×

Saturation and elementary equivalence of \(C^\ast\)-algebras. (English) Zbl 1477.46057

Summary: We study the saturation properties of several classes of \(C^\ast\)-algebras. Saturation has been shown by I. Farah and B. Hart [C. R. Math. Acad. Sci., Soc. R. Can. 35, No. 2, 35–56 (2013; Zbl 1300.46047)] to unify the proofs of several properties of coronas of \(\sigma\)-unital \(C^\ast\)-algebras; we extend their results by showing that some coronas of non-\(\sigma\)-unital \(C^\ast\)-algebras are countably degree-1 saturated. We then relate saturation of the abelian \(C^\ast\)-algebra \(C(X)\), where \(X\) is 0-dimensional, to topological properties of \(X\), particularly the saturation of \(CL(X)\). We also characterize elementary equivalence of the algebras \(C(X)\) in terms of \(CL(X)\) when \(X\) is 0-dimensional, and show that elementary equivalence of the generalized Calkin algebras of densities \(\aleph_\alpha\) and \(\aleph_\beta\) implies elementary equivalence of the ordinals \(\alpha\) and \(\beta\).

MSC:

46L05 General theory of \(C^*\)-algebras
03C65 Models of other mathematical theories
03C50 Models with special properties (saturated, rigid, etc.)
03C20 Ultraproducts and related constructions
46L10 General theory of von Neumann algebras
54C35 Function spaces in general topology

Citations:

