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25994 results in Abstract analysis

Page 898 of 1300

  • ROBERT WINSTON KEITH ODONI (1947–2002)

  • Stephen D. Cohen
  • Journal:
    Glasgow Mathematical Journal / Volume 45 / Issue 3 / September 2003
    Published online by Cambridge University Press:
    10 September 2003, pp. 569-575
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  • Born in London on 14 July 1947, Robert Winston Keith Odoni was the eldest of the five children of Walter Anthony and Lois Marie Theresa Odoni. The family name is Italian: grandfather Alfred Odoni had come to England from Switzerland in the 1920s and set up a manufacturing business in Birmingham, later transferring to London. Robert's father continued the business (then making cycle stands) until the 1970s. His mother was from Boulder in Colorado: they had met in America whilst he was on wartime naval service.

  • TRANSLATION GENERALIZED QUADRANGLES FOR WHICH THE TRANSLATION DUAL ARISES FROM A FLOCK

  • KOEN THAS
  • Journal:
    Glasgow Mathematical Journal / Volume 45 / Issue 3 / September 2003
    Published online by Cambridge University Press:
    10 September 2003, pp. 457-474
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  • It is shown that each finite translation generalized quadrangle (TGQ) $\mathcal{S}$, which is the translation dual of the point-line dual of a flock generalized quadrangle, has a line $[\infty]$ each point of which is a translation point. This leads to the fact that the full group of automorphisms of $\mathcal{S}$ acts $2$-transitively on the points of $[\infty]$, and the observation applies to the point-line duals of the Kantor flock generalized quadrangles, the Roman generalized quadrangles and the recently discovered Penttila-Williams generalized quadrangle. Moreover, by previous work of the author, the non-classical generalized quadrangles (GQ's) which have two distinct translation points, are precisely the TGQ's of which the translation dual is the point-line dual of a non-classical flock GQ.

    We emphasize that, for a long time, it has been thought that every non-classical TGQ which is the translation dual of the point-line dual of a flock GQ has only one translation point. There are important consequences for the theory of generalized ovoids (or eggs) in PG$(4n - 1,q)$, the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG$(3,q)$, subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles. Several problems on these topics will be solved completely.

  • 0 - Introduction

  • Gilles Pisier, Texas A & M University and Université de Paris VI (Pierre et Marie Curie)
  • Book:
    Introduction to Operator Space Theory
    Published online:
    05 October 2013
    Print publication:
    25 August 2003, pp 1-14

  • Summary

    The theory of operator spaces is very recent. It was developed after Ruan's thesis (1988) by Effros and Ruan and Blecher and Paulsen. It can be described as a noncommutative Banach space theory. An operator space is simply a Banach space given together with an isometric linear embedding into the space B(H) of all bounded operators on a Hilbert space H. In this new category, the objects remain Banach spaces but the morphisms become the completely bounded maps (instead of the bounded linear ones). The latter appeared in the early 1980s following Stinespring's pioneering work (1955) and Arveson's fundamental results (1969) on completely positive maps. We study completely bounded (in short c.b.) maps in Chapter 1. This notion became important in the early 1980s through the independent work of Wittstock [Wit1–2], Haagerup [H4], and Paulsen [Pa2]. These authors independently discovered, within a short time interval, the fundamental factorization and extension property of c.b. maps (see Theorem 1.6).

    For the reader who might wonder why c.b. maps are the “right” morphisms for the category of operator spaces, here are two arguments that come to mind: Consider E1 ⊂ B(H1) and E2 ⊂ B(H2) and let π: B (H1) → B(H2) be a C*-morphism (i.e. a *-homomorphism) such that π(E1) ⊂ E2. Then, quite convincingly, u = π∣E1 : E1 → E2 should be an “admissible” morphism in the category of operator spaces. Let us call these morphisms of the “first kind.”

Page 898 of 1300