2009
DOI: 10.1007/s10958-009-9465-4
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Projective modules in classical and quantum functional analysis

Abstract: Along with the classical version, there are two "quantized" versions of the theory of operator algebras. In these lectures, the fundamental homological notion of a projective module is described in the framework of these three theories. Our initial definitions of projectivity do not go far from their prototypes in abstract algebra; however, the principal results concern essentially functional-analytic objects and, as a rule, have no purely algebraic analogs.

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Cited by 4 publications
(13 citation statements)
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“…That is, if (iv) does not hold, then (iii) cannot hold either. ■ Remark 5.8 Implication (iii) ⇒ (iv) in Theorem 5.3 answers Helemskii's question, for operator algebras, in [16,Section 7] in the positive since it is easy to see that his relative structure is equivalent to ours. Relative homological algebra is common in the ring theory setting, cf., e.g., [18,Chapter IX] or [22,Section V.7].…”
mentioning
confidence: 85%
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“…That is, if (iv) does not hold, then (iii) cannot hold either. ■ Remark 5.8 Implication (iii) ⇒ (iv) in Theorem 5.3 answers Helemskii's question, for operator algebras, in [16,Section 7] in the positive since it is easy to see that his relative structure is equivalent to ours. Relative homological algebra is common in the ring theory setting, cf., e.g., [18,Chapter IX] or [22,Section V.7].…”
mentioning
confidence: 85