Lectures on Noncommutative Geometry
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Noncommutative Uniform Algebras
Noncommutative Uniform Algebras Mati Abel and Krzysztof Jarosz Abstract. We show that a real Banach algebra A such that a2 = a 2 , for a A is a subalgebra of the algebra C (X) of continuous quaternion valued functions on a compact setk kX. ∈ H ° ° ° ° 1. Introduction A well known Hirschfeld-Zelazkoú Theorem [5](seealso[7]) states that for a complex Banach algebra A the condition (1.1) a2 = a 2 , for a A k k ∈ implies that (i) A is commutative, and further° ° implies that (ii) A must be of a very speciÞc form, namely it ° ° must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X)the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A,thatA is isometrically isomorphic with a subalgebra of CH (X) - the algebra of continuous quaternion valued functions on acompactsetX. That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeld-Zelazkoú Theorem, the analogous results can be stated for none unital algebras as one can formally add aunittosuchanalgebra. -
Isospectralization, Or How to Hear Shape, Style, and Correspondence
Isospectralization, or how to hear shape, style, and correspondence Luca Cosmo Mikhail Panine Arianna Rampini University of Venice Ecole´ Polytechnique Sapienza University of Rome [email protected] [email protected] [email protected] Maks Ovsjanikov Michael M. Bronstein Emanuele Rodola` Ecole´ Polytechnique Imperial College London / USI Sapienza University of Rome [email protected] [email protected] [email protected] Initialization Reconstruction Abstract opt. target The question whether one can recover the shape of a init. geometric object from its Laplacian spectrum (‘hear the shape of the drum’) is a classical problem in spectral ge- ometry with a broad range of implications and applica- tions. While theoretically the answer to this question is neg- ative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and opti- mization. In this paper, we introduce a numerical proce- dure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of an- other. We implement the isospectralization procedure using modern differentiable programming techniques and exem- plify its applications in some of the classical and notori- Without isospectralization With isospectralization ously hard problems in geometry processing, computer vi- sion, and graphics such as shape reconstruction, pose and Figure 1. Top row: Mickey-from-spectrum: we recover the shape of Mickey Mouse from its first 20 Laplacian eigenvalues (shown style transfer, and dense deformable correspondence. in red in the leftmost plot) by deforming an initial ellipsoid shape; the ground-truth target embedding is shown as a red outline on top of our reconstruction. -
2 Hilbert Spaces You Should Have Seen Some Examples Last Semester
2 Hilbert spaces You should have seen some examples last semester. The simplest (finite-dimensional) ex- C • A complex Hilbert space H is a complete normed space over whose norm is derived from an ample is Cn with its standard inner product. It’s worth recalling from linear algebra that if V is inner product. That is, we assume that there is a sesquilinear form ( , ): H H C, linear in · · × → an n-dimensional (complex) vector space, then from any set of n linearly independent vectors we the first variable and conjugate linear in the second, such that can manufacture an orthonormal basis e1, e2,..., en using the Gram-Schmidt process. In terms of this basis we can write any v V in the form (f ,д) = (д, f ), ∈ v = a e , a = (v, e ) (f , f ) 0 f H, and (f , f ) = 0 = f = 0. i i i i ≥ ∀ ∈ ⇒ ∑ The norm and inner product are related by which can be derived by taking the inner product of the equation v = aiei with ei. We also have n ∑ (f , f ) = f 2. ∥ ∥ v 2 = a 2. ∥ ∥ | i | i=1 We will always assume that H is separable (has a countable dense subset). ∑ Standard infinite-dimensional examples are l2(N) or l2(Z), the space of square-summable As usual for a normed space, the distance on H is given byd(f ,д) = f д = (f д, f д). • • ∥ − ∥ − − sequences, and L2(Ω) where Ω is a measurable subset of Rn. The Cauchy-Schwarz and triangle inequalities, • √ (f ,д) f д , f + д f + д , | | ≤ ∥ ∥∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ 2.1 Orthogonality can be derived fairly easily from the inner product. -
Regularity Conditions for Banach Function Algebras
Regularity conditions for Banach function algebras Dr J. F. Feinstein University of Nottingham June 2009 1 1 Useful sources A very useful text for the material in this mini-course is the book Banach Algebras and Automatic Continuity by H. Garth Dales, London Mathematical Society Monographs, New Series, Volume 24, The Clarendon Press, Oxford, 2000. In particular, many of the examples and conditions discussed here may be found in Chapter 4 of that book. We shall refer to this book throughout as the book of Dales. Most of my e-prints are available from www.maths.nott.ac.uk/personal/jff/Papers Several of my research and teaching presentations are available from www.maths.nott.ac.uk/personal/jff/Beamer 2 2 Introduction to normed algebras and Banach algebras 2.1 Some problems to think about Those who have seen much of this introductory material before may wish to think about some of the following problems. We shall return to these problems at suitable points in this course. Problem 2.1.1 (Easy using standard theory!) It is standard that the set of all rational functions (quotients of polynomials) with complex coefficients is a field: this is a special case of the “field of fractions" of an integral domain. Question: Is there an algebra norm on this field (regarded as an algebra over C)? 3 Problem 2.1.2 (Very hard!) Does there exist a pair of sequences (λn), (an) of non-zero complex numbers such that (i) no two of the an are equal, P1 (ii) n=1 jλnj < 1, (iii) janj < 2 for all n 2 N, and yet, (iv) for all z 2 C, 1 X λn exp (anz) = 0? n=1 Gap to fill in 4 Problem 2.1.3 Denote by C[0; 1] the \trivial" uniform algebra of all continuous, complex-valued functions on [0; 1]. -
