Complex Numbers Of Absolute Value Research Articles

Multiplier systems for Siegel modular groups

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

Representations of abelian algebraic groups

There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension

Modulatable orthogonal sequences and their application to SSMA systems

A set of N-1 orthogonal sequences of period N/sup 2/ is proposed, where N is a natural number. Each orthogonal sequence proposed can be modulated by N complex numbers of absolute value 1, so the modulated sequence is also orthogonal. When N is an odd prime number, the absolute value of the cross-correlation function between any two of the N-1 orthogonal sequences is constant and satisfies the mathematical lower bound. This property of the cross-correlation function is not changed when each of the two orthogonal sequences is modulated by N complex numbers of absolute value 1. Two spread-spectrum multiple-access (SSMA) systems using these sequences are proposed. One system is an asynchronous SSMA system, using the proposed sequences unmodulated. The cochannel interference peak between any two channels in this system realizes the mathematical lower bound for an asynchronous SSMA system using a set of orthogonal sequences. The other system is a synchronous SSMA system without cochannel interference which uses the modulated form of the proposed sequences. >

A Pointwise Ergodic Theorem for the Group of Rational Rotations

Let f be a bounded, measurable function defined on the multiplicative group $\Omega$ of complex numbers of absolute value 1, and define \begin{equation}\tag {$(1)$}{{f_n}(\omega ) = \frac {1} {n}\sum \limits _{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,\end{equation} where ${z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $n(k)$ is an increasing sequence of positive integers with $n(k)$ dividing $n(k’)$ whenever $k < k’$, then ${f_{n(k)}}$ converges almost surely as $k \to \infty$.

On unitary/antiunitary projective representations of groups

Unitary/antiunitary projective representations of groups (i.e., projective representations of groups where unitary as well as antiunitary operators in a separable complex Hilbert space are considered) are studied in a systematic way. Particular emphasis is put on continuous unitary/antiunitary projective representations of a Polish group G. It is shown that every continuous unitary/antiunitary projective representation of G can be lifted to a Borel unitary/antiunitary multiplier representation of G (namely, to a representation “up to a factor” which is a Borel mapping) and that this, in turn, can be derived from a continuous unitary/antiunitary (ordinary) representation of a Polish group obtained from an extension of G by the multiplicative group of all complex numbers of absolute value 1.

Automorphisms of commutative Banach algebras

This paper presents a new proof of the theorem of Kamowitz and Scheinberg which states that if α \alpha is an element of infinite order of the automorphism group of a commutative semisimple Banach algebra then the spectrum of α \alpha contains all complex numbers of absolute value 1. The proof depends on the fact that the only closed translation invariant subalgebras of l ∞ ( − ∞ , + ∞ ) {l^\infty }( - \infty , + \infty ) (pointwise multiplication) for which the restriction of the shift has a complex number of absolute value 1 in its resolvent set are certain spaces of periodic sequences.

The punctured plane is isomorphic to the unit circle

This note shows that the multiplicative group of non-zero complex numbers and the multiplicative group of complex numbers of absolute value 1 are isomorphic as groups.

Deceptive convergence of Fourier series on 𝑆𝑈(2)

measure on G will be denoted by jt, and we normalize jt so that ti(G)= 1. By a representation of G we will mean a finite-dimensional unitary representation of G. Let U be a faithful representation of G and let V= U 03 U be the direct sum of U and its complex conjugate representation. Let To(U) be the space of constant functions on G, let T1(U) be the linear space spanned by To(U) and the coordinate functions of V, and for each integer n ? 1 let Tn(U) be the space spanned by all functions ff,2-fn where J E T1(U) for 1 ? i? n. We will call Tn(U) the space of U-trigonometric polynomials of degree ? n. For each nonnegative integer n let Un be the orthogonal projection of L2(G) onto Tn(U). There is a natural way to extend Un to a projection from L1(G) onto Tn(U) (see (2.1)) and we will denote this extension by Un. Iff is a function in L1(G) we will call the sequence {Unf} the U-Fourier series for f, and we will call Unf the nth partial sum of the U-Fourier series for f If f E L2(G, y), then Unf -?f in L2(G, [u) as n -s o. If x E G then the U-Fourier series for f at x is defined to be the sequence {Unf(x)}. If G= T is the group of complex numbers of absolute value 1, and U is the 1-dimensional representation of T on C defined by

Unimodular contractions in Hilbert space

Let be a unitary operator on a Hubert space H. Then in particular, (i) is a contraction, i.e. || T 1; and (ii) The spectrum of is a subset of the unit circle, i.e. Sp (T)cC, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras.

Locally compact topologies for groups

The investigations leading to this paper were suggested by the papers of Hewitt [3] and Ross [5]. In [3] Hewitt was interested in proving that if an abelian group is locally compact in two topologies, one strictly stronger than the other, there is a character continuous in one topology and discontinuous in the other (actually a special case of a theorem of Kaplansky-see Theorem 1.1 of [2]). Actually Hewitt proved a stronger result. His arguments were based on the fact that both the additive group of reals, and the multiplicative group of complex numbers of absolute value 1 have the property that every stronger locally compact group topology is discrete. A natural question to ask is what other groups have this property. The answer is very simple (2.1 of this paper). We consider in the second section of this paper the obvious generalization. Namely which groups have the property that there are only finitely many stronger locally compact group topologies. The investigations in the first section of this paper were suggested by the paper of Ross [5]. Ross was considering the same question as Hewitt, and was led to consider the relationship between two locally compact group topologies on a group G such that G has the same closed subgroups in the two topologies. We investigate this further in the first section of this paper, and are able to say that many of the properties of G as a topological group can be recovered once we know the closed subgroups.

