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Birkhäuser

Functional Identities

  • Book
  • © 2007

Overview

Authors:
  1. Matej Brešar

    1. Department of Mathematics and Computer Science FNM, University of Maribor, Maribor, Slovenia

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  2. Mikhail A. Chebotar

    1. Department of Mathematical Sciences, Kent State University, Kent, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Wallace S. Martindale

    1. Department of Mathematics, University of Massachusetts, Amherst, USA

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  • First monograph devoted to functional identities and accessible to a wider audience
  • Touching a variety of mathematical areas such as ring theory, algebra and operator theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Frontiers in Mathematics (FM)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. An Introductory Course

    1. Front Matter

      Pages 1-1
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    2. What is a Functional Identity?

      Pages 3-28

    3. The Strong Degree and the FI-Degree

      Pages 29-45

  3. The General Theory

    1. Front Matter

      Pages 47-47
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    2. Constructing d-Free Sets

      Pages 49-85

    3. Functional Identities on d-Free Sets

      Pages 87-110

    4. Functional Identities in (Semi)prime Rings

      Pages 111-144

  4. Applications

    1. Front Matter

      Pages 145-145
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    2. Lie Maps and Related Topics

      Pages 147-188

    3. Linear Preserver Problems

      Pages 189-219

    4. Further Applications to Lie Algebras

      Pages 221-233

  5. Back Matter

    Pages 235-272
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Keywords

About this book

A functional identity (FI) can be informally described as an identical relation involving(arbitrary)elementsinaringtogetherwith(“unknown”)functions;more precisely,elementsaremultipliedbyvaluesoffunctions.ThegoalofthegeneralFI theory is to determine the form of these functions, or, when this is not possible, to determine the structure of the ring admitting the FI in question. This theory has turnedouttobeapowerfultoolfor solvingavarietyofproblemsindi?erentareas. It is not always easy to recognize that the problem in question can be interpreted through some FI; often this is the most intriguing part of the process. But once one succeeds in discovering an FI that ?ts into the general theory, this abstract theory then as a rule yields the desired conclusions at a high level of generality. Among classical algebraic concepts, the one of a polynomial identity (PI) seems to be, at least on the surface, the closest one to the concept of an FI. In fact, a PI is formally just a very special example of an FI (where functions are polynomials).However,the theoryof PI’shasquite di?erent goalsthan the theory of FI’s. One could say, especially from the point of view of applications, that the twotheoriesarecomplementaryto eachother.Under somenaturalrestrictions,PI theorydealswithringsthatareclosetoalgebrasoflowdimensions,whileFItheory gives de?nitive answers in algebras of su?ciently large or in?nite dimensions.

Authors and Affiliations

  • Department of Mathematics and Computer Science FNM, University of Maribor, Maribor, Slovenia

    Matej Brešar

  • Department of Mathematical Sciences, Kent State University, Kent, USA

    Mikhail A. Chebotar

  • Department of Mathematics, University of Massachusetts, Amherst, USA

    Wallace S. Martindale

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