Maxwell's Equations in Differential Form
M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 71 CHAPTER Maxwell’s Equations in Differential Form 3 In Chapter 2 we introduced Maxwell’s equations in integral form. We learned that the quantities involved in the formulation of these equations are the scalar quantities, elec- tromotive force, magnetomotive force, magnetic flux, displacement flux, charge, and current, which are related to the field vectors and source densities through line, surface, and volume integrals. Thus, the integral forms of Maxwell’s equations,while containing all the information pertinent to the interdependence of the field and source quantities over a given region in space, do not permit us to study directly the interaction between the field vectors and their relationships with the source densities at individual points. It is our goal in this chapter to derive the differential forms of Maxwell’s equations that apply directly to the field vectors and source densities at a given point. We shall derive Maxwell’s equations in differential form by applying Maxwell’s equations in integral form to infinitesimal closed paths, surfaces, and volumes, in the limit that they shrink to points. We will find that the differential equations relate the spatial variations of the field vectors at a given point to their temporal variations and to the charge and current densities at that point. In this process we shall also learn two important operations in vector calculus, known as curl and divergence, and two related theorems, known as Stokes’ and divergence theorems. 3.1 FARADAY’S LAW We recall from the previous chapter that Faraday’s law is given in integral form by d E # dl =- B # dS (3.1) CC dt LS where S is any surface bounded by the closed path C.In the most general case,the elec- tric and magnetic fields have all three components (x, y,and z) and are dependent on all three coordinates (x, y,and z) in addition to time (t). -
Inverse and Implicit Function Theorems for Noncommutative
Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains Mark E. Mancuso Abstract Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity. Keywords: Noncommutive functions, operator noncommutative functions, free anal- ysis, inverse and implicit function theorems, strong operator topology, dilation theory. MSC (2010): Primary 46L52; Secondary 47A56, 47J07. INTRODUCTION Polynomials in d noncommuting indeterminates can naturally be evaluated on d-tuples of square matrices of any size. The resulting function is graded (tuples of n × n ma- trices are mapped to n × n matrices) and preserves direct sums and similarities. Along with polynomials, noncommutative rational functions and power series, the convergence of which has been studied for example in [9], [14], [15], serve as prototypical examples of a more general class of functions called noncommutative functions. The theory of non- commutative functions finds its origin in the 1973 work of J. L. Taylor [17], who studied arXiv:1804.01040v2 [math.FA] 7 Aug 2019 the functional calculus of noncommuting operators. Roughly speaking, noncommutative functions are to polynomials in noncommuting variables as holomorphic functions from complex analysis are to polynomials in commuting variables. -
Isospectral Towers of Riemannian Manifolds
New York Journal of Mathematics New York J. Math. 18 (2012) 451{461. Isospectral towers of Riemannian manifolds Benjamin Linowitz Abstract. In this paper we construct, for n ≥ 2, arbitrarily large fam- ilies of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada's method. Contents 1. Introduction 451 2. Genera of quaternion orders 453 3. Arithmetic groups derived from quaternion algebras 454 4. Isospectral towers and chains of quaternion orders 454 5. Proof of Theorem 4.1 456 5.1. Orders in split quaternion algebras over local fields 456 5.2. Proof of Theorem 4.1 457 6. The Sunada construction 458 References 461 1. Introduction Let M be a closed Riemannian n-manifold. The eigenvalues of the La- place{Beltrami operator acting on the space L2(M) form a discrete sequence of nonnegative real numbers in which each value occurs with a finite mul- tiplicity. This collection of eigenvalues is called the spectrum of M, and two Riemannian n-manifolds are said to be isospectral if their spectra coin- cide. Inverse spectral geometry asks the extent to which the geometry and topology of M is determined by its spectrum. Whereas volume and scalar curvature are spectral invariants, the isometry class is not. Although there is a long history of constructing Riemannian manifolds which are isospectral but not isometric, we restrict our discussion to those constructions most Received February 4, 2012. -
