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Combinatorial Set Theory of C*-algebras pp 3–46Cite as

C∗-algebras, Abstract, and Concrete

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Abstract

In this chapter we introduce the abstract C∗-algebras and work towards the Gelfand–Naimark–Segal Theorem (Theorem 1.10.1). Along the way we discuss abelian C∗-algebras and Gelfand–Naimark and Stone dualities, continuous functional calculus, positivity in C∗-algebras, approximate units, and quasi-central approximate units.

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Notes

  1. 1.

    The density character of a topological space is the minimal cardinality of a dense subset.

  2. 2.

    Any similarity with the weakly compact large cardinals [148] is accidental—“weakly compact” means “compact in the weak topology”.

  3. 3.

    Here, and elsewhere, we use logician’s convention that \(\bar a\) stands for a tuple (a0, …, an−1) of an unspecified length n.

  4. 4.

    Warning: In the theory of operator algebras “contractive” is synonymous with “of norm ≤ 1”.

  5. 5.

    Not to be confused with the spectrum of an operator!

  6. 6.

    That is \(|b|+\frac 1n\), certainly not (|b| + 1)∕n!

  7. 7.

    For those readers who may prefer a precise definition of F = (Fi,j) by its matrix entries:Fi,i := Fi+1 − Fi, Fi,i+1 = Fi+1,i := (Fi(1 − Fi))1∕2, and Fi,j := 0 if |i − j|≥ 2.

  8. 8.

    See however footnote 2 in Chapter 12.

  9. 9.

    Used in the proof of Theorem 1.3.1.

  10. 10.

    Used in the proof of Lemma 1.5.7.

  11. 11.

    Used in the proof of Lemma 1.3.4.

  12. 12.

    Used in the proof of Lemma 1.4.7.

  13. 13.

    Parts of this exercise will be used throughout this book.

  14. 14.

    Used in the proof of Proposition 1.6.8, twice.

  15. 15.

    Used in the proof of Lemma 5.6.2.

  16. 16.

    Used in the proof of Lemma 3.6.3.

  17. 17.

    Used in the proof of Proposition 14.1.6.

  18. 18.

    Used in the proof of Lemma 1.8.4.

  19. 19.

    Used in the proof of Corollary 3.2.6.

  20. 20.

    Used in the proofs of Proposition 3.6.5 and Lemma 5.3.6.

  21. 21.

    Used in the proof of Lemma 4.3.2.

  22. 22.

    Used in the proof of Lemma 1.8.4.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Ilijas Farah

  2. Matematički Institut SANU, Beograd, Serbia

    Ilijas Farah

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  1. Ilijas Farah
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Farah, I. (2019). C∗-algebras, Abstract, and Concrete. In: Combinatorial Set Theory of C*-algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27093-3_1

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