Real Analysis Quals - Helpful Information
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:1508.00389V3 [Math.OA]
LIFTING THEOREMS FOR COMPLETELY POSITIVE MAPS JAMES GABE Abstract. We prove lifting theorems for completely positive maps going out of exact C∗- algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X is a second countable topological space, A and B are separable, nuclear C∗- algebras over X, and the action of X on A is continuous, then E(X; A, B) =∼ KK(X; A, B) naturally. As an application, we show that a separable, nuclear, strongly purely infinite C∗-algebra A absorbs a strongly self-absorbing C∗-algebra D if and only if I and I ⊗ D are KK- equivalent for every two-sided, closed ideal I in A. In particular, if A is separable, nuclear, and strongly purely infinite, then A ⊗O2 =∼ A if and only if every two-sided, closed ideal in A is KK-equivalent to zero. 1. Introduction Arveson was perhaps the first to recognise the importance of lifting theorems for com- pletely positive maps. In [Arv74], he uses a lifting theorem to give a simple and operator theoretic proof of the fact that the Brown–Douglas–Fillmore semigroup Ext(X) is actually a group. This was already proved by Brown, Douglas, and Fillmore in [BDF73], but the proof was somewhat complicated and very topological in nature. All the known lifting theorems at that time were generalised by Choi and Effros [CE76], when they proved that any nuclear map going out of a separable C∗-algebra is liftable. This result, together with the dilation theorem of Stinespring [Sti55] and the Weyl–von Neumann type theorem of Voiculescu [Voi76], was used by Arveson [Arv77] to prove that the (generalised) Brown– Douglas–Fillmore semigroup Ext(A) defined in [BDF77] is a group for any unital, separable, nuclear C∗-algebra A. -
Continuous Selections of Multivalued Mappings 3
Continuous selections of multivalued mappings Duˇsan Repovˇsand Pavel V. Semenov Abstract This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II which was pub- lished in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. We remark that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topo- logical spaces) since this topics is covered by another survey in this volume. 1 Preliminaries A selection of a given multivalued mapping F : X Y with nonempty values F (x) = , for every x X, is a mapping Φ : X → Y (in general, also multivalued)6 which∅ for every ∈x X, selects a nonempty→ subset Φ(x) F (x). When all Φ(x) are singletons, a selection∈ is called singlevalued and is identified⊂ with the usual singlevalued mapping f : X Y, f(x) = Φ(x). As a rule, we shall use small letters f, g, h, φ, ψ, ... for singlevalued→ { mappings} and capital letters F, G, H, Φ, Ψ, ... for multivalued mappings. There exist a great number of theorems on existence of selections in the category of topological spaces and their continuous (in various senses) map- arXiv:1401.2257v1 [math.GN] 10 Jan 2014 Duˇsan Repovˇs Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, P. O. Box 2964, Ljubljana, Slovenia 1001 e-mail: [email protected] Pavel V. -
Selection Theorems
SELECTION THEOREMS STEPHANIE HICKS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS NIPISSING UNIVERSITY SCHOOL OF GRADUATE STUDIES NORTH BAY, ONTARIO © Stephanie Hicks August 2012 I hereby declare that I am the sole author of this Major Research Paper. I authorize Nipissing University to lend this Major Research Paper to other institutions or individuals for the purpose of scholarly research. I further authorize Nipissing University to reproduce this Major Research Paper by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. v Acknowledgements I would like to thank several important individuals who played an integral role in the development of this paper. Most importantly, I would like to thank my fiancé Dan for his ability to believe in my success at times when I didn’t think possible. Without his continuous support, inspiration and devotion, this paper would not have been possible. I would also like to thank my parents and my sister for their encouragement and love throughout the years I have spent pursuing my post secondary education. Many thanks are due to my advisor Dr. Vesko Valov for his expertise and guidance throughout this project. Additionally, I would like to thank my external examiner Vasil Gochev and my second reader Dr. Logan Hoehn for their time. Lastly, I would like to thank Dr. Wenfeng Chen and Dr. Murat Tuncali for passing on their invaluable knowledge and encouragement to pursue higher education throughout my time as a Mathematics student at Nipissing University. -
1 Lecture 09
