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A Simple C*-Algebra Generated by Two Finite-Order Unitaries

Published online by Cambridge University Press:  20 November 2018

Man-Duen Choi
Man-Duen Choi*
Affiliation:
University of Toronto, Toronto, Ontario
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We present an example which illustrates several peculiar phenomena that may occur in the theory of C*-algebras. In particular, we show that a C*-subalgebra of a nuclear (amenable) C*-algebra need not be nuclear (amenable).

The central object of this paper is a pair of abstract unitary matrices,

acting on a common Hilbert space. For an explicit construction, we may decompose an infinite-dimensional Hilbert space H into H = H0 ⴲ H1 , H1 = Hα ⴲ Hβ with dim H0 = dim H1 = dim Hα = dim Hβ, letting u, v Є B(H) be any two unitary operators such that

and u2 = 1, v3 = 1. Whereas many choices of u, v are possible, it might be surprising to see that C*(u, v), the C*-algebra generated by u and v, is algebraically unique; namely, if (u1,V1) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C*(u1, v1) (Theorem 2.6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 867 - 880
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Akemann, C. A. and Ostrand, P. A., Computing norms in group C*-algebras, Amer. J. Math. 98 (1976), 1015–1047.Google Scholar
2. Anderson, J., A C*-algebra A for which Ext A is not a group, Ann. of Math. 107 (1978), 455–458.Google Scholar
3. Apostal, C., Foias, C., and Voiculescu, D., Some results on non-quasi-triangular operators, IV, Rev. Roumaine Math. Pures Appl. 18 (1973), 487–514.Google Scholar
4. Bunce, J., Characterizations of amenable and strongly amenable C*''-algebras, Pacific J. Math. 43 (1972), 563–572.Google Scholar
5. Bunce, J., Finite operators and amenable C*-algebras, Proc. Amer. Math. Soc. 56 (1976), 145–151.Google Scholar
6. Bunce, J. and Paschke, W. L., Quasi-expectations and amenable von Neumann algebras, Proc. Amer. Math. Soc. 71 (1978), 232–236.Google Scholar
7. Choi, M. D., A Schwarz inequality for positive linear maps on C*-algebras, I1L J. Math. 18 (1974), 565–574.Google Scholar
8. Choi, M. D. and Effros, E. G., Injectivity and operator spaces, J. Func. Anal. 24 (1977), 156–209.Google Scholar
9. Choi, M. D. and Effros, E. G., Nuclear C*-algebras and the approximation property, Amer. J. Math. 100 (1978), 61–79.Google Scholar
10. Choi, M. D. and Effros, E. G., Separable nuclear C*-algebras and injectivity, Duke Math. J. 43 (1976), 309–322.Google Scholar
11. Choi, M. D. and Effros, E. G., Nuclear C*-algebras and injectivity: the general case, Indiana U. Math. J. 26 (1977), 443–446.Google Scholar
12. Choi, M. D. and Effros, E. G., Lifting problems and the cohomology of C*-algcbros, Can. J. Math. 29 (1977), 1092–1111.Google Scholar
13. Connes, A., On the cohomology of operator algebras, J. Func. Anal. 28 (1978), 248–253.Google Scholar
14. Cnntz, J.: Simple C*-algebras generated by isometrics, Commun. Math. Phys. 57 (1977), 173–185.Google Scholar
15. Effros, E. G., Aspects of non-commutative order , Notes for a lecture given at the Second U.S.-Japan Seminar on C*-algebras and Applications to Physics, 1977.Google Scholar
16. Johnson, B. E., Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).Google Scholar
17. Lance, E. C., On nuclear C*-algebras, J. Func. Anal. 12 (1973) 157–176.Google Scholar
18. Powers, R. T., Simplicity of the C*-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151–156.Google Scholar
19. Rosenberg, J., Amenability of cross products of C*-algebras, Comm. Math. Phys. 57 (1977), 187–191.Google Scholar
20. Stinespring, W. F., Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.Google Scholar
21. Takesaki, M., On the crossnorm of the direct product of C*-algebras, Tôhoku Math. J. 16 (1964), 111–122.Google Scholar
22. Thayer, F. J., Quasi-diagonal C*-algebras, J. Func. Anal. 25 (1977), 50–57.Google Scholar
23. Tomiyama, J., Tensor products and projections of norm one in von Neumann algebras, Lecture Notes, Univ. of Copenhagen, 1970.Google Scholar
24. Wassermann, S., On tensor products of certain group C*-algebras, J. Func. Anal. 28 (1976), 239–254.Google Scholar