## Tohoku Mathematical Journal

### Simultaneous similarity, bounded generation and amenability

Gilles Pisier

#### Abstract

We prove that a discrete group $G$ is amenable if and only if it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $\pi$. Analogously, a $C^*$-algebra $A$ is nuclear if and only if any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $L(A\otimes_{\max} B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $L(A\otimes_{\max} B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non-Abelian free group, then $A$ must be nuclear. We also show that $L(A\otimes_{\max} B)\le d$ if and only if the canonical quotient map from the unital free product $A\,{\ast}\, B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length at most $d$.

#### Article information

Source
Tohoku Math. J. (2), Volume 59, Number 1 (2007), 79-99.

Dates
First available in Project Euclid: 16 April 2007

https://projecteuclid.org/euclid.tmj/1176734749

Digital Object Identifier
doi:10.2748/tmj/1176734749

Mathematical Reviews number (MathSciNet)
MR2321994

Zentralblatt MATH identifier
1160.46037

#### Citation

Pisier, Gilles. Simultaneous similarity, bounded generation and amenability. Tohoku Math. J. (2) 59 (2007), no. 1, 79--99. doi:10.2748/tmj/1176734749. https://projecteuclid.org/euclid.tmj/1176734749

#### References

• J. Anderson, Extreme points in sets of positive linear maps on $\mathcalB(\mathcalH)$, J. Funct. Anal. 31 (1979), 195--217.
• D. Blecher and C. Le Merdy, Operator algebras and their modules, Oxford University Press, Oxford, 2004.
• F. Boca, Free products of completely positive maps and spectral sets, J. Funct. Anal. 97 (1991), 251--263.
• F. Boca, Completely positive maps on amalgamated product $C^*$-algebras, Math. Scand. 72 (1993), 212--222.
• M. Bożejko and G. Fendler, Herz-Schur multipliers and uniformly bounded representations of discrete groups, Arch. Math. (Basel) 57 (1991), 290--298.
• E. Christensen, Perturbations of operator algebras. II, Indiana Univ. Math. J. 26 (1977), 891--904.
• E. Christensen, E. Effros and A. M. Sinclair, Completely bounded multilinear maps and $C^*$-algebraic cohomology, Invent. Math. 90 (1987), 279--296.
• A. Connes, Classification of injective factors. Cases $\textitII_1$, $\textitII_\infty$, $\textitIII_\lambda$, $\lambda \not=1$, Ann. of Math. (2) 104 (1976), 73--115.
• E. Effros and Z. J. Ruan, Operator spaces, Oxford University Press, Oxford, 2000.
• U. Haagerup, Solution of the similarity problem for cyclic representations of $C^*$-algebras, Ann. of Math. (2) 118 (1983), 215--240.
• U. Haagerup, All nuclear $C^*$-algebras are amenable, Invent. Math. 74 (1983), 305--319.
• U. Haagerup and G. Pisier, Bounded linear operators between $C^*$-algebras, Duke Math. J. 71 (1993), 889--925.
• C. Lance, On nuclear $C\sp*$-algebras, J. Funct. Anal. 12 (1973), 157--176.
• C. Le Merdy, A strong similarity property of nuclear $C^*$-algebras, Rocky Mountain J. Math. 30 (2000), 279--292.
• M. Nagisa and S. Wada, Simultaneous unitarizability and similarity problem, Sci. Math. 2 (1999), 255--261 (electronic).
• A. Paterson, Amenability, Math. Surv. Monogr., vol. 29, American Mathematical Society, Providence, R.I., 1988.
• V. Paulsen, Completely bounded maps and operator algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge University Press, Cambridge, 2002.
• G. Pisier, Introduction to operator space theory, London Math. Soc. Lecture Note Ser., 294, Cambridge University Press, Cambridge, 2003.
• G. Pisier, Similarity problems and completely bounded maps, second, expanded edition, includes the solution to `The Halmos problem', Lecture Notes Math., vol. 1618, Springer, Berlin, 2001.
• G. Pisier, The operator Hilbert space $\mathrm OH$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), viii+103 pp.
• G. Pisier, Joint similarity problems and the generation of operator algebras with bounded length, Integral Equations Operator Theory 31 (1998), 353--370.
• G. Pisier, The similarity degree of an operator algebra, St. Petersburg Math. J. 10 (1999), 103--146.
• G. Pisier, A similarity degree characterization of nuclear $C^*$-algebras, Publ. Res. Inst. Math. Sci. 42 (2006), to appear.
• G. Pisier, Are unitarizable groups amenable? Infinite groups: geometric, combinatorial and dynamical spaces, 323--362, Progr. Math., 248, Birkhäuser, Basel, 2005.
• I. Raeburn and A. M. Sinclair, The $C\sp *$-algebra generated by two projections, Math. Scand. 65 (1989), 278--290.
• M. Takesaki, Theory of operator algebras I, Springer, New York, 1979.
• D. Voiculescu, K. Dykema and A. Nica, Free random variables, CRM Monogr. Ser. 1, American Mathematical Society, Providence, R.I., 1992.
• S. Wassermann, On tensor products of certain group $C^*$-algebras, J. Funct. Anal. 23 (1976), 239--254.