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let's recall the conception of \equicontinuity" for family of functions that we learned in classical analysis: A family of continuous functions defined on a region Ω, Λ = ffαg ⊂ C(Ω); is called an equicontinuous family if 8 > 0; 9δ > 0 such that for any x1; x2 2 Ω with jx1 − x2j < δ; and for any fα 2 Λ, we have jfα(x1) − fα(x2)j < . This conception of equicontinuity can be easily generalized to maps between topological vector spaces. For simplicity we only consider linear maps between topological vector space , in which case the continuity (and in fact the uniform continuity) at an arbitrary point is reduced to the continuity at 0. Definition 1.1. Let X; Y be topological vector spaces, and Λ be a family of continuous linear operators. We say Λ is an equicontinuous family if for any neighborhood V of 0 in Y , there is a neighborhood U of 0 in X, such that L(U) ⊂ V for all L 2 Λ. Remark 1.2. Equivalently, this means if x1; x2 2 X and x1 − x2 2 U, then L(x1) − L(x2) 2 V for all L 2 Λ. Remark 1.3. If Λ = fLg contains only one continuous linear operator, then it is always equicontinuous. 1 If Λ is an equicontinuous family, then each L 2 Λ is continuous and thus bounded. In fact, this boundedness is uniform: Proposition 1.4. Let X,Y be topological vector spaces, and Λ be an equicontinuous family of linear operators from X to Y . -
Functional Analysis Solutions to Exercise Sheet 6
Institut für Analysis WS2019/20 Prof. Dr. Dorothee Frey 21.11.2019 M.Sc. Bas Nieraeth Functional Analysis Solutions to exercise sheet 6 Exercise 1: Properties of compact operators. (a) Let X and Y be normed spaces and let T 2 K(X; Y ). Show that R(T ) is separable. (b) Let X, Y , and Z be Banach spaces. Let T 2 K(X; Y ) and S 2 L(Y; Z), where S is injective. Prove that for every " > 0 there is a constant C ≥ 0 such that kT xk ≤ "kxk + CkST xk for all x 2 X. Solution: (a) Since bounded sets have a relatively compact image under T , for each n 2 N the sets T (nBX ) are totally bounded, where nBX := fx 2 X : kxk ≤ ng. Then it follows from Exercise 1 of Exercise sheet 3 that T (nBX ) is separable. But then ! [ [ T (nBX ) = T nBX = T (X) = R(T ) n2N n2N is also separable, as it is a countable union of separable sets. The assertion follows. (b) First note that by using a homogeneity argument, one finds that this result is true, ifand only it is true for x 2 X with kxk = 1. Thus, it remains to prove the result for x 2 X with kxk = 1. We argue by contradiction. If the result does not hold, then there is an " > 0 such that for all C ≥ 0 there exists and x 2 X with kxk = 1 such that kT xk > " + CkST xk. By choosing C = n for n 2 N, we find a sequence (xn)n2N in X with kxnk = 1 such that kT xnk > " + nkST xnk: (1) Since (xn)n2N is bounded and T is compact, there exists a subsequence (T xnj )j2N with limit y 2 Y . -
ARZEL`A-ASCOLI's THEOREM in UNIFORM SPACES Mateusz Krukowski 1. Introduction. Around 1883, Cesare Arzel`
DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018020 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 1, January 2018 pp. 283{294 ARZELA-ASCOLI'S` THEOREM IN UNIFORM SPACES Mateusz Krukowski∗ L´od´zUniversity of Technology, Institute of Mathematics W´olcza´nska 215, 90-924L´od´z, Poland Abstract. In the paper, we generalize the Arzel`a-Ascoli'stheorem in the setting of uniform spaces. At first, we recall the Arzel`a-Ascolitheorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper intro- duces the notion of the extension property which, similarly as equicontinuity, equates different topologies on C(X; Y ). This property enables us to prove the Arzel`a-Ascoli'stheorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki's theorem, we establish the relative compactness of 0 n subfamily of C(R; D (R )). 1. Introduction. Around 1883, Cesare Arzel`aand Giulio Ascoli provided the nec- essary and sufficient conditions under which every sequence of a given family of real-valued continuous functions, defined on a closed and bounded interval, has a uniformly convergent subsequence. Since then, numerous generalizations of this result have been obtained. For instance, in [10], p. 278 the compact families of C(X; R), with X a compact space, are exactly those which are equibounded and equicontinuous. The space C(X; R) is given with the standard norm kfk := sup jf(x)j; x2X where f 2 C(X; R). -
A General Product Measurability Theorem with Applications to Variational Inequalities
