of p-adic L-functions Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 2 / 1 Related topics (for discussions, not included in the text of these materials) 6) Other approaches of constructing p-adic L-functions by Mazur-Tate-Teitelbaum, J.Coates, P.Colmez, H.Hida ... (using modular symbols by Manin-Mazur and their generalizations; Euler systems, work of D.Delbourgo, T.Ochiai, L.Berger, ..., overconvergent modular symbols by R.Pollack, G.Stevens, H.Darmon, ...) 7) Relations to the Iwasawa Theory 8) Applications of p-adic L-functions to Diophantine geometry 9) Open questions and problems in the theory of p-adic L-functions (Basic sources: Coates 60th Birthday Volume, Bourbaki talks by P.Colmez, J.Coates ...) Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 3 / 1 Lecture N◦1. -
A Finiteness Theorem for Odd Perfect Numbers
A finiteness theorem for odd perfect numbers Paul Pollack University of Georgia June 23, 2020 1 of 23 Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. 2 of 23 Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). 2 of 23 . In the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort | of which the most exemplary form is that type of number which is called perfect. -
ON CERTAIN P-ADIC BANACH LIMITS of P-ADIC TRIANGULAR MATRIX ALGEBRAS
International Journal of Pure and Applied Mathematics Volume 85 No. 3 2013, 435-455 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v85i3.1 ijpam.eu ON CERTAIN p-ADIC BANACH LIMITS OF p-ADIC TRIANGULAR MATRIX ALGEBRAS R.L. Baker University of Iowa Iowa City, Iowa 52242, USA Abstract: In this paper we investigate the class of p-adic triangular UHF (TUHF) Banach algebras.A p-adic TUHF Banach algebra is any unital p-adic Banach algebra of the form = n, where ( n) is an increasing sequence T T T T of p-adic Banach subalgebras of such that each n contains the identity of T S T T and is isomorphic as an Ωp-algebra to Tpn (Ωp) for some pn, where Tpn (Ωp) is the algebra of upper triangular pn pn matrices over the p-adic field Ωp. The main × result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra, provided that the algebra satisfies certain local dimensionality conditions. AMS Subject Classification: 12J25, 12J99, 46L99, 46S10 Key Words: p-adic Banach algebras, p-adic Banach limits, p-adic UHF algebras, p-adic Triangular UHF algebras 1. Introduction In the article, Triangular UHF algebras [2], the author of the present paper introduced a very concrete class of non-selfadjoint operator algebras which are currently called standard triangular UHF operator algebras. Briefly: a standard triangular UHF (TUHF) operator algebra is any unital Banach algebra that is isometrically isomorphic to a Banach algebra inductive limit of the form = T lim(Tp ; σp p ).