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2 - Preliminaries

Published online by Cambridge University Press:  19 January 2010

Kalyan B. Sinha
Affiliation:
Indian Statistical Institute, New Delhi
Debashish Goswami
Affiliation:
Indian Statistical Institute, Kolkata
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Summary

In this chapter we shall introduce all the basic materials and preliminary notions needed later on in this book.

C* and von Neumann algebras

For the details on the material of this section, the reader may be referred to [125], [40] and [76].

C*-algebras

An abstract normed *-algebra A is said to be a pre C*-algebra if it satisfies the C*-property : ‖x*x‖ = ‖x2. If A is furthermore complete under the norm topology, one says that A is a C*-algebra. The famous structure theorem due to Gelfand, Naimark and Segal (GNS) asserts that every abstract C*-algebra can be embedded as a norm-closed *-subalgebra of B(H) (the set of all bounded linear operators on some Hilbert space H). In view of this, we shall fix a complex Hilbert space H and consider a concrete C*-algebra A inside B(H). The algebra A is said to be unital or nonunital depending on whether it has an identity or not. However, even any nonunital C*-algebra always has a net (sequence in case the algebra is separable in the norm topology) of approximate identity, that is, an nondecreasing net eμ of positive elements such that eμaa for all aA. Note that the set of compact operators on an infinite dimensional Hilbert space H, to be denoted by K(H), is an example of nonunital C*-algebra.

We now briefly discuss some of the important aspects of C*-algebra theory. First of all, let us mention the following remarkable characterization of commutative C*-algebras.

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Publisher: Cambridge University Press
Print publication year: 2007

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  • Preliminaries
  • Kalyan B. Sinha, Indian Statistical Institute, New Delhi, Debashish Goswami
  • Book: Quantum Stochastic Processes and Noncommutative Geometry
  • Online publication: 19 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511618529.004
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  • Preliminaries
  • Kalyan B. Sinha, Indian Statistical Institute, New Delhi, Debashish Goswami
  • Book: Quantum Stochastic Processes and Noncommutative Geometry
  • Online publication: 19 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511618529.004
Available formats
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Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Preliminaries
  • Kalyan B. Sinha, Indian Statistical Institute, New Delhi, Debashish Goswami
  • Book: Quantum Stochastic Processes and Noncommutative Geometry
  • Online publication: 19 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511618529.004
Available formats
×