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 90, pp. 1{12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A GENERAL PRODUCT MEASURABILITY THEOREM WITH APPLICATIONS TO VARIATIONAL INEQUALITIES KENNETH L. KUTTLER, JI LI, MEIR SHILLOR Abstract. This work establishes the existence of measurable weak solutions to evolution problems with randomness by proving and applying a novel the- orem on product measurability of limits of sequences of functions. The mea- surability theorem is used to show that many important existence theorems within the abstract theory of evolution inclusions or equations have straightfor- ward generalizations to settings that include random processes or coefficients. Moreover, the convex set where the solutions are sought is not fixed but may depend on the random variables. The importance of adding randomness lies in the fact that real world processes invariably involve randomness and variabil- ity. Thus, this work expands substantially the range of applications of models with variational inequalities and differential set-inclusions. 1. Introduction This article concerns the existence of P-measurable solutions to evolution prob- lems in which some of the input data, such as the operators, forcing functions or some of the system coefficients, is random. Such problems, often in the form of evolutionary variational inequalities, abound in many fields of mathematics, such as optimization and optimal control, and in applications such as in contact me- chanics, populations dynamics, and many more. The interest in random inputs in variational inequalities arises from the uncertainty in the identification of the sys- tem inputs or parameters, possibly due to the process of construction or assembling of the system, and to the imprecise knowledge of the acting forces or environmental processes that may have stochastic behavior. -
Nonlinear Differential Inclusions of Semimonotone and Condensing Type in Hilbert Spaces
Bull. Korean Math. Soc. 52 (2015), No. 2, pp. 421–438 http://dx.doi.org/10.4134/BKMS.2015.52.2.421 NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE AND CONDENSING TYPE IN HILBERT SPACES Hossein Abedi and Ruhollah Jahanipur Abstract. In this paper, we study the existence of classical and gen- eralized solutions for nonlinear differential inclusions x′(t) ∈ F (t, x(t)) in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condens- ing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper. 1. Introduction Our aim is to study the nonlinear differential inclusion of first order x′(t) ∈ F (t, x(t)), (1.1) x(0) = x0, in a Hilbert spaces in which F may be condensing or semimonotone set-valued map. An approach to investigating the existence of solution for a differential inclusion is to reduce it to a problem for ordinary differential equations. In this setting, we are interested to know under what conditions the solution of the corresponding ordinary differential equation belongs to the right side of the given differential inclusion; in other words, under what conditions a continuous function such as f(·, ·) exists so that f(t, x) ∈ F (t, x) for every (t, x) and x′(t)= f(t, x(t)). -
REVIEW on SPACES of LINEAR OPERATORS Throughout, We Let X, Y Be Complex Banach Spaces. Definition 1. B(X, Y ) Denotes the Vector
REVIEW ON SPACES OF LINEAR OPERATORS Throughout, we let X; Y be complex Banach spaces. Definition 1. B(X; Y ) denotes the vector space of bounded linear maps from X to Y . • For T 2 B(X; Y ), let kT k be smallest constant C such that kT xk ≤ Ckxk, or equivalently kT k = sup kT xk : kxk=1 Lemma 2. kT k is a norm on B(X; Y ), and makes B(X; Y ) into a Banach space. Proof. That kT k is a norm is easy, so we check completeness of B(X; Y ). 1 Suppose fTjgj=1 is a Cauchy sequence, i.e. kTi − Tjk ! 0 as i; j ! 1 : For each x 2 X, Tix is Cauchy in Y , since kTix − Tjxk ≤ kTi − Tjk kxk : Thus lim Tix ≡ T x exists by completeness of Y: i!1 Linearity of T follows immediately. And for kxk = 1, kTix − T xk = lim kTix − Tjxk ≤ sup kTi − Tjk j!1 j>i goes to 0 as i ! 1, so kTi − T k ! 0. Important cases ∗ • Y = C: we denote B(X; C) = X , the \dual space of X". It is a Banach space, and kvkX∗ = sup jv(x)j : kxk=1 • If T 2 B(X; Y ), define T ∗ 2 B(Y ∗;X∗) by the rule (T ∗v)(x) = v(T x) : Then j(T ∗v)(x)j = jv(T x)j ≤ kT k kvk kxk, so kT ∗vk ≤ kT k kvk. Thus ∗ kT kB(Y ∗;X∗) ≤ kT kB(X;Y ) : • If X = H is a Hilbert space, we saw we can identify X∗ and H by the relation v $ y where v(x) = hx; yi : Then T ∗ is defined on H by hx; T ∗yi = hT x; yi : Note the identification is a conjugate linear map between X∗ and H: cv(x) = chx; yi = hx; cy¯ i : 1 2 526/556 LECTURE NOTES • The case of greatest interest is when X = Y ; we denote B(X) ≡ B(X; X). -
Arxiv:1706.03690V2 [Math.OA] 9 Apr 2018 Whether Alone
