Skip to main content Accessibility help
×
  1. Home
  2. Browse subjects
  3. Mathematics
  4. Abstract analysis
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

25994 results in Abstract analysis

Page 913 of 1300

  • 1 - Introduction

  • Vern Paulsen, University of Houston
  • Book:
    Completely Bounded Maps and Operator Algebras
    Published online:
    24 November 2009
    Print publication:
    06 February 2003, pp 1-8

  • Summary

    It is assumed throughout this book that the reader is familiar with operator theory and the basic properties of C*-algebras (see for example [76] and [8, Chapter 1]). We concentrate primarily on giving a self-contained exposition of the theory of completely positive and completely bounded maps between C*-algebras and the applications of these maps to the study of operator algebras, similarity questions, and dilation theory. In particular, we assume that the reader is familiar with the material necessary for the Gelfand–Naimark–Segal theorem, which states that every C*-algebra has a one-to-one, ∗-preserving, norm-preserving representation as a norm-closed, ∗-closed algebra of operators on a Hilbert space.

    In this chapter we introduce some of the key concepts that will be studied in this book.

    As well as having a norm, a C*-algebra also has an order structure, induced by the cone of positive elements. Recall that an element of a C*-algebra is positive if and only if it is self-adjoint and its spectrum is contained in the nonnegative reals, or equivalently, if it is of the form a*a for some element a. Since the property of being positive is preserved by ∗-isomorphism, if a C*-algebra is represented as an algebra of operators on a Hilbert space, then the positive elements of the C*-algebra coincide with the positive operators that are contained in the representation of the algebra.

  • 3 - Completely Positive Maps

  • Vern Paulsen, University of Houston
  • Book:
    Completely Bounded Maps and Operator Algebras
    Published online:
    24 November 2009
    Print publication:
    06 February 2003, pp 26-42

  • Summary

    Let A be a C*-algebra, and let be a subspace. Then we shall call an operator space. Clearly, Mn can be regarded as a subspace of Mn(A), and we let Mn have the norm structure that it inherits from the (unique) norm structure on the C*-algebra Mn(A). We make no attempt at this time to define a norm structure on Mn without reference to A. Thus, one thing that distinguishes from an ordinary normed space is that it comes naturally equipped with norms on Mn for all n ≥ 1. Later in this book we shall give a more axiomatic definition of operator spaces, at which time we shall begin to refer to subspaces of C*-algebras as concrete operator spaces. For now we simply stress that by an operator space we mean a concrete subspace of a C*-algebra, together with this extra “baggage” of a well-defined sequence of norms on Mn. Similarly, if S ⊆ A is an operator system, then we endow Mn(S) with the norm and order structure that it inherits as a subspace of Mn(A).

    As before, if B is a C*-algebra and ϕ: S → B is a linear map, then we define ϕn: Mn(S) → Mn(B) by ϕn((ai, j)) = (ϕ(ai, j)). We call ϕ n-positive if ϕn is positive, and we call ϕ completely positive if ϕ is n-positive for all n.

  • Corrigendum

  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 133 / Issue 1 / February 2003
    Published online by Cambridge University Press:
    12 July 2007, p. 1
    Print publication:
    February 2003
    • Article
      • You have access
    • PDF
    • Export citation

  • Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

  • Kenji Nishihara
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 133 / Issue 1 / February 2003
    Published online by Cambridge University Press:
    12 July 2007, pp. 177-196
    Print publication:
    February 2003

  • Consider the Cauchy problem for a one-dimensional compressible flow through porous media,Hsiao and Liu showed that the solution (υ, u) behaves as the diffusion wave (ῡ, ū), i.e. the solution of the porous-media equation due to the Darcy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When υ0(x) has the same constant state at x = ±∞, the convergence rate ‖(υ − ῡ)(·,t)‖L∞ = O(t−1) obtained is ‘optimal’, since ‖ῡ(·,t)‖∞ = O(t−1/2). However, this ‘optimal’ convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the ‘truly optimal’ convergence rate by choosing suitably located diffusion waves.

Page 913 of 1300