Zbl 1300.46047
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arveson, W., Notes on extensions of \(C^\ast \)-algebras, Duke Math. J., 44, 2, 329-355 (1977) · Zbl 0368.46052
[2] Baldwin, J., Amalgamation, absoluteness, and categoricity (2012), available at · Zbl 1278.03073
[3] Bankston, P., Expressive power in first-order topology, J. Symbolic Logic, 49, 478-487 (1984) · Zbl 0576.03023
[4] Bankston, P., Reduced coproducts of compact Hausdorff spaces, J. Symbolic Logic, 52, 404-424 (1987) · Zbl 0634.54007
[5] Ben Yaacov, I.; Berenstein, A.; Henson, C. W.; Usvyatsov, A., Model theory for metric structures, (Model Theory with Applications to Algebra and Analysis, vol. 2. Model Theory with Applications to Algebra and Analysis, vol. 2, London Math. Soc. Lecture Note Ser., vol. 350 (2008), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 315-427 · Zbl 1233.03045
[6] Ben Yaacov, I.; Iovino, J., Model theoretic forcing in analysis, Ann. Pure Appl. Logic, 158, 3, 163-174 (2009) · Zbl 1160.03010
[7] Blackadar, B., Operator Algebras, Encyclopaedia Math. Sci., vol. 122 (2006), Springer-Verlag: Springer-Verlag Berlin
[8] Breuer, M., Fredholm theories in von Neumann algebras. I, Math. Ann., 178, 243-254 (1968) · Zbl 0162.18701
[9] Breuer, M., Fredholm theories in von Neumann algebras. II, Math. Ann., 180, 313-325 (1969) · Zbl 0175.44102
[10] Carlson, K.; Cheung, E.; Farah, I.; Gerhardt-Bourke, A.; Hart, B.; Mezuman, L.; Sequeira, N.; Sherman, A., Omitting types and AF algebras, Arch. Math. Logic, 53, 157-169 (2014) · Zbl 1348.03035
[11] Chang, C. C.; Keisler, H. J., Model Theory (1990), North-Holland · Zbl 0697.03022
[12] Coskey, S.; Farah, I., Automorphisms of corona algebras, and group cohomology, Trans. Amer. Math. Soc., 366, 3611-3630 (2014) · Zbl 1303.46052
[13] Dias, R. R.; Tall, F. D., Indestructibility of compact spaces, Topology Appl., 160, 2411-2426 (2013) · Zbl 1295.54026
[14] Doner, J. E.; Mostowski, A.; Tarski, A., The elementary theory of well-ordering—a metamathematical study, (Stud. Logic Found. Math., vol. 96 (1978)), 1-54 · Zbl 0461.03003
[15] Eagle, C. J., Omitting types in infinitary \([0, 1]\)-valued logic, Ann. Pure Appl. Logic, 165, 913-932 (2014) · Zbl 1282.03015
[16] Eagle, C. J.; Farah, I.; Kirchberg, E.; Vignati, A., Quantifier elimination in C*-algebras (2015), arXiv preprint
[17] Eagle, C. J.; Goldbring, I.; Vignati, A., The pseudoarc is a co-existentially closed continuum (2015), arXiv preprint · Zbl 1338.54141
[18] Elliott, G., On the classification of inductive limits of sequences of semi-simple finite dimensional algebras, J. Algebra, 38, 29-44 (1976) · Zbl 0323.46063
[19] Farah, I., All automorphisms of the Calkin algebra are inner, Ann. of Math., 173, 619-661 (2011) · Zbl 1250.03094
[20] Farah, I.; Hart, B., Countable saturation of corona algebras, C. R. Math. Rep. Acad. Sci., Canada, 35, 2, 35-56 (2013) · Zbl 1300.46047
[21] Farah, I.; Hart, B.; Sherman, D., Model theory of operator algebras II: model theory, Israel J. Math., 201, 477-505 (2014) · Zbl 1301.03037
[22] Farah, I.; McKenney, P., Homeomorphisms of Cech-Stone remainders: the zero-dimensional case (2012), arXiv preprint · Zbl 1397.03058
[23] Farah, I.; Shelah, S., Rigidity of continuous quotients (2014), preprint
[24] Ghasemi, S., \(SAW^\ast \)-algebras are essentially non-factorizable (2012), preprint
[25] Grove, K.; Pedersen, G., Sub-Stonean spaces and corona sets, J. Funct. Anal., 56, 1, 124-143 (1984) · Zbl 0539.54029
[26] Gurevic, R., On ultracoproducts of compact Hausdorff spaces, J. Symbolic Logic, 53, 294-300 (1988) · Zbl 0652.03026
[27] Henson, C. W.; Iovino, J., Ultraproducts in analysis, (Analysis and Logic. Analysis and Logic, London Math. Soc. Lecture Note Ser., vol. 262 (2003), Cambridge Univ. Press)
[28] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis, vol. I, Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren Math. Wiss., vol. 115 (1979), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0416.43001
[29] Kirchberg, E.; Rørdam, M., Central sequence \(C^\ast \)-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math., 695, 175-214 (2014) · Zbl 1307.46046
[30] Kunen, K., Set Theory: An Introduction to Independence Proofs (1980), North-Holland Pub. Co.: North-Holland Pub. Co. Amsterdam · Zbl 0443.03021
[31] Masumoto, S., The countable chain condition for \(C^\ast \)-algebras, 2015, submitted for publication, preprint · Zbl 1373.46049
[32] Mijajlović, Ž., Saturated Boolean algebras with ultrafilters, Publ. Inst. Math. (Beograd) (N.S.), 26, 40, 175-197 (1979) · Zbl 0436.03022
[33] Parovičenko, I. I., On a universal bicompactum of weight ℵ, Dokl. Akad. Nauk SSSR, 150, 36-39 (1963)
[34] Pedersen, G. K., The corona construction, (Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference. Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference, Indianapolis, IN, 1988. Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference. Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference, Indianapolis, IN, 1988, Pitman Res. Notes Math. Ser., vol. 225 (1990)), 49-92
[35] Phillips, J., \(K\)-theory relative to a semifinite factor, Indiana Univ. Math. J., 39, 2, 339-354 (1990) · Zbl 0770.46029
[36] Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math., 10, 224-233 (1971) · Zbl 0224.02045
[37] Voiculescu, D. V., Countable degree-1 saturation of certain \(C^\ast \)-algebras (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.