Exact C*-Algebras, Tensor Products, and the Classification of Purely Infinite Algebras

Our survey (and the reference list) does not reflect the history of tensor products of operator algebras. Here we make use of tensor product functors on the category of C-algebras as a unifying principle. An application of our theory to the classification problem of Elliott [13] can be found at the end of this paper. Throughout the paper algebra means C*-algebra and vN-algebra means von Neumann algebra. L (H) is the algebra of bounded operators on a Hilbert space of infinite dimension.

[1]  Eberhard Kirchberg,et al.  On non-semisplit extensions, tensor products and exactness of groupC*-algebras , 1993 .

[2]  D. Voiculescu.  Around quasidiagonal operators , 1993 .

[3]  G. Kasparov.  THE OPERATOR K-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS , 1981 .

[4]  S. Wassermann.  Exact C*-algebras and related topics , 1994 .

[5]  E. Kirchberg.  On Restricted Perturbations in Inverse Images and a Description of Normalizer Algebras in C*-Algebras , 1995 .

[6]  Simon Wassermann.  C*-algebras associated with groups with Kazhdan's property T , 1991 .

[7]  S. Wassermann.  On tensor products of certain group C∗-algebras , 1976 .

[8]  U. Haagerup.  Quasitraces on Exact $C^\ast$-Algebras are Traces , 1994 .

[9]  Z. Ruan.  Amenability of Hopf von Neumann Algebras and Kac Algebras , 1996 .

[10]  Exact operator spaces , 1993, math/9308204.

[11]  E. Kirchberg,et al.  OnC*-algebras having linear, polynomial and subexponential growth , 1992 .

[12]  E. Kirchberg.  Commutants of unitaries in UHF algebras and functorial properties of exactness. , 1994 .

[13]  S. Adams.  Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups , 1994 .

[14]  M. Rørdam.  Classification of inductive limits of Cuntz algebras. , 1993 .

[15]  Raoul Robert,et al.  Statistical Mechanics and Hydrodynamical Turbulence , 1995 .

[16]  E. Kirchberg.  On Subalgebras of the Car-Algebra , 1995 .

[17]  R. Douglas,et al.  Relative K homology and C* algebras , 1991 .

[18]  C. Batty,et al.  C*-Tensor Norms and Slice Maps , 1980 .

[19]  Man-Duen Choi.  A schwarz inequality for positive linear maps on $C^{\ast}$-algebras , 1974 .

[20]  Vern I. Paulsen,et al.  Tensor products of operator spaces , 1991 .

[21]  U. Haagerup,et al.  Approximation properties for group *-algebras and group von Neumann algebras , 1994 .

[22]  J. Cuntz.  K-theory for certain C-algebras , 1981 .

[23]  George A. Elliott.  The Classification Problem for Amenable C*-Algebras , 1995 .

[24]  B. Blackadar,et al.  K-Theory for Operator Algebras , 1986 .

[25]  C. Lance.  On nuclear C∗-algebras , 1973 .

[26]  M. Rieffel.  Strong morita equivalence of certain transformation groupC*-algebras , 1976 .

[27]  E. Effros,et al.  ON APPROXIMATION PROPERTIES FOR OPERATOR SPACES , 1990 .

[28]  S. Wassermann.  Tensor Products of Free-Group C* -Algebras , 1990 .

[29]  E. Kirchberg.  Discrete groups with Kazhdan's propertyT and factorization property are residually finite , 1994 .

[30]  Jon Kraus.  The slice map problem and approximation properties , 1991 .

[31]  G. Pisier,et al.  Bilinear forms on exact operator spaces andB(H)⊗B(H) , 1995 .

[32]  Joachim Cuntz,et al.  SimpleC*-algebra generated by isometries , 1977 .

[33]  G. Skandalis.  Une Notion de nucléarité en K-théorie (d'après J. Cuntz) , 1988 .

[34]  Lifting problems and local reflexivity for $C^\ast$-algebras , 1985 .