Etant donné un opérateur T sur un espace LP (1 < p < ∞), la théorie ergodique s'intéresse à la convergence presque sûre des moyennes de Césaro

des itérés d'un point f de Lp par T. On dit que T vérifie le théorème ergodique si cette convergence a lieu pour tout f de Lp.

Parmi les nombreux résultats sur cette question (cf. [21]) nous citerons d'abord ceux de A. Ionescu-Tulcea ([19]) et R. Chacon-S. A. McGrath ([10]) que l'on peut réunir dans l'assertion suivante:

“Si 1 < p < ∞ et T une isométrie positive de Lp, ou si 1 < p ≠ 2 < ∞ et si T est une isométrie surjective de Lp, alors T vérifie le théorème ergodique”.

Nous nous intéressons ici à une version vectorielle de ce théorème. Plus précisément, si X est un espace de Banach réel, un opérateur linéare T sur l'espace LP(X) est dit vérifier le théorème ergodique vectoriel si la suite des moyennes de Césaro

converge presque sûrement en norme dans X, quel que soit f ∊ Lp(X).

In this paper we study unit preserving *-endomorphisms of and type II1 factors. A *-endomorphism α which has the property that the intersection of the ranges of αn for n = 1 , 2 , … consists solely of multiples of the unit are called shifts. In Section 2 it is shown that shifts of can be characterized up to outer conjugacy by an index n = ∞ 1, 2 , …. In Section 3 shifts of R the hyperfinite II1 factor are studied. An outer conjugacy invariant of a shift of R is the Jones index [R: α(R)]. In Section 3 a class of shifts of index 2 are studied. These are called binary shifts. It is shown that there are uncountably many binary shifts which are pairwise non conjugate and among the binary shifts there are at least a countable infinity of shifts which are pairwise not outer conjugate.

A function f in Hp on the unit disc U of the complex plane has the uniform growth

We consider in this paper a subspace of Hp with better uniform growth

For the previous results on see [5, 6, 7]. We start with proving an inequality on Hp which is related to the Hardy-Stein identity (Theorem 2.1) in Section 2. This is applied in the subsequent section to prove some space imbedding theorems related to (Theorems 3.1 and 3.5). These theorems have some known theorems as their corollaries. Finally we prove some coefficient relations on in the last section.

The authors wish to thank Professor Patrick Ahern for the helpful conversations during his visit to Korea. Actually he suggested to the first author the possibility of Theorem 2.1 some years ago.

In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C*-algebra is said to be of class i, i ∊ {1, 2, 3}, if the range of any normalized trace on its K0 group has rank i as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras.

The mod torsion Hurewicz map

for compact Lie groups provides a useful and efficient means of studying G. In effect, it measures how far G fails to be a product of spheres. For the Hopf-Samelson theorem (see [17]) tells us that

In other words

Serre pointed out that there exists a map

inducing this Q isomorphism.

Conditions for a subgroup, F, of PSL(2, R) to be discrete have been investigated by a number of authors. Jørgensen's inequality [5] gives an elegant necessary condition for discreteness for subgroups of PSL(2, C). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21] studied two generator discrete subgroups of PSL(2, R) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20].

In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL(2, R) is discrete if and only if every non-elementary subgroup, G, generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL(2, R) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).

The concepts of Choquet boundary and Shilov boundary are well-established in the context of a complex function algebra (see [2] for example). There have been a few attempts to develop the concept of a Shilov boundary for real algebras, [4], [6] and [7]. But there seems to be none to develop the concept of Choquet boundary for real algebras.

The aim of this paper is to develop the theory of Choquet boundary of a real function algebra (see Definition (1.8)) along the lines of the corresponding theory for a complex function algebra.

In the first section we define a real-part representing measure for a continuous linear functional ϕ on a real function algebra A with the property ║ϕ║ = 1 = ϕ(1). The elements of A are functions on a compact, Hausdorff space X. The Choquet boundary is then defined as the set of those points x ∊ X such that the real part of the evaluation functional, Re(ex), has a unique real part representing measure.

A finite group G is a spherical space form group if it admits a fixed point free representation τ:G → U(k) for some k; for the remainder of this paper, we assume G is such a group. The eta invariant of Atiyah et al [2] defines Q/Z valued invariants of equivariant bordism. In [6], we showed the eta invariant completely detects the reduced equivariant unitary bordism groups and completely detects all but the 2-primary part of the reduced equivariant SpinC bordism groups . The coefficient ring is without torsion; all the torsion in is of order 2. The prime 2 plays a distinguished role in the discussion of equivariant SpinC bordism and is quite different from at the prime 2. Let ker*(η, G) denote the kernel of all eta invariants and let ker*(SW, G) denote the kernel of the Z2-equivariant Stiefel-Whitney numbers (see Section 1 for details). Then:

THEOREM 0.1. Let. If M = ker*(η, G) ∩ ker*(SW, G), M = 0.

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).

Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.

(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2)

exists a.e. for all.

In [5] C. Akemann and G. Pedersen defined four concepts of semicontinuity for elements of A**, the enveloping W*-algebra of a C*-algebra A. For three of these the associated classes of lower semicontinuous elements are , and (notation explained in Section 2), and we will call these the classes of strongly lsc, middle lsc, and weakly lsc elements, respectively. There are three corresponding concepts of continuity: The strongly continuous elements are the elements of A itself, the middle continuous elements are the multipliers of A, and the weakly continuous elements are the quasi-multipliers of A. It is natural to ask the following questions, each of which is three-fold.

(Q1) Is every lsc element the limit of a monotone increasing net of continuous elements?

(Q2) Is every positive lsc element the limit of an increasing net of positive continuous elements?

