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Set theory and \(C^*\)-algebras. (English) Zbl 1127.46043

It is well-known that many problems in topology, infinitary combinatorics, but also in analysis and algebra, cannot be solved without the use of extra set-theoretic hypotheses such as the continuum hypothesis CH, Martin’s axiom, or Jensen’s diamond principle.
The paper under review, mainly written for set theorists, shows where and how such extra set-theoretic hypotheses can be applied to \(C^*\)-algebras. Beginning with the definition (and examples) of \(C^*\)-algebras, some consistency results (mainly involving CH) are given, and at the end, the Calkin algebra, as a basic object of interest for set theorists, is considered.

MSC:

46L05 General theory of \(C^*\)-algebras
03E50 Continuum hypothesis and Martin’s axiom
03E75 Applications of set theory
03E35 Consistency and independence results
46L30 States of selfadjoint operator algebras
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References:

[1] Set theory: An introduction to independence proofs (1980) · Zbl 0443.03021
[2] DOI: 10.1090/S0002-9947-03-03353-1 · Zbl 1042.46033 · doi:10.1090/S0002-9947-03-03353-1
[3] Transactions of the American Mathematical Society 249 pp 303– (1979)
[4] DOI: 10.1007/BF01404917 · Zbl 0647.46053 · doi:10.1007/BF01404917
[5] Mathematica Scandinavica 42 pp 101– (1978) · Zbl 0397.46052 · doi:10.7146/math.scand.a-11739
[6] DOI: 10.2307/2372748 · Zbl 0086.09704 · doi:10.2307/2372748
[7] DOI: 10.1006/jfan.1996.0047 · Zbl 0864.47008 · doi:10.1006/jfan.1996.0047
[8] Fundamentals of the theory of operator algebras I (1997)
[9] Perfect C*-algebras 55 (1985) · Zbl 0584.46045
[10] Mathematica Scandinavica 41 pp 117– (1977) · Zbl 0377.46049 · doi:10.7146/math.scand.a-11707
[11] Analysis and probability: Wavelets, signals, fractals (2006) · Zbl 1104.42001
[12] DOI: 10.4153/CJM-1986-063-7 · Zbl 0704.46043 · doi:10.4153/CJM-1986-063-7
[13] Studia Mathematica 46 pp 197– (1973)
[14] Journal of Operator Theory 2 pp 159– (1979)
[15] Journal of the London Mathematical Society 36 pp 165– (1987)
[16] DOI: 10.2977/prims/1195179059 · Zbl 0593.46047 · doi:10.2977/prims/1195179059
[17] Illinois Journal of Mathematics 37 pp 1– (1993)
[18] DOI: 10.1090/S0894-0347-1990-1057041-5 · doi:10.1090/S0894-0347-1990-1057041-5
[19] DOI: 10.1016/S0012-365X(03)00253-X · Zbl 1040.46040 · doi:10.1016/S0012-365X(03)00253-X
[20] DOI: 10.1016/S0022-1236(03)00196-4 · Zbl 1031.46063 · doi:10.1016/S0022-1236(03)00196-4
[21] DOI: 10.1090/S0002-9939-98-04233-6 · Zbl 0891.47028 · doi:10.1090/S0002-9939-98-04233-6
[22] DOI: 10.2307/1970319 · Zbl 0152.33002 · doi:10.2307/1970319
[23] Mathematical quantization (2001) · Zbl 0999.81002
[24] Recent advances in operator theory and related topics 127 pp 305–
[25] Quantum field theory in curved spacetime and black hole thermodynamics (1994) · Zbl 0842.53052
[26] Proceedings of the American Mathematical Society 112 pp 949– (1991)
[27] DOI: 10.1016/0022-1236(72)90005-5 · Zbl 0227.46070 · doi:10.1016/0022-1236(72)90005-5
[28] Topology Proceedings 9 pp 269– (1984)
[29] DOI: 10.1016/0166-8641(93)90127-Y · Zbl 0785.03033 · doi:10.1016/0166-8641(93)90127-Y
[30] Séminaire Bourbaki 1 pp 331– (1995)
[31] DOI: 10.1016/B978-0-444-86580-9.50014-8 · doi:10.1016/B978-0-444-86580-9.50014-8
[32] Journal für die Reine und Angewandte Mathematik 348 pp 72– (1984)
[33] Theory of operator algebras I (1979)
[34] An introduction to independence for analysts (1987) · Zbl 0629.03030
[35] DOI: 10.1090/S0002-9939-1988-0935111-X · doi:10.1090/S0002-9939-1988-0935111-X
[36] A course in functional analysis (1990) · Zbl 0706.46003
[37] Classification theory and the number of nonisomorphic models (1990)
[38] Operator algebras and applications, Part I pp 521– (1982)
[39] Proper forcing (1982) · Zbl 0495.03035
[40] C*-algebras and W*-algebras (1971)
[41] DOI: 10.1073/pnas.0507888103 · Zbl 1160.46333 · doi:10.1073/pnas.0507888103
[42] Operator algebras and quantum statistical methanics 2: Equilibrium states, models in quantum statistical mechanics (1997)
[43] Subfactors and knots (1991)
[44] Transactions of the AmericanMathematical Society 298 pp 747– (1986)
[45] DOI: 10.1016/0022-1236(72)90078-X · Zbl 0237.46070 · doi:10.1016/0022-1236(72)90078-X
[46] DOI: 10.1073/pnas.0401489101 · Zbl 1064.46034 · doi:10.1073/pnas.0401489101
[47] Set theory (2003)
[48] DOI: 10.1017/S0017089500030755 · Zbl 0820.46057 · doi:10.1017/S0017089500030755
[49] DOI: 10.1090/S0002-9947-1978-0515547-X · doi:10.1090/S0002-9947-1978-0515547-X
[50] DOI: 10.2307/2372501 · Zbl 0053.25903 · doi:10.2307/2372501
[51] DOI: 10.1305/ndjfl/1093635236 · Zbl 0702.03030 · doi:10.1305/ndjfl/1093635236
[52] Proceedings of the London Mathematical Society 23 pp 547– (1971)
[53] Transactions of the American Mathematical Society 300 pp 557– (1987)
[54] C*-algebras and their automorphism groups (1979) · Zbl 0416.46043
[55] DOI: 10.1512/iumj.1978.27.27070 · Zbl 0393.46047 · doi:10.1512/iumj.1978.27.27070
[56] Uspehi Matematiceskih Nauk (N.S.) 6 pp 160– (1951)
[57] Proceedings of the American Mathematical Society 97 pp 413– (1986)
[58] Uspehi Matematiceskih Nauk (N.S.) 3 pp 52– (1948)
[59] Mathematical theory of quantum fields (1999)
[60] DOI: 10.1016/0022-1236(79)90061-2 · Zbl 0422.46049 · doi:10.1016/0022-1236(79)90061-2
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