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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The density character of a topological space is the minimal cardinality of a dense subset.
- 2.
Any similarity with the weakly compact large cardinals [148] is accidental—“weakly compact” means “compact in the weak topology”.
- 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.
Warning: In the theory of operator algebras “contractive” is synonymous with “of norm ≤ 1”.
- 5.
Not to be confused with the spectrum of an operator!
- 6.
That is \(|b|+\frac 1n\), certainly not (|b| + 1)∕n!
- 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.
- 9.
Used in the proof of Theorem 1.3.1.
- 10.
Used in the proof of Lemma 1.5.7.
- 11.
Used in the proof of Lemma 1.3.4.
- 12.
Used in the proof of Lemma 1.4.7.
- 13.
Parts of this exercise will be used throughout this book.
- 14.
Used in the proof of Proposition 1.6.8, twice.
- 15.
Used in the proof of Lemma 5.6.2.
- 16.
Used in the proof of Lemma 3.6.3.
- 17.
Used in the proof of Proposition 14.1.6.
- 18.
Used in the proof of Lemma 1.8.4.
- 19.
Used in the proof of Corollary 3.2.6.
- 20.
- 21.
Used in the proof of Lemma 4.3.2.
- 22.
Used in the proof of Lemma 1.8.4.
References
Arveson, W.: Notes on extensions of C∗-algebras. Duke Math. J. 44, 329–355 (1977)
Arveson, W.: A Short Course on Spectral Theory. Graduate Texts in Mathematics, vol. 209. Springer, New York (2002)
Bice, T., Vignati, A.: C∗-algebra distance filters. Adv. Oper. Theory 3(3), 655–681 (2018)
Blackadar, B.: Operator Algebras: Theory of C∗-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, vol. 122. Operator Algebras and Non-commutative Geometry, III. Springer, Berlin (2006)
Brown, N., Ozawa, N.: C∗-Algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence (2008)
Dales, H.G., Woodin, W.H.: An Introduction to Independence for Analysts. London Mathematical Society Lecture Note Series, vol. 115. Cambridge University Press, Cambridge (1987)
Husemoller, D.: Fibre Bundles. Graduate Texts in Mathematics, vol. 20, 3rd edn. Springer, New York (1994)
Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Perspectives in Mathematical Logic. Springer, Berlin (1995)
Kirchberg, E., Rørdam, M.: Infinite non-simple C∗-algebras: absorbing the cuntz algebra \(\mathcal O_\infty \). Adv. Math. 167(2), 195–264 (2002)
Kishimoto, A.: The representations and endomorphisms of a separable nuclear C∗-algebra. Int. J. Math. 14(03), 313–326 (2003)
Latrémolière, F.: The quantum Gromov-Hausdorff propinquity. Trans. Am. Math. Soc. 368(1), 365–411 (2016) MR 3413867
Murphy, G.J.: C∗-Algebras and Operator Theory. Academic Press, Boston (1990)
Ozawa, N.: An invitation to the similarity problems after Pisier (operator space theory and its applications). Kyoto Univ. Res. Inf. Repository 1486, 27–40 (2006)
Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)
Pedersen, G.K.: C∗-algebras and their automorphism groups. London Mathematical Society Monographs, vol. 14. Academic Press, London (1979)
Pedersen, G.K.: Analysis Now. Graduate Texts in Mathematics, vol. 118. Springer, New York (1989)
Runde, V.: The structure of discontinuous homomorphisms from non-commutative C∗-algebras. Glasg. Math. J. 36(2), 209–218 (1994). MR 1279894
Thiel, H., Winter, W.: The generator problem for Z-stable C∗-algebras. Trans. Am. Math. Soc. 366(5), 2327–2343 (2014)
Voiculescu, D.: Around quasidiagonal operators. Integr. Equ. Oper. Theory 17(1), 137–149 (1993)
Weaver, N.: Mathematical Quantization. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-27093-3_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-27091-9
Online ISBN: 978-3-030-27093-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)