A NEW PROOF OF KIRCHBERG’S O2-STABLE CLASSIFICATION JAMES GABE Abstract. I present a new proof of Kirchberg’s O2-stable classification theorem: two ∗ separable, nuclear, stable/unital, O2-stable C -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are home- omorphic. Many intermediate results do not depend on pure infiniteness of any sort. 1. Introduction After classifying all AT-algebras of real rank zero [Ell93], Elliott initiated a highly am- bitious programme of classifying separable, nuclear C∗-algebras by K-theoretic and tracial invariants. During the past three decades much effort has been put into verifying such classification results. In the special case of simple C∗-algebras the classification has been verified under the very natural assumptions that the C∗-algebras are separable, unital, sim- ple, with finite nuclear dimension and in the UCT class of Rosenberg and Schochet [RS87]. The purely infinite case is due to Kirchberg [Kir94] and Phillips [Phi00], and the stably finite case was recently solved by the work of many hands; in particular work of Elliott, Gong, Lin, and Niu [GLN15], [EGLN15], and by Tikuisis, White, and Winter [TWW15]. This makes this the right time to gain a deeper understanding of the classification of non-simple C∗-algebras which is the main topic of this paper. The Cuntz algebra O2 plays a special role in the classification programme as this has the properties of being separable, nuclear, unital, simple, purely infinite, and is KK-equivalent ∗ to zero. Hence if A is any separable, nuclear C -algebra then A ⊗ O2 has no K-theoretic nor tracial data to determine potential classification, and one may ask if such C∗-algebras are classified by their primitive ideal space alone. -
Functional Analysis
Functional Analysis Lecture Notes of Winter Semester 2017/18 These lecture notes are based on my course from winter semester 2017/18. I kept the results discussed in the lectures (except for minor corrections and improvements) and most of their numbering. Typi- cally, the proofs and calculations in the notes are a bit shorter than those given in class. The drawings and many additional oral remarks from the lectures are omitted here. On the other hand, the notes con- tain a couple of proofs (mostly for peripheral statements) and very few results not presented during the course. With `Analysis 1{4' I refer to the class notes of my lectures from 2015{17 which can be found on my webpage. Occasionally, I use concepts, notation and standard results of these courses without further notice. I am happy to thank Bernhard Konrad, J¨orgB¨auerle and Johannes Eilinghoff for their careful proof reading of my lecture notes from 2009 and 2011. Karlsruhe, May 25, 2020 Roland Schnaubelt Contents Chapter 1. Banach spaces2 1.1. Basic properties of Banach and metric spaces2 1.2. More examples of Banach spaces 20 1.3. Compactness and separability 28 Chapter 2. Continuous linear operators 38 2.1. Basic properties and examples of linear operators 38 2.2. Standard constructions 47 2.3. The interpolation theorem of Riesz and Thorin 54 Chapter 3. Hilbert spaces 59 3.1. Basic properties and orthogonality 59 3.2. Orthonormal bases 64 Chapter 4. Two main theorems on bounded linear operators 69 4.1. The principle of uniform boundedness and strong convergence 69 4.2. -
Extremely Amenable Groups and Banach Representations
Extremely Amenable Groups and Banach Representations A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Javier Ronquillo Rivera May 2018 © 2018 Javier Ronquillo Rivera. All Rights Reserved. 2 This dissertation titled Extremely Amenable Groups and Banach Representations by JAVIER RONQUILLO RIVERA has been approved for the Department of Mathematics and the College of Arts and Sciences by Vladimir Uspenskiy Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 Abstract RONQUILLO RIVERA, JAVIER, Ph.D., May 2018, Mathematics Extremely Amenable Groups and Banach Representations (125 pp.) Director of Dissertation: Vladimir Uspenskiy A long-standing open problem in the theory of topological groups is as follows: [Glasner-Pestov problem] Let X be compact and Homeo(X) be endowed with the compact-open topology. If G ⊂ Homeo(X) is an abelian group, such that X has no G-fixed points, does G admit a non-trivial continuous character? In this dissertation we discuss some reformulations of this problem and its connections to other mathematical objects such as extremely amenable groups. When G is the closure of the group generated by a single map T ∈ Homeo(X) (with respect to the compact-open topology) and the action of G on X is minimal, the existence of non-trivial continuous characters of G is linked to the existence of equicontinuous factors of (X, T ). In this dissertation we present some connections between weakly mixing dynamical systems, continuous characters on groups, and the space of maximal chains of subcontinua of a given compact space.