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Wang, Kun

On invariants of \(\mathrm C^{\ast}\)-algebras with the ideal property. (English) Zbl 1464.46065

J. Noncommut. Geom. 12, No. 3, 1199-1225 (2018).
A C*-algebra has the ideal property if each of its closed two-sided ideals is generated by projections. This class of C*-algebras includes both simple, unital C*-algebras, as well as C*-algebras of real rank zero.
The paper shows that for C*-algebras with the ideal property, the extended Elliott invariant and the Stevens invariant can be recovered from each other, which means that both invariants contain the same information.
Both invariants consist of the scaled, ordered \(K_0\)-group and the \(K_1\)-group. Additionally, they encode different aspects of tracial data: The extended Elliott invariant contains the non-cancellative cone of extended-valued lower-semicontinuous traces (as studied in [G. A. Elliott et al., Am. J. Math. 133, No. 4, 969–1005 (2011; Zbl 1236.46052)]) together with the natural pairing between \(K_0\) and the cone of traces. On the other hand, the Stevens invariant contains for each projection the cancellative cone of finite traces on the unital corner given by the projection, together with natural pairing maps between \(K_0\) and these cones of traces, and with natural restriction maps between the cones of traces.
Reviewer: Hannes Thiel (Münster)

Cited in 2 Documents

MSC:

46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras

Keywords:

classification of C*-algebras; ideal property; extended Elliott invariant; Stevens invariant

Citations:

Zbl 1236.46052
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Full Text: DOI arXiv

References:

