An entropy-theoretic proof of the Sucheston zero-one law

The purpose of this note is to provide a new proof to the following reformulation of the Sucheston zero-one law: An automorphism T of a Lebesgue space X is mixing if and only if, for every subsequence A of the sequence of natural numbers and every partition α of X having finite entropy, there exists a subsequence B = {b(j)} of A such that Λ_{r = 1}^∞ V_{j = r}^∞T^b(j)(α) is the trivial partition.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

On Strong Uniform Distribution II

 Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers an sequence if for every f in ? we have

Continuity properties of decomposable probability measures on euclidean spaces

It is shown that every full e^A decomposable probability measure on R^k, where A is a linear operator all of whose eigenvalues have negative real part, is either absolutely continuous with respect to Lebesgue measure or continuous singular with respect to Lebesgue measure. This result is used to characterize the continuity properties of random variables that are limits of solutions of certain stochastic difference equations.     

A generalization to the non-separable case of Takesaki's duality theorem forC *-algebras

Takesaki [5] poses the question of how much information about aC ^*-algebraA is contained in its representation theory. He gives it a precise meaning in the following setting: One can furnish the set Rep (A:H) of all representations ofA in a suitable Hilbert spaceH with a topology, with an action of the unitary groupG ofB(H) on it, and with an addition. The setA ^F of operator fields Rep (A:H)B(H) commuting with the action ofG and addition, called the admissible operator fields, turn out to form aW ^*-algebra isomorphic to the bidual ofA with Arens multiplication or with the universal enveloping von Neumann algebra ofA. Takesaki shows in the separable case thatA can be identified inA ^F as the set of continuous admissible operator fields, and leaves the same question open for arbitraryC ^*-algebras. Changing the structures on Rep(A:H) slightly, it is shown here that this result obtains in the general case as well. The proof proceeds along the lines set up in [5] but makes no use of the representation theory of NGCR algebras.     

A theorem connected with results of Steinhaus and Smítal

We will prove that if A and B are subsets of the real line, each having positive outer Lebesgue measure, then A + B, the set of all numbers a + b with a ϵ A and b ϵ B, is “full,” in the sense of outer Lebesgue measure, in some interval K. This result is related to theorems of Steinhaus and Smítal.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Die struktur endlicher schwach abgeschlossener Jordanalgebren Teil II. Diskrete Jordanalgebren

In the preceding note [6] we reduced the study of continuous finite weakly closed Jordan algebras to real associative W^*-algebras of type II₁. Here we treat the remaining case of discrete finite weakly closed Jordan algebras and describe them completely by finite dimensional simple formally real Jordan algebras and by simple formally real Jordan algebras of quadratic forms of real Hilbert spaces. Jacobsons theory of Jordan algebras with minimum condition combined with W^*-algebra techniques constitutes an essential tool in the proof.     

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

The generalized Eckart-Young principal axis theorem

A rectangular matrix [a_pq ] is said to be diagonal if a_pq = 0 when p ≠ q. We present a simple proof of the following theorem of Wiegmann, but in principle given earlier by Eckart and Young: THEOREM If {A_i) is a set of complex r – s matrices such thatA A andA A are Hermitian for all i andj, then there exist unitary matrices P and Q such that for each i the matrixpA Q is real and diagonal Special cases of the above are well known and extremely useful. For example, the case n = 1 yields the classical singular value decomposition.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Construction of polynomial superalgebras

In the algebraA, defined equationally, we construct a basis (standard) with with a suitable law of multiplication and we give the transition formula from this basis to the original one so thatA has the structure of a Z₂-graded space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 257–261, February, 1990.     

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+A^t = J ? I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).     

On the positivity of the upper Lyapunov exponent in one-parameter families of linear differential systems

For a special class of linear differential systems affinely depending on a real parameter m, we obtain tests for the positivity of the upper characteristic exponent λ(A _μ ) for all μ in some set J of positive Lebesgue measure.     

STRONG LAWS FOR α-MIXING SEQUENCE PROCESSES INDEXED BY SETS

ＳＴＲＯＮＧＬＡＷＳＦＯＲα－ＭＩＸＩＮＧＳＥＱＵＥＮＣＥＰＲＯＣＥＳＳＥＳＩＮＤＥＸＥＤＢＹＳＥＴＳ￥ＸＵＢＩＮＧＡｂｓｔｒａｃｔ：ＬｅｔＪ＝｛１，２，．．．｝ｄａｎｄｌｅｔ｛Ｘｊ，ｊ∈Ｊ｝ｂｅａｎａ－ｍｉｘｉｎｇｓｅｑｕｅｎｃｅｗｈｉｃｈｉｓｎｏ...     

An elementary proof of the strong law of large numbers

Summary In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.