A global dimension theorem for quantized Banach algebras

Volosova, N.

doi:10.1090/S0077-1554-09-00174-5

A global dimension theorem for quantized Banach algebras
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by N. V. Volosova
Translated by: Alex Martsinkovsky

Trans. Moscow Math. Soc. 2009, 207-235

DOI: https://doi.org/10.1090/S0077-1554-09-00174-5

Published electronically: December 3, 2009

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Abstract:

We prove that for a commutative quantized ($\stackrel {h}{\otimes }$ and $\stackrel {o}{\otimes }$) algebra with infinite spectrum, the maximum of its left and right global homological dimensions and, as a consequence, its homological bidimension are strictly greater than one. This result is a quantum analog of the global dimension theorem of A. Ya. Helemskii.

References

David P. Blecher, A completely bounded characterization of operator algebras, Math. Ann. 303 (1995), no. 2, 227–239. MR 1348798, DOI 10.1007/BF01460988
David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973, DOI 10.1093/acprof:oso/9780198526599.001.0001
Jean De Cannière and Uffe Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500. MR 784292, DOI 10.2307/2374423
Edward G. Effros, Advances in quantized functional analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 906–916. MR 934293
Edward G. Effros and Zhong-Jin Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), no. 2, 163–187. MR 1060634, DOI 10.1142/S0129167X90000113
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Ryszard Engelking, Topologia ogólna, Biblioteka Matematyczna [Mathematics Library], vol. 47, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1975 (Polish). MR 0500779
A. Ja. Helemskiĭ, The homological dimension of normed modules over Banach algebras, Mat. Sb. (N.S.) 81 (123) (1970), 430–444 (Russian). MR 0262831
A. Ja. Helemskiĭ, The global dimension of a functional Banach algebra is different from one, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 95–96 (Russian). MR 0305072
A. Ja. Helemskiĭ, A certain method of computing and estimating the global homology dimension of Banach algebras, Mat. Sb. (N.S.) 87(129) (1972), 122–135 (Russian). MR 0300087
A. Ya. Helemskii, The lowest values of global homological dimension of functional Banach algebras, Trudy Sem. Petrovsk. 3 (1978), 223–242. (Russian)
A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West. MR 1093462, DOI 10.1007/978-94-009-2354-6
A. Ya. Khelemskiĭ, Banakhovy i polinormirovannye algebry: obshchaya teoriya, predstavleniya, gomologii, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1031991
A. Ya. Helemskii, Wedderburn-type theorems for operator algebras and modules: traditional and “quantized” homological approaches, Topological homology, Nova Sci. Publ., Huntington, NY, 2000, pp. 57–92. MR 1782510
A. Ya. Helemskii, Projective modules in the classical and quantum functional analysis, PIMS Distinguished Chair Lecture Notes, July–August, 2003.
A. Ya. Khelemskiĭ, Quantum versions of vector duality and of the exponential law in the framework of a nonmatrix approach, Mat. Sb. 197 (2006), no. 12, 133–156 (Russian, with Russian summary); English transl., Sb. Math. 197 (2006), no. 11-12, 1841–1863. MR 2437084, DOI 10.1070/SM2006v197n12ABEH003825
A. Ya. Helemskii, Quantum functional analysis, MCNMO, Moscow, 2009. (Russian).
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
Herbert Kamowitz, Cohomology groups of commutative Banach algebras, Trans. Amer. Math. Soc. 102 (1962), 352–372. MR 170219, DOI 10.1090/S0002-9947-1962-0170219-7
Vern I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), no. 1, 1–17. MR 733029, DOI 10.1016/0022-1236(84)90014-4
Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
Gilles Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), no. 2, 351–369. MR 1415321, DOI 10.1090/S0894-0347-97-00227-0
Sandra Pott, An account on the global homological dimension theorem of A. Ya. Helemskii, Ann. Univ. Sarav. Ser. Math. 9 (1999), no. 3, i–iv and 155–194. MR 1721449
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Zhong-Jin Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449–1474. MR 1363075, DOI 10.2307/2375026
Yu. V. Selivanov, Lower bounds for homological dimensions of Banach algebras, Mat. Sb. 198 (2007), no. 9, 133–160 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 9-10, 1351–1377. MR 2360795, DOI 10.1070/SM2007v198n09ABEH003887
N. V. Volosova, Description of projective ideals in quantized algebras of continuous functions on compacta, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (2005), 20–24, 80 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull. 60 (2005), no. 5, 19–23 (2006). MR 2216841