Decomposition rank of approximately subhomogeneous C*-algebras

Abstract It is shown that every Jiang–Su stable approximately subhomogeneous C * {{\mathrm{C}^{*}}} -algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous C * {{\mathrm{C}^{*}}} -algebras. A key step in the proof is that subhomogeneous C * {{\mathrm{C}^{*}}} -algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product C * {{\mathrm{C}^{*}}} -algebras associated to minimal homeomorphisms with mean dimension zero.

[1]  Stefan Friedl.  Algebraic topology , 2020, Graduate Studies in Mathematics.

[2]  Jorge Castillejos,et al.  Nuclear dimension of simple stably projectionless C∗-algebras , 2019, Analysis & PDE.

[3]  A. Tikuisis,et al.  Nuclear dimension of simple $$\mathrm {C}^*$$-algebras , 2019, Inventiones mathematicae.

[4]  G. Elliott,et al.  The classification of simple separable unital Z-stable locally ASH algebras , 2017 .

[5]  A. Tikuisis,et al.  Quasidiagonality of nuclear C*-algebras , 2015, 1509.08318.

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

[7]  A. Tikuisis,et al.  Covering Dimension of C*-Algebras and 2-Coloured Classification , 2015, Memoirs of the American Mathematical Society.

[8]  G. Elliott,et al.  The classification of simple separable unital locally ASH algebras , 2015, 1506.02308.

[9]  Huaxin Lin.  Crossed products and minimal dynamical systems , 2015, Journal of Topology and Analysis.

[10]  Huaxin Lin,et al.  Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras , 2014, 1501.00135.

[11]  G. Elliott,et al.  The C∗-algebra of a minimal homeomorphism of zero mean dimension , 2014, 1406.2382.

[12]  A. Tikuisis.  High-dimensional Z-stable AH algebras , 2014, 1406.0883.

[13]  Dominic Enders.  Subalgebras of finite codimension in semiprojective C*-algebras , 2014, 1405.2750.

[14]  Wilhelm Winter,et al.  Nuclear dimension and $$\mathcal Z$$Z-stability , 2014, 1403.0747.

[15]  W. Winter.  Classifying crossed product C*-algebras , 2013, 1308.5084.

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

[17]  Luis Santiago.  Reduction of the dimension of nuclear C*-algebras , 2012, 1211.7159.

[18]  A. Tikuisis,et al.  Decomposition rank of -stable C∗-algebras , 2012, 1210.1386.

[19]  José R. Carrión.  Classification of a class of crossed product C⁎-algebras associated with residually finite groups , 2011 .

[20]  Leonel Robert,et al.  Classification of inductive limits of 1-dimensional NCCW complexes , 2010, 1007.1964.

[21]  Bhishan Jacelon,et al.  A simple, monotracial, stably projectionless C*‐algebra , 2010, J. Lond. Math. Soc..

[22]  W. Winter,et al.  Quasitraces are Traces: A Short Proof of the Finite-Nuclear-Dimension Case , 2010, 1005.2229.

[23]  W. Winter.  Decomposition rank and $\mathcal{Z}$ -stability , 2010 .

[24]  Karen R. Strung,et al.  Minimal dynamics and Z-stable classification , 2010, 1001.1268.

[25]  Andrew S. Toms.  K-theoretic rigidity and slow dimension growth , 2009, 0910.2061.

[26]  Andrew S. Toms,et al.  Minimal dynamics and the classification of C*-algebras , 2009, Proceedings of the National Academy of Sciences.

[27]  P. W. Ng,et al.  Nuclear dimension and the corona factorization property , 2009, 0904.0716.

[28]  Andrew S. Toms,et al.  Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture , 2009, 0903.4133.

[29]  Wilhelm Winter,et al.  Decomposition rank and Z-stability , 2008, 0806.2948.

[30]  Huaxin Lin.  Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras , 2008, 0806.0636.

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

[32]  G. Pedersen,et al.  Limits and C*-algebras of low rank or dimension , 2007, 0708.2727.

[33]  Andrew S. Toms,et al.  Regularity properties in the classification program for separable amenable C*-algebras , 2007, 0704.1803.

[34]  Andrew S. Toms,et al.  The Elliott conjecture for Villadsen algebras of the first type , 2006, math/0611059.

[35]  P. W. Ng,et al.  A Note On Subhomogeneous C*-Algebras , 2006, math/0601069.

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

[37]  N. Christopher Phillips,et al.  Crossed products by minimal homeomorphisms , 2004, math/0408291.

[38]  E. Kirchberg,et al.  Purely infinite C*-Algebras: Ideal-preserving zero homotopies , 2003, math/0312286.

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

[40]  W. Winter,et al.  COVERING DIMENSION AND QUASIDIAGONALITY , 2002, math/0207164.

[41]  N. Phillips,et al.  Recursive subhomogeneous algebras , 2001, math/0101156.

[42]  N. Christopher Phillips.  Cancellation and stable rank for direct limits of recursive subhomogeneous algebras , 2001, math/0101157.

[43]  I. Putnam.  On the $K$-Theory of $C^*$-Algebras of Principal Groupoids , 1998 .

[44]  Terry A. Loring,et al.  Lifting solutions to perturbing problems in C*-algebras , 1996 .

[45]  T. Giordano,et al.  Topological orbit equivalence and C*-crossed products. , 1995 .

[46]  G. Elliott,et al.  The structure of the irrational rotation C*-algebra , 1993 .

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

[48]  I. Putnam,et al.  The C∗-algebras associated with minimal homeomorphisms of the Cantor set , 1989 .

[49]  Claude L. Schochet,et al.  The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor , 1987 .

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

[51]  in simple , 2023 .

[52]  G. Elliott,et al.  ON THE CLASSIFICATION OF SIMPLE UNITAL C *-ALGEBRAS WITH FINITE DECOMPOSITION RANK , 2015 .

[53]  Stuart White,et al.  Nuclear dimension and Z -stability , 2015 .

[54]  Karen R. Strung,et al.  MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION , 2011 .

[55]  Søren Eilers,et al.  STABILITY OF ANTICOMMUTATION RELATIONS : AN APPLICATION OF NONCOMMUTATIVE CW COMPLEXES , 1998 .

[56]  J. Hatzenbuhler,et al.  DIMENSION THEORY , 1997 .

[57]  Sze-Tsen Hu,et al.  Theory of Retracts. , 1968 .

[58]  Karol Borsuk,et al.  Theory Of Retracts , 1967 .