Nonamenable simple $C^*$ -algebras with tracial approximation

We construct two types of unital separable simple C∗-algebras A1 z and A C2 z , one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, Ai z has a unique tracial state, (K0(A Ci z ),K0(A Ci z )+, [1ACi z ]) = (Z,Z+, 1) and K1(A Ci z ) = {0} (i = 1, 2). We show that A Ci z (i = 1, 2) is essentially tracially in the class of separable Z-stable C∗-algebras of nuclear dimension 1. Ai z has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple non-exact C∗-algebras which are essentially tracially in the class of simple separable nuclear Z-stable C∗-algebras and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

[1]  Joachim Cuntz,et al.  Dimension Functions on Simple C *-Algebras , 2005 .

[2]  E. Kirchberg,et al.  Non-simple Purely Innnite C -algebras , 1999 .

[3]  Claude L. Schochet.  Topological methods for C-algebras II: geometric resolutions and the K?unneth formula , 1982 .

[4]  Huaxin Lin.  An Introduction to the Classification of Amenable C-Algebras , 2001 .

[5]  Huaxin Lin,et al.  On classification of simple non-unital amenable C*-algebras, II , 2017, 1702.01073.

[6]  Mikael Rørdam,et al.  Classification of Nuclear, Simple C*-algebras , 2002 .

[7]  E. Kirchberg,et al.  Generalized inductive limits of finite-dimensional C*-algebras , 1997 .

[8]  E. Effros,et al.  The completely positive lifting problem for C*-algebras , 1976 .

[9]  M. Rørdam.  On the structure of simple C∗-algebras tensored with a UHF-algebra, II , 1991 .

[10]  W. Winter.  Nuclear dimension and -stability of pure C ∗ -algebras , 2010, 1006.2731.

[11]  B. Blackadar.  A simple *-algebra with no nontrivial projections , 1980 .

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

[13]  B. Blackadar.  Operator Algebras , 1998 .

[14]  G. Elliott,et al.  The cone of lower semicontinuous traces on a C*-algebra , 2008, 0805.3122.

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

[16]  Eberhard Kirchberg,et al.  Non-simple purely infinite C∗-algebras: the Hausdorff case , 2004 .

[17]  Huaxin Lin,et al.  A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. , 2018, 1812.11590.

[18]  K. Goodearl.  Notes on a class of simple $C^*$-algebras with real rank zero , 1992 .

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

[20]  Huaxin Lin.  The Tracial Topological Rank of C*‐Algebras , 2001 .

[21]  Huaxin Lin.  Traces and simple C^*-algebras with tracial topological rank zero , 2004 .

[22]  Lawrence G. Brown,et al.  Stable isomorphism of hereditary subalgebras of $C^*$-algebras. , 1977 .

[23]  Huaxin Lin,et al.  On infinite simple C∗-algebras , 1991 .

[24]  L. Robert.  REMARKS ON ${\mathcal{Z}}$-STABLE PROJECTIONLESS C*-ALGEBRAS , 2011, Glasgow Mathematical Journal.

[25]  D. Handelman,et al.  Dimension functions and traces on C∗-algebras , 1982 .

[26]  G. Elliott,et al.  Simple stably projectionless C*-algebras with generalized tracial rank one , 2017, Journal of Noncommutative Geometry.

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

[28]  Wilhelm Winter,et al.  The Jiang–Su algebra revisited , 2008, 0801.2259.

[29]  Nonnuclear subalgebras of AF algebras , 2000 .

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

[31]  Wilhelm Winter,et al.  Strongly self-absorbing C*-algebras , 2005, math/0502211.

[32]  G. Elliott,et al.  On the classification of simple amenable C*-algebras with finite decomposition rank , 2015, 1507.07876.

[33]  J. Cuntz.  The structure of multiplication and addition in simple $C^*$-algebras. , 1977 .

[34]  Yasuhiko Sato,et al.  Decomposition rank of UHF-absorbing c* -algebras , 2013, 1303.4371.

[35]  Yasuhiko Sato,et al.  Strict comparison and Z-absorption of nuclear C*-algebras , 2011 .