[1] B. Blackadar and D. Handelman, Dimension functions and traces on C-algebras, {\it J. Funct.} {\it Anal.}, 45 (1982), no. 3, 297-340.Zbl 0513.46047 MR 650185 · Zbl 0513.46047
[2] M. Dadarlat, Reduction to dimension three of local spectra of real rank zero C-algebras, {\it J. Reine Angew. Math.}, 460 (1995), 189-212.Zbl 0815.46065 MR 1316577 · Zbl 0815.46065
[3] M. Dadarlat and G. Gong, A classification result for approximately homogeneous Calgebras of real rank zero, {\it Geom. Funct. Anal.}, 7 (1997), no. 4, 646-711.Zbl 0905.46042 MR 1465599 · Zbl 0905.46042
[4] M. Dadarlat and T. Loring, Classifying C-algebras via ordered, mod p K-theory, {\it Math.} {\it Ann.}, 305 (1996), no. 4, 601-616.Zbl 0857.46039 MR 1399706 · Zbl 0857.46039
[5] S. Eilers, A complete invariant for AD algebras with bounded dimension drop in K1, {\it J.} {\it Funct. Anal.}, 139 (1996), no. 2, 325-348.Zbl 0866.46045 MR 1402768 · Zbl 0866.46045
[6] G. A. Elliott, On the classification of inductive limits of sequences of semi-simple finitedimensional algebras, {\it J. Algebra}, 38 (1976), no. 1, 29-44.Zbl 0323.46063 MR 397420 · Zbl 0323.46063
[7] G. A. Elliott, A classification of certain simple C-algebras, in {\it Quantum and non-} {\it commutative analysis (Kyoto, 1992)}, 373-385, Math. Phys. Stud., 16, Kluwer Acad. Publ., Dordrecht, 1993.Zbl 0843.46045 MR 1276305 · Zbl 0843.46045
[8] G. A. Elliott, A classification of certain simple C-algebras. II, {\it J. Ramanujan Math. Soc.}, 12(1997), no. 1, 97-134.Zbl 0954.46035 MR 1462852 · Zbl 0954.46035
[9] G. A. Elliott and G. Gong, On the classification of C-algebras of real rank zero. II, {\it Ann.} {\it of Math. (2)}, 144 (1996), no. 3, 497-610.Zbl 0867.46041 MR 1426886 · Zbl 0867.46041
[10] G. A. Elliott and G. Gong, On inductive limits of matrix algebras over the two-torus, {\it Amer.} {\it J. Math.}, 118 (1996), no. 2, 263-290.Zbl 0847.46032 MR 1385277 · Zbl 0847.46032
[11] G. A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit C-algebra. II. The isomorphism theorem, {\it Invent. Math.}, 168 (2007), no. 2, 249-320.Zbl 1129.46051 MR 2289866 1220K. Wang · Zbl 1129.46051
[12] G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Homomorphisms, homotopies and approximations by circle algebras, {\it C. R. Math. Rep. Acad. Sci. Canada}, 16 (1994), no. 1, 45-50.Zbl 0832.55008 MR 1276344 · Zbl 0832.55008
[13] G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Abelian C-subalgebras of C-algebras of real rank zero and inductive limit C-algebras, {\it Duke Math. J.}, 85 (1996), no. 3, 511-554. Zbl 0869.46030 MR 1422356 · Zbl 0869.46030
[14] G. A. Elliott, L. Robert, and L. Santiago, The cone of lower semicontinuous traces on a C-algebra, {\it Amer. J. Math.}, 133 (2011), no. 4, 969-1005.Zbl 1236.46052 MR 2823868 · Zbl 1236.46052
[15] B. Fuchssteiner and W. Lusky, {\it Convex Cones}, North-Holland Mathematics Studies, 56, North-Holland Publishing Co., Amsterdam-New York, 1981.Zbl 0478.46002 MR 640719 · Zbl 0478.46002
[16] G. Gong, On inductive limits of matrix algebras over higher dimensional spaces. Part II, {\it Math. Scand.}, 80 (1997), no. 1, 56-100.Zbl 0901.46054 MR 1466905 · Zbl 0901.46054
[17] G. Gong, On the classification of C-algebras of real rank zero and unsuspended E-equivalence types, {\it J. Funct. Anal.}, 152 (1998), no. 2, 281-329.Zbl 0921.46058 MR 1607999 · Zbl 0921.46058
[18] G. Gong, On the classification of simple inductive limits C-algebras. I. The reduction theorem, {\it Doc. Math.}, 7 (2002), 255-461.Zbl 1024.46018 MR 2014489 · Zbl 1024.46018
[19] G. Gong, C. Jiang, and L. Li, A classification of inductive limit C-algebras with the ideal property.ariXv:1607.07581v1 · Zbl 1529.46039
[20] G. Gong, C. Jiang, L. Li, and C. Pasnicu, A T structure for AH algebras with the ideal property and torsion free K-theory, {\it J. Funct. Anal.}, 258 (2010), no. 6, 2119-2143. Zbl 1283.46039 MR 2578465 · Zbl 1283.46039
[21] K. Ji and C. L. Jiang, A complete classification of AI algebras with the ideal property, {\it Canad. J. Math.}, 63 (2011), no. 2, 381-412.Zbl 1266.46047 MR 2809060 · Zbl 1266.46047
[22] C. Jiang and K. Wang, A complete classification of limits of splitting interval algebras with the ideal property, {\it J. Ramanujan Math. Soc.}, 27 (2012), no. 3, 305-354.Zbl 1273.46042 MR 2987231 · Zbl 1273.46042
[23] X. Jiang and H. Su, A classification of Simple Limits of Splitting Interval Algebras, {\it J.} {\it Funct. Anal.}, 151 (1997), no. 1, 50-76.Zbl 0921.46057 MR 1487770 · Zbl 0921.46057
[24] E. Kirchberg and M. Rordam, Infinite non-simple C*-algebras: absorbing the Cuntz algebra O1, {\it Adv. Math.}, 167 (2002), no. 2, 195-264.Zbl 1030.46075 MR 1906257 · Zbl 1030.46075
[25] L. Li, Classification of simple C-algebras: inductive limits of matrix algebras over trees., {\it Mem. Amer. Math. Soc.}, 127 (1997), no. 605, vii+123pp.Zbl 0883.46039 MR 1376744 · Zbl 0883.46039
[26] H. Lin, {\it An Introduction to the classification of amenable }C{\it -algebras}, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.Zbl 1013.46055 MR 1884366 · Zbl 1013.46055
[27] C. Pasnicu, Shape equivalence, non-stable K-theory and AH algebras, {\it Pacific J. Math.}, 192(2000), no. 1, 159-182.Zbl 1092.46513 MR 1741023 · Zbl 1092.46513
[28] C. Pasnicu, The ideal property and tensor products of C-algebras, {\it Rev. Roumaine Math.} {\it Pures Appl.}, 49 (2004), no. 2, 153-162.Zbl 1126.46308 MR 2060886 · Zbl 1126.46308
[29] C. Pasnicu, The ideal property, the projection property, continuous fields and crossed products, {\it J. Math. Anal. Appl.}, 323 (2006), no. 2, 1213-1224.Zbl 1112.46043 MR 2260174 On invariants of C-algebras with the ideal property1221 · Zbl 1112.46043
[30] C. Pasnicu and N. C. Phillips, Permanence properties for crossed products and fixed point algebras of finite groups, {\it Trans. Amer. Math. Soc.}, 366 (2014), no. 9, 4625-4648. Zbl 1316.46050 MR 3217695 · Zbl 1316.46050
[31] C. Pasnicu and N. C. Phillips, The weak ideal property and topological dimension zero. arXiv:1601.00039 · Zbl 1394.46043
[32] C. Pasnicu and M. Rørdam, Tensor products of C-algebras with the ideal property, {\it J.} {\it Funct. Anal.}, 177 (2000), no. 1, 130-137.Zbl 0989.46032 MR 1789946 · Zbl 0989.46032
[33] G. K. Pedersen, C{\it -algebras and their automorphism groups}, London Mathematical Society Monographs, 14, Academic Press, Inc., London-New York, 1979.Zbl 0416.46043 MR 548006 · Zbl 0416.46043
[34] F. Perera and A. S. Toms, Recasting the Elliott conjecture, {\it Math. Ann.}, 338 (2007), no. 3, 669-702.Zbl 1161.46035 MR 2317934 · Zbl 1161.46035
[35] R. R. Phelps, {\it Lectures on Choquet’s theorem}. Second edition, Lecture Notes in Mathematics, 1757, Springer-Verlag, Berlin, 2001.Zbl 0997.46005 MR 1835574 · Zbl 0997.46005
[36] S. Razak, On the classification of simple stably projectionless C-algebras, {\it Canad. J.} {\it Math.}, 54 (2002), no. 1, 138-224.Zbl 1038.46051 MR 1880962 · Zbl 1038.46051
[37] M. Rørdam, On the structure of simple C-algebras tensored with a UHF-algebra. II, {\it J.} {\it Funct. Anal.}, 107 (1992), no. 2, 255-269.Zbl 0810.46067 MR 1172023 · Zbl 0810.46067
[38] M. Rørdam, The stable and the real rank of Z-absorbing C-algebras, {\it Int. J. Math.}, 15 (2004), no. 10, 1065-1084.Zbl 1077.46054 MR 2106263 · Zbl 1077.46054
[39] K. H. Stevens, The classification of certain non-simple approximate interval algebras, in {\it Operator algebras and their applications. II (Waterloo, ON, 1994/1995)}, 105-148, Fields Inst. Commun., 20, Amer. Math. Soc., Providence, RI, 1998.Zbl 0926.46047 MR 1643184 · Zbl 0926.46047
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