Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 70 (2009) 2009, Pages 207–235 S 0077-1554(09)00174-5 Article electronically published on December 3, 2009 A GLOBAL DIMENSION THEOREM FOR QUANTIZED BANACH ALGEBRAS N. V. VOLOSOVA Abstract. We prove that for a commutative quantized ( h ⊗ and o ⊗) 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. 1. Introduction The goal of this paper is to extend the so-called global dimension theorem (see [11], [12, Prop. V.2.21] and [25]) — one of the main results in the homological theory of Banach algebras — to quantized algebras. It asserts that the global (homological) dimension dgA of a commutative Banach algebra with infinite spectrum is strictly larger than one. As a consequence, we have a bound on the homological bidimension for the corresponding classes of quantized algebras. The history of the homological theory of Banach algebras goes back to 1962 when H. Kamovitz [19] defined homology groups of a Banach algebra A with coefficients in a bimodule X — a Banach analog of Hochschild cohomology, one of the key concepts of homological algebra. In 1970, A. Ya. Helemskii [8], using such tools from pure algebra as resolutions and derived functors, proposed a more general approach to the homology of Banach algebras; at the same time, the notions of homological dimension of a module over an algebra and the global homological dimension of an algebra were introduced. It turned out that the homological theory of Banach algebras has a number of features not found in its purely-algebraic counterpart. One such result without a purely-algebraic analog is the global dimension theorem, proved by A.Ya. Helemskii in 1972 [9] (a complete proof can be found in [11]; see also [25] for a detailed exposition). By now, the estimate dgA> 1 has also been obtained for some other classes of Banach algebras; we also know that, under certain conditions, global dimension is “well-behaved” under projective tensor product [28]. We shall prove a similar theorem for quantized algebras with an additional, compared to Banach algebras, structure: quantum norm; such algebras are studied in the so-called quantized functional analysis. That branch of mathematics emerged in the beginning of the 1980s when, in the papers by Haagerup, Paulsen, and Wittstock, the concept of completely bounded map was introduced [21]. The term “quantized functional analysis” was used by E.Effros in 1986 [4]. There are two essentially equivalent approaches to what should be understood by a quantum norm on a linear space E. The former (to which we adhere in this paper) deals with a family of norms on the space of matrices with entries in E. That theory is developed in [2, 6, 22]. In the other approach, one works with only one norm, rather than a family, defined on a larger space, namely F⊗ E, where F is the 2000 Mathematics Subject Classification. Primary 46M18; Secondary 46H05, 46J20. Supported by the RFFI (Project No. 05–01–00982 and Project No. 08–01–00867). c 2009 American Mathematical Society 207 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use