Strongly self-absorbing C*-algebras

Say that a separable, unital C*-algebra V ? C is strongly self absorbing if there exists an isomorphism V such that lx> ai>e approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras ?2, Ooo, the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of ?00 with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra V we characterise when a separable C*-algebra absorbs V tensorially (i.e., is P-stable), and prove closure properties for the class of separable Testable C* algebras. Finally, we compute the possible If-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C*-algebras.

[1]  AF-embeddings into C∗-algebras of real rank zero☆ , 2003, math/0310340.

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

[3]  Huaxin Lin.  Classification of Simple Tracially AF C *-Algebras , 2001, Canadian Journal of Mathematics.

[4]  N. Christopher Phillips.  A Classification Theorem for Nuclear Purely Infinite Simple C -Algebras 1 , 1995 .

[5]  Mikael Rørdam,et al.  Classification of Nuclear C*-Algebras. Entropy in Operator Algebras , 2001 .

[6]  E. Effros,et al.  $C^{\ast}$-algebras with approximately inner flip. , 1978 .

[7]  Andrew S. Toms.  On the Independence of K-Theory and Stable Rank for Simple C*-Algebras , 2005, math/0509427.

[8]  G. Gong.  On the classification of simple inductive limit $C^*$-algebras. I: The reduction theorem , 2002, Documenta Mathematica.

[9]  Mikael Rordam,et al.  The stable and the real rank of Z-absorbing C*-algebras , 2004 .

[10]  J. Villadsen.  On the stable rank of simple C*-algebras , 1999 .

[11]  W. Winter.  On the Classification of Simple Z‐Stable C*‐Algebras with Real Rank Zero and Finite Decomposition Rank , 2005, math/0502181.

[12]  K. Thomsen.  Limits of certain subhomogeneous C*-algebras , 1997 .

[13]  Covering dimension for nuclear C∗-algebras , 2001, math/0107218.

[14]  Wilhelm Winter,et al.  COVERING DIMENSION AND QUASIDIAGONALITY , 2002 .

[15]  Simple nuclear C∗-algebras of tracial topological rank one , 2004, math/0401240.

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

[17]  Xinhui Hongbing Jiang,et al.  On a simple unital projectionless C*-algebra , 1999 .

[18]  M. Takesaki,et al.  Analyticity and the Unruh effect: a study of local modular flow , 2024, Journal of High Energy Physics.

[19]  Invariant Means and Finite Representation Theory of $c*$-Algebras , 2003, math/0304009.

[20]  Xinhui Jiang,et al.  A Classification of Simple Limits of Splitting Interval Algebras , 1997 .

[21]  M. Rørdam.  THE STABLE AND THE REAL RANK OF ${\mathcal Z}$-ABSORBING C*-ALGEBRAS , 2004 .

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

[23]  On topologically finite-dimensional simple C*-algebras , 2003, math/0311501.

[24]  E. Kirchberg,et al.  Embedding of exact C^*-algebras in the Cuntz algebra *_2 , 2000 .

[25]  Mikael Rrd Am.  THE STABLE AND THE REAL RANK OF Z-ABSORBING C -ALGEBRAS , 2004 .

[26]  J. Villadsen,et al.  Simple C*-Algebras with Perforation , 1998 .

[27]  Decomposition Rank of Subhomogeneous C*‐Algebras , 2002, math/0210420.

[28]  W. Winter.  Simple C∗-algebras with locally finite decomposition rank , 2006, math/0602617.

[29]  G. Gong,et al.  Obstructions to 𝒵-Stability for Unital Simple C *-Algebras , 2000, Canadian Mathematical Bulletin.

[30]  Cancellation and stable rank for direct limits of recursive subhomogeneous algebras , 2001, math/0101157.

[31]  Perturbation of Hausdorff Moment Sequences, and an Application to the Theory of C*-Algebras of Real Rank Zero , 2005, math/0502239.

[32]  M. Dadarlat.  Reduction to dimension three of local spectra of real rank zero C*-algebras. , 1995 .

[33]  G. Elliott,et al.  On the classification of simple inductive limit C*-algebras, II: The isomorphism theorem , 2007 .

[34]  Cristian Ivanescu.  On the classification of simple approximately subhomogeneous C*-algebras not necessarily of real rank zero , 2003, math/0312044.

[35]  G. Pedersen.  C-Algebras and Their Automorphism Groups , 1979 .

[36]  George A. Elliott,et al.  On the classification of inductive limits of sequences of semisimple finite-dimensional algebras , 1976 .

[37]  Jim McGregor,et al.  Simple C , 1997 .

[38]  Mikael Rordam,et al.  A simpleC*-algebra with a finite and an infinite projection , 2002, math/0204339.

[39]  Recursive subhomogeneous algebras , 2001, math/0101156.

[40]  George A. Elliott,et al.  On the classification of C*-algebras of real rank zero. , 1993 .

[41]  L. Lubitz,et al.  Symptomatic zinc deficiency in breast‐fed term and premature infants , 1998, Journal of paediatrics and child health.