Summary
In this chapter, we first study inequalities satisfied by any bounded linear operator u: A→Y on a C*-algebra with values in a Banach space Y. The case when Y is another C*-algebra is of particular interest. Then we turn to homomorphisms u: A →* B(H) and prove that, if u is cyclic (= has a cyclic vector), boundedness implies complete boundedness. Hence bounded cyclic homomorphisms are similar to *-representations. This extends to homomorphisms with finite cyclic sets. We also include the positive solution to the similarity problem for C*-algebras without tracial states and for nuclear C*-algebras. Finally, we show that for a given C*-algebra, the similarity problem and the derivation problem are equivalent.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes and Remarks on Chapter 7
Pedersen G. C’-algebras and their automorphism groups. Academic Press, London, 1979.
Kadison R. and Ringrose J. Fundamentals of the theory of operator algebras, vol. II. Academic Press. New-York, 1986.
Haagerup U.: Solution of the similarity problem for cyclic representations of C’-algebras. Annals of Math. 118 (1983) 215–240.
Haagerup U. and Pisier G.: Linear operators between C’-algebras. Duke Math. J. 71 (1993) 889–925.
Pisier G. Grothendieck’s theorem for noncommutative C’-algebras with an appendix on Grothendieck’s constants. J. Funct. Anal. 29 (1978) 397–415.
Lust-Piquard F. and Pisier G.: Non commutative Khintchine and Paley inequalities. Arkiv fr Mat. 29 (1991) 241–260.
Haagerup U. and Pisier G.: Linear operators between C’-algebras. Duke Math. J. 71 (1993) 889–925.
Christensen E.: Extensions of derivations. J. Funct. Anal. 27 (1978) 234–247.
Christensen E.: Extensions of derivations II. Math. Scand. 50 (1982) 111–122.
Christensen E.: On non self adjoint representations of operator algebras. Amer. J. Math. 103 (1981) 817–834.
Kadison R. A note on the similarity problem. J. Operator Theory 26 (1991) 389–405.
Kadison R.: On an inequality of Haagerup-Pisier. J. Op. Theory 29 (1993) 57–68.
Bunce J. W.: The similarity problem for representations of C’-algebras. Proc. Amer. Math. Soc. 81 (1981) 409–414.
Kirchberg E.: The derivation and the similarity problem are equivalent. Preprint. August 94.
Christensen E.: Similarities of Ill factors with property T. Journal Operator Theory 15 (1986) 281–288.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pisier, G. (1996). The similarity problem for cyclic homomorphisms on a C*-algebra. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21537-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-21537-1_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60322-1
Online ISBN: 978-3-662-21537-1
eBook Packages: Springer Book Archive