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On the homotopy equivalence of simple AI-algebras
Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 165-191 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $A$ and $B$ be simple unital AI-algebras (an AI-algebra is an inductive limit of $C^*$-algebras of the form $\bigoplus_i^kC([0,1],M_{N_i})$. It is proved that two arbitrary unital homomorphisms from $A$ into $B$ such that the corresponding maps $\mathrm K_0A\to\mathrm K_0B$ coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for $A$ and $B$ to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same $\mathrm K$-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, which is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix $h$ can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with $h$. 
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     title = {On the homotopy equivalence of simple {AI-algebras}},
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     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_2_a0/}
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O. Yu. Aristov. On the homotopy equivalence of simple AI-algebras. Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 165-191. http://geodesic.mathdoc.fr/item/SM_1999_190_2_a0/

[1] Elliott G. A., “A classification of certain simple $C^*$-algebras”, Quantum and Non-commutative Analysis, eds. H. Araki et al., Kluwer, Dordrecht, 1993, 373–385 | MR | Zbl

[2] Elliott G. A., “On the classification of inductive limits of semisimple finite-dimensional algebras”, J. Algebra, 38 (1976), 29–44 | DOI | MR | Zbl

[3] Merfi Dzh., $C^*$-algebry i teoriya operatorov, Faktorial, M., 1997

[4] Elliott G. A., “The classification problem for amenable $C^*$-algebras”, Proc. International Congress of Mathematicians, ICM '94, V. 2 (August 3–11, 1994, Zürich, Switzerland), ed. S. D. Chatterji, Birkhäuser, Basel, 1995, 922–932 | MR | Zbl

[5] Blackadar B., $\mathrm K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, 5, Springer-Verlag, New York, 1986 | MR | Zbl

[6] Thomsen K., “Inductive limits of interval algebras: the tracial state space”, Amer. J. Math., 116 (1994), 605–620 | DOI | MR | Zbl

[7] Thomsen K., “On the range of the Elliott invariant”, J. Funct. Anal., 221 (1994), 255–274 | DOI | MR

[8] Villadsen J., “The range of the Elliott invariant”, J. Reine Angew. Math., 462 (1995), 31–55 | MR | Zbl

[9] Elliott G. A., “On the classification of $C^*$-algebras real rank zero”, J. Reine Angew. Math., 443 (1993), 179–219 | MR | Zbl

[10] Manuilov V. M., “Diagonalizatsiya operatorov nad nepreryvnymi polyami $C^*$-algebr”, Matem. sb., 188:6 (1997), 99–118 | MR | Zbl

[11] Manuilov V. M., “O pochti kommutiruyuschikh operatorakh”, Funkts. analiz i ego prilozh., 31:3 (1997), 80–82 | MR | Zbl

[12] Bratteli U., Robinson D., Operatornye algebry i kvantovaya statisticheskaya mekhanika, Mir, M., 1982 | MR | Zbl

[13] Diksme Zh., $C^*$-algebry i ikh predstavleniya, Nauka, M., 1974 | MR

[14] Khelemskii A. Ya., Banakhovy i polinormirovannye algebry. Obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989 | MR

[15] Choi M.-D., Elliott G. A., “Density of the self-adjoint elements with finite spectrum in an irrational rotation $C^*$-algebra”, Math. Scand., 67 (1990), 73–86 | MR | Zbl

[16] Davidson K., “Almost commuting hermitian matrices”, Math. Scand., 56 (1985), 222–241 | MR

[17] Kishimoto A., Kumjian A., “The Ext class of an approximately inner automorphism”, Trans. Amer. Math. Soc., 350:10 (1998), 4127–4148 | DOI | MR | Zbl

[18] Glimm J., “On a certain class of operator algebras”, Trans. Amer. Math. Soc., 95 (1960), 318–340 | DOI | MR | Zbl

[19] Bratteli O., Kishimoto A., “Noncommutative spheres. III: Irrational rotations”, Comm. Math. Phys., 147 (1992), 605–624 | DOI | MR | Zbl

[20] Thomsen K., “On isomorphisms of inductive limit $C^*$-algebras”, Proc. Amer. Math. Soc., 113 (1991), 947–953 | DOI | MR | Zbl

[21] Elliott G. A., Gong G., “On inductive limits of matrix algebras over two-torus”, Amer. J. Math., 118 (1996), 263–290 | DOI | MR | Zbl

[22] Thomsen K., “From traces to states on the $\mathrm K_0$ group of a simple $C^*$-algebra”, Bull. London Math. Soc., 28 (1996), 66–72 | DOI | MR | Zbl

[23] Blackadar B., “Shape theory for $C^*$-algebras”, Math. Scand., 56 (1985), 249–275 | MR | Zbl

[24] Loring T., “Stable relations. II: Corona semiprojectivity and dimension drop $C^*$-algebras”, Pacific J. Math., 172 (1996), 461–475 | MR | Zbl

[25] Loring T., “$C^*$-algebras generated by stable relations”, J. Funct. Anal., 112 (1993), 159–203 | DOI | MR | Zbl

[26] Blackadar B., Rordam M., “Extending states on preodered semigroups and the exsistence of quasitraces on $C^*$-algebras”, J. Algebra, 152 (1992), 240–247 | DOI | MR | Zbl

[27] Effros E. G., Handelman D. E., Shen C.-L., “Dimension groups and their affine representation”, Amer. J. Math., 102 (1980), 385–407 | DOI | MR | Zbl