Some theorems on Fourier coefficients

where P is given by (1.1)? If one allows the coefficients Cn to be complex numbers of absolute value 1, an affirmative answer to the question is furnished by the partial sums of the series ZE inlogneinO; this example is due to Hardy and Littlewood [4, pp. 116-118]. A theorem of Salem and Zygmund [2, pp. 270, 278] shows, roughly speaking, that (N log N)12 is the most probable order of magnitude for ||P||X if C-n = ? 1. During the summer of 1958, Salem drew my attention to the problem stated in the first paragraph. It turns out that an affirmative answer can be given by a construction which uses nothing more sophisticated than the parallelogram law

Isomorphism of AW ∗ -Algebras

H. A. Dye, in his paper On the geometry of projections in certain operator algebras,' has shown that the structure of the projection lattice of an AW*-algebra determines the structure of the algebra. He then goes on to show that, in the case of a factor not of type 12n, the structure of projection lattice is determined by the unitary structure. The crucial step in his proof lies in the study of the family of unitary operators p (Xe +1e) where e is a fixed projection and X takes on all complex numbers of absolute value 1. Following his method by examining the operators p(Xe + 1 e) in greater detail we are able to extend his result. However, since we rely heavily on the existence of four orthogonal equivalent projections, we are not able to get any information about algebras of type I2n+l1 We first collect a few results about AW*-algebras which are pertinent to our discussion. (a) If e, f are orthogonal equivalent projections then there exists a unitary operator u such that f=ueu-1 and u(1 -e-f) = (1 -e-f)u 1 e -f. In fact, if v is a partial isometry with vv =e, v* =f, then u=v+v*+(1-e-f) will do. (b) A projection e is called properly infinite if, for every central projection g, ge is either infinite or zero. Given any projection e there is a central projection g such that ge is properly infinite (or zero) and (1 -g)e is finite. (c) By the central cover of a projection e we mean the least central projection containing e. Let e be properly infinite and f finite. Then, if e, f have the same central cover, e contains infinitely many orthogonal projections equivalent to f. Let M, N be AW*-algebras having no component of type Inv p an algebraic isomorphism between the unitary groups M. and N.. De-

A theorem on power series whose coefficients have given signs

This theorem answers in the negative the question: Is there a universal scrambling {I ek}I Ei,k = ?1, turning every power series EakZk with positive coefficients into a power series >.-kakZk having the circle of convergence as natural boundary? This problem was raised by Mrs. Turan, and I am indebted to Dr. P. Erdos for communicating it to me. An example (?4) shows that the semi-circle in the statement of the theorem can not be replaced by a larger arc. A question which remains open is to find a corresponding theorem for the case that {11} is a given sequence of complex numbers of absolute value one.

The homology of Kummer manifolds

The theory of Kummer surfaces is a classical topic in algebraic geometry.1 A generalization to higher dimensions has been given by W. Wirtinger.2 The varieties introduced by Wirtinger may be called Kummer varieties and have algebraic dimension n with 22n ordinary double points. If these double points are desingularized in a standard manner by means of dilatations, one obtains nonsingular varieties whose underlying manifolds will be called Kummer manifolds of complex dimension n (topological dimension 2n). The question of the torsion of Kummer varieties has been discussed recently by A. Andreotti;3 since this author, however, did not desingularize the varieties under consideration, his results give no answer to the basic question whether they admit nonramified coverings or not. This question will be settled here by showing that the desingularized model mentioned above is simply connected ; moreover, its homology groups will be determined for all dimensions.4 From a purely topological point of view all Kummer manifolds of a given dimension are homeomorphic to one another and may be defined as follows. Let T denote the 2«-dimensional torus (n ^ 2) regarded as the 2w-fold product of the complex numbers of absolute value one with itself. For a subset s of the integers 1, ■ ■ ■ , 2n let p, denote that point of T defined by

Linear functions on almost periodic functions

Our aim is, first, to present two realizations of the space 1* of all bounded complex linear functionals on 21, and, second, to use these realizations for the study of positive definite functions which are not necessarily continuous. In this fashion, we obtain a generalization of Bochner's representation theorem for continuous positive definite functions as well as various facts concerning positive definite functions and their structure. 0.2 Throughout the present paper, the symbol R denotes the real numbers, considered either as an additive group or as a field; K the field of complex numbers; T the multiplicative group of complex numbers of absolute value 1; Tm the complete Cartesian product of m groups each identical with T, m being any cardinal number greater than 1. The characteristic function of a subset B of a set X is denoted by XB. If G is any locally compact Abelian group, we denote the group of all continuous characters of G by the symbol G]. G] is given the usual compact-open topology. If X is any topological space, we denote the set of all complex-valued continuous functions on X which are bounded in absolute value by the symbol C(X). The space of all trigonometric polynomials 0.1.1 is denoted by 93; the space of all almost periodic continuous functions on R by 2f. For a normed complex linear space V, we denote the space of all bounded complex linear functionals on V by the symbol V*.