(Q3) If h ≧ k, where h is lsc and k is usc, does there exist a continuous x such that h ≧ x ≧ k?

We present here multiparameter results about positive operators acting on a weakly sequentially complete Banach lattice. Sections 1, 2 and 3 generalize results obtained by M. A. Akcoglu and the second author in the case of a contraction. Even in that case, the classical L1 theory extends to Banach lattices only under an additional monotonicity assumption (C), introduced in [3], without which the TL (or stochastic) ergodic theorem fails. The example proving this in [4] also shows that, without (C), the decomposition of the space into the “positive” part P, the largest support of a T-invariant element, and the “null” part N on which the TL limit is zero (see, e.g., [22], p. 141), also fails. If T is not a contraction but only mean-bounded, then the space decomposes into the “remaining” part Y, the largest support of a T*-invariant element, and the “disappearing part“ Z (see, e.g., [22], p. 172). Here we obtain, for Banach lattices and in the multiparameter case, a unified proof of both decompositions, and of the TL ergodic theorem.

Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such that

uniformly on F as n → ∞.

The transformations we will discuss in this paper are the Hankel transformation Hυ defined for f ∊ C0, the collection of continuous functions compactly supported in (0, ∞), by

(1.1)

and the and transformations defined for such f by

(1.2)

and

(1.3)

where Jv >and Yv are the Bessel functions of the first and second kinds respectively, and Hv is the Struve function; for the theory of these functions see [1, Chapter VII].

These transformations were studied extensively by one of us in [5] and [6] on the spaces defined in [7; Sections 1 & 5]. In those papers the boundedness of the three transformations was fully given on the spaces for 1 < p < ∞, but not for p = 1. Also inversion formulae were given for the transformations only for portions of their respective ranges of boundedness.

Let G be a reductive group over a field of characteristic zero. Fix a Borel subgroup B of G which contains a maximal torus T. For each dominant weight X we have an irreducible representation V(X) of G with highest weight X. For two dominant representation X1 and X2 we have a decomposition

This decomposition is determined by the element

of the group ring of the group of characters of T.

The objective of this paper is to compute r(X1, X2) for all pairs X1 and X2 of fundamental weights. This will be used to compute the equations for cones over homogeneous spaces. This problem immediately reduces to the case when G has simple type; An, Bn, Cn, Dn, E6, E7, E8, F4 and G2. We will give complete details for the classical types. For the case An we will work with GLn.

Fix a point on a Riemannian manifold, and consider the tangent space V at the point equipped with inner product g. The Riemann curvature tensor R and its first covariant derivative ∇R at the point are tensors in and . If we take all the known symmetries of these tensors we can define subspaces and such that R ∊ Curv and ∇R ∊ ∇Curv. Also, the orthogonal group O(g) acts naturally on all these spaces. The two fundamental problems of the linear algebra of the spaces Curv and ∇Curv are: (1) find the decomposition into irreducible representations of O(g), with corresponding projection operators, (2) give a description of the structure of the O(g) orbits, by means of orbit invariant functions and a canonical form for elements of each orbit.

Let L be a finitely generated extension of a field K having characteristic p ≠ 0. A rank t higher derivation on L over K is a sequence

of additive maps of K into K such that

d0 is the identity map and dt(x) = 0, i > 0, x ∊ K. [6] contains the relevant background material on higher derivations. By Zorn's Lemma, there are maximal separable extensions of K in L. A maximal separable extension D of K in L is called distinguished if

Dieudonné [4] established that any finitely generated extension always has distinguished subfields. L has the same dimension over any distinguished subfield [5], and this dimension is called the order of inseparability of L/K. The least n such that K(LP)n is separable over K is called the inseparable exponent of L/K, inex(L/K).

In order to compute the group K*(Ω3S3X; Z/2) when X is a finite, torsion free CW-complex we apply the techniques developed by Snaith in [38], [39], [40], [41] which were used in [42] to determine the Atiyah-Hirzebruch spectral sequence ( [11], [1, Part III])

for X as above. Roughly speaking the method consists in defining certain classes in K*(Ω3S3X; Z/2) via the π-equivariant mod 2 K-homology of S2 × Y2,

([35]), π the cyclic group of order 2 (acting antipodally on S2, by permuting factors in Y2, and diagonally on S2 × Y2), Y a finite subcomplex of Ω3S3X, and then showing that the classes so produced map under the edge homomorphism to cycles (in the E1-term of the Atiyah-Hirzebruch spectral sequence for

which determine certain homology classes of H*(Ω3S3X; Z/2), thus exhibiting these as infinite cycles of the spectral sequence

Let An be the automorphism group of the free group Fn of rank n, and let Kn be the normal subgroup of An consisting of those elements which induce the identity automorphism in the commutator quotient group . The group Kn has been called the group of IA automorphisms of Fn (see e.g. [1]). It was shown by Magnus [7] using earlier work of Nielsen [11] that Kn is finitely generated, with generating set the automorphisms

and

where x1, x2, …, xn, is a chosen basis of Fn.

Let X and Y be 1-connected spaces having the homotopy type of cw-complexes.

Definition 0.1. A continuous map f:X → Y is Ω-split if Ωf:ΩX → ΩY admits a retraction up to homotopy.

In [6] we prove the following “mapping theorem”:

THEOREM 0.1. (a) If f is Ω-split, then cat(X) ≦ cat(Y);

(b) If π*(f) is split injective and ΩY has the homotopy type of a product of Eilenberg-MacLane spaces, then f is Ω-split.

Let Bτ denote the class of entire functions of exponential type τ (>0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143]

1.1

where t, γ are real numbers and is the so called conjugate function of f. Let us put

1.2

The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε > 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.