Abstract: Necessary and sufficient conditions on $P$ for the unit balls of $BL(\mathbf{R})$ and $\operatorname{Lip}(\mathbf{R})$ to be functional $P$-Donsker classes are obtained.

Abstract: Necessary and sufficent conditions are obtained for LU-factorization of operators on 11. In particular it is shown that uniform invertibility of the compressions of the operator is not sufficient to insure an LU-factorization of the operator, thus answering a question of de Boor, Jia, and Pinkus. The question of when a bounded linear operator on Ip, 1 < p < oo, has an LU-factorization has been much studied recently. Barkar and Gohberg [2] have shown that if A is an operator on lp which has an LU-factorization, then A and its compressions An = PnAPn are uniformly invertible, i.e. supn { -1AJ 1, IIAA'II < xo. In the other direction, various classes of operators such as invertible, diagonally dominant operators on 11 [7] and invertible, totally positive operators [3, 1] on lp have been shown to have LU-factorizations. For these kinds of operators it is known [1] that their compressions satisfy a stronger condition than uniform invertibility; namely, that the inverses of the compressions are order bounded, i.e. IlsupnIA-I1 11 < oo. Left open, then, is the possibility (first raised in [3] with a negative expectation) that uniform invertibility might be sufficient for a matrix operator on loo to have an LU-factorization. In this paper an example is given that shows that uniform invertibility is not sufficient for factoring an operator on lo (or 11). However, we also show that uniform invertibility of the compressions is sufficient to ensure an L U-factorization when the operator has an inverse whose columns decay at a certain rate away from the diagonal. Among the operators with this property are the banded operators. We wish to express thanks to the referee for several helpful suggestions. We now fix some terminology and notation. If x = (xi) is an element of 11 we denote its usual projection onto the span of the first n basis vectors by Pnx. A bounded linear operator A on 11 is said to be upper (respectively lower) triangular if PnAPn = APn (respectively PnA) for all n. We say that A is unit upper (lower) triangular if it is upper (lower) triangular and its diagonal entries in the matrix representation for A relative to the usual basis e, of 11 are all ones. An operator A is said to have an LU-factorization (relative to the usual basis e, of 11) if there exist invertible operators L and U so that A = LU and the operators L, L-1 are unit lower triangular while U, U-1 are upper triangular. An operator A is said to be Received by the editors June 10, 1985. 1980 Mathematics Subject Classification. Primary 47A68; Secondary 46E40. Kev words and phrases. LU-factorization, banded, triangular operator. ?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Abstract: Given a totally real algebraic number field K, we investigate when totally positive units, U?, are squares, u£. In particular, we prove that the rank of U? /Uji is bounded above by the minimum of (1) the 2-rank of the narrow class group of K and (2) the rank of Ul /U? as L ranges over all (finite) totally real extension fields of K. Several applications are also provided. 1. Notation and preliminaries. Let K be an algebraic number field and let CK denote the ideal class group in the ordinary or "wide" sense. Let CK+) denote the "narrow" ideal class group of A". Thus \CK\ = hK, the "wide" class number of K, and \CK+)\ = h(K+), the "narrow" class number of K. We denote the Hubert class field of K by A"(1); i.e., Gal(A"(1)/A") s CK, and we denote the "narrow" Hilbert class field by A~(+); i.e., Gal(A~(+)/A") s CK+\ Moreover we adopt the "bar" convention to mean "modulo squares"; for example, CK = CK/C\. Let UK denote the group of units of the ring of algebraic integers of K. When K is totally real, we let Ux denote the subgroup of totally positive units; i.e., those units u such that ua > 0 for all embeddings a of A' into R. Finally, for any finite abelian group A with \A\ = 2d, d is called the 2-rank of A, which we denote by dim2 A. 2. Results. We are concerned with the question: (*) When is U? = U21 We begin by observing that dim2(?7?) = 0 if and only if A"( + )=A"(1) [6, Theorem 3.1, p. 203]. In particular, when A" is a real finite Galois extension of 2-power degree over Q, then dim2(Ux) = 0 if and only if N(UK) = (±1) [3, Theorem 1, p. 166]. For example, when A is a real quadratic field, then dim2(U?) = 0 if and only if the norm of the fundamental unit is -1. Necessary and sufficient conditions (in terms of the arithmetic of the underlying quadratic field K ) for the existence of a fundamental unit of norm -1 are unknown (see [8]). This indicates the difficulty of solving (*) for the simplest even degree case. In this regard one may ask whether (*) is equivalent to such a norm statement for other fields. In a recent letter to the authors, V. Ennola answered (*) for cyclic cubic fields K as follows: Let e be a norm positive unit of A" such that -1 and the conjugates of e generate the unit group. Then dim2(Ux) = 0 if and only if e is not totally positive. However, as with Received by the editors February 18, 1985, and, in revised form, September 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R80, 11R27, 11R29; Secondary 11R37,11R32. 1 This author's research is supported by N.S.E.R.C. Canada. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Abstract: Resumé De nombreux travaux effectués durant ces 20 dernières années par les agrostologues de l'Institut d'Elevage et de Médeeine Vétérinaire des pays Tropicaux (I. E. M. V. T. — Maisons‐Alfort) ont conduit à définir, en République du Mali, divers groupements végétaux. Un séjour de quatre années eonsécutives (1974 à a 1977) a permis, par traitement informatique des données recueillies, de nommner diverses unités phytosociologiques dans une région du Mali central et ceci dans les milieux secs et humides. Seules les unités se trouvant dans les milieux humides seront présentées dans cet article; elles sont formées par: un Ordre, deux Allianees, deux Sous‐Alliances, et huit Associations.

Abstract: We consider two pairs (A,B) and (G,F) where A ⊂ B is a subalgebra over a commutative ring K with unit, F is a B-module and G ⊂ F is a sub-A-module. A relative extension of (A,B) by (G,F) is a commutative diagram bsoth rows being Hochschild extensions. The equivalence classes of relative extensions form a K-module Ex(A,B;G,F) while equipped with the usual (Baer) sum and scalar multiplication, and it is isomorphic to a fibre product modulo an operation: Here, the K-modules of cocycles and cochains are defined in the well-known Hochschild manner for (commutative unitary K-) algebras. Using this isomorphism, we establish some long exact sequences of relative derivation and extension modules which interlock in an interesting way.

Abstract: We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K 0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K 0 of a unit-regular ring or even as K 0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K 0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].