Abstract
In this chapter we give basic examples of constructions C∗-algebras: direct sums and products, inductive limits, stabilization, suspension, cone, hereditary C∗-subalgebras, quotients, tensor products, full and reduced group C∗-algebras, and full and reduced crossed products. Finite-dimensional C∗-algebras and ∗-homomorphisms between them are classified by Bratteli diagrams. Universal C∗-algebras given by generators and relations are studied in some detail. After a discussion of automorphisms of C∗-algebras, we conclude with a section on C∗-algebras of real rank zero.
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Notes
- 1.
This is a special case of the tensor product, defined in Section 2.4.1 below.
- 2.
The general definition of a corner will be given once we introduce the multiplier algebra (Definition 13.1.7). According to this definition, \(\mathscr K(K)\) is a corner of \(\mathscr K(H)\) for every closed subspace K of H.
- 3.
We will discuss relations, and universal algebras generated by relations, at length in Section 2.3.
- 4.
These relations can be realized only by unbounded self-adjoint operators; see, e.g., [125, p. 63].
- 5.
- 6.
In this exercise {0} is not considered to be a C∗-algebra.
- 7.
Used in the proof of Proposition 4.4.2.
- 8.
Used in the proof of Theorem 15.4.5.
- 9.
Used in the proof of Proposition 17.5.4.
- 10.
Used in the proof of Proposition 17.5.4.
- 11.
Used in the proof of Theorem 5.5.4.
- 12.
Used in the proof of Theorem 6.1.3.
- 13.
K-theoretic methods [208] can be used to prove, e.g., that \(M_n(\mathscr O_3)\cong \mathscr O_3\) if and only if n is odd.
- 14.
All this works for locally compact abelian groups, but we defined \(\mathrm {C}^*_r(\varGamma )\) for discrete Γ only.
- 15.
Used in the proof of Theorem 14.4.7.
- 16.
Warning: The operator algebraist’s definition of an ordered abelian group differs from model theorist’s definition of an ordered abelian group, as in the former the ordering is not required to be linear.
References
Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In: Chatzidakis, Z., et al. (eds.) Model Theory with Applications to Algebra and Analysis, vol. II, pp. 315–427. London Mathematical Society Lecture Notes Series, no. 350. London Mathematical Society, London (2008)
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)
Cuntz, J.: Simple C∗-algebras generated by isometries. Commun. Math. Phys. 57(2), 173–185 (1977)
Davidson, K.R.: C∗-Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)
Farah, I., Hart, B., Lupini, M., Robert, L., Tikuisis, A., Vignati, A., Winter, W.: Model theory of C∗-algebras, Memoirs AMS (to appear)
Farah, I., Katsura, T.: Nonseparable UHF algebras II: Classification. Math. Scand. 117(1), 105–125 (2015)
Hall, B.C.: Quantum Theory for Mathematicians. Graduate Texts in Mathematics, vol. 267. Springer, Berlin (2013)
Junge, M., Pisier, G.: Bilinear forms on exact operator spaces and B(H) ⊗ B(H). Geom. Funct. Anal. 5(2), 329–363 (1995)
Loring, T.A.: Lifting solutions to perturbing problems in C∗-algebras. Fields Institute Monographs, vol. 8. American Mathematical Society, Providence (1997)
Ozawa, N., Pisier, G.: A continuum of C∗-norms on \(\mathcal B(H)\otimes \mathcal B(H)\) and related tensor products. Glasg. Math. J. 58(02), 433–443 (2016)
Phillips, N.C.: An introduction to crossed product C∗-algebras and minimal dynamics. Preprint, available at http://pages.uoregon.edu/ncp/Courses, 5 Feb 2017
Rørdam, M.: Classification of Nuclear C∗-Algebras. Encyclopaedia of Mathematical Sciences, vol. 126. Springer, Berlin (2002)
Rørdam, M., Larsen, F., Laustsen, N.J.: An introduction to K-theory for C∗ algebras. London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)
Williams, D.P.: Crossed Products of C∗-Algebras. Mathematical Surveys and Monographs, vol. 134. American Mathematical Society, Providence (2007)
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Farah, I. (2019). Examples and Constructions of C∗-algebras. In: Combinatorial Set Theory of C*-algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27093-3_2
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