Chapter 2 C -Algebras
Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e. -
Algebra Homomorphisms and the Functional Calculus
Pacific Journal of Mathematics ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARK PHILLIP THOMAS Vol. 79, No. 1 May 1978 PACIFIC JOURNAL OF MATHEMATICS Vol. 79, No. 1, 1978 ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARC THOMAS Let b be a fixed element of a commutative Banach algebra with unit. Suppose σ(b) has at most countably many connected components. We give necessary and sufficient conditions for b to possess a discontinuous functional calculus. Throughout, let B be a commutative Banach algebra with unit 1 and let rad (B) denote the radical of B. Let b be a fixed element of B. Let έ? denote the LF space of germs of functions analytic in a neighborhood of σ(6). By a functional calculus for b we mean an algebra homomorphism θr from έ? to B such that θ\z) = b and θ\l) = 1. We do not require θr to be continuous. It is well-known that if θ' is continuous, then it is equal to θ, the usual functional calculus obtained by integration around contours i.e., θ{f) = -±τ \ f(t)(f - ]dt for f eέ?, Γ a contour about σ(b) [1, 1.4.8, Theorem 3]. In this paper we investigate the conditions under which a functional calculus & is necessarily continuous, i.e., when θ is the unique functional calculus. In the first section we work with sufficient conditions. If S is any closed subspace of B such that bS Q S, we let D(b, S) denote the largest algebraic subspace of S satisfying (6 — X)D(b, S) = D(b, S)f all λeC. -
The Weyl Functional Calculus F(4)
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 4, 240-267 (1969) The Weyl Functional Calculus ROBERT F. V. ANDERSON Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by Edward Nelson Received July, 1968 I. INTRODUCTION In this paper a functional calculus for an n-tuple of noncommuting self-adjoint operators on a Banach space will be proposed and examined. The von Neumann spectral theorem for self-adjoint operators on a Hilbert space leads to a functional calculus which assigns to a self- adjoint operator A and a real Borel-measurable function f of a real variable, another self-adjoint operatorf(A). This calculus generalizes in a natural way to an n-tuple of com- muting self-adjoint operators A = (A, ,..., A,), because their spectral families commute. In particular, if v is an eigenvector of A, ,..., A, with eigenvalues A, ,..., A, , respectively, and f a continuous function of n real variables, f(A 1 ,..., A,) u = f(b , . U 0. This principle fails when the operators don’t commute; instead we have the Uncertainty Principle, and so on. However, the Fourier inversion formula can also be used to define the von Neumann functional cakuius for a single operator: f(4) = (37F2 1 (Sf)(&) exp( -2 f A,) 4 E’ where This definition generalizes naturally to an n-tuple of operators, commutative or not, as follows: in the n-dimensional Fourier inversion 240 THE WEYL FUNCTIONAL CALCULUS 241 formula, the free variables x 1 ,..,, x, are replaced by self-adjoint operators A, ,..., A, . -
Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
Published in: Journal of Geometric Mechanics doi:10.3934/jgm.2017014 Volume 9, Number 3, September 2017 pp. 335{390 GEOMETRY OF MATRIX DECOMPOSITIONS SEEN THROUGH OPTIMAL TRANSPORT AND INFORMATION GEOMETRY Klas Modin∗ Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Abstract. The space of probability densities is an infinite-dimensional Rie- mannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher{Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry|the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multi- variate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher{Rao geometries are discussed. The link to matrices is obtained by considering OMT and information ge- ometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent descrip- tion of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decomposi- tions (smooth and linear category), entropy gradient flows, and the Fisher{Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. -
Some C*-Algebras with a Single Generator*1 )
transactions of the american mathematical society Volume 215, 1976 SOMEC*-ALGEBRAS WITH A SINGLEGENERATOR*1 ) BY CATHERINE L. OLSEN AND WILLIAM R. ZAME ABSTRACT. This paper grew out of the following question: If X is a com- pact subset of Cn, is C(X) ® M„ (the C*-algebra of n x n matrices with entries from C(X)) singly generated? It is shown that the answer is affirmative; in fact, A ® M„ is singly generated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. If A is a separable C*-algebra with identity, then A ® K and A ® U are shown to be sing- ly generated, where K is the algebra of compact operators in a separable, infinite- dimensional Hubert space, and U is any UHF algebra. In all these cases, the gen- erator is explicitly constructed. 1. Introduction. This paper grew out of a question raised by Claude Scho- chet and communicated to us by J. A. Deddens: If X is a compact subset of C, is C(X) ® M„ (the C*-algebra ofnxn matrices with entries from C(X)) singly generated? We show that the answer is affirmative; in fact, A ® Mn is singly gen- erated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. Working towards a converse, we show that A ® M2 need not be singly generated if A is generated by a set con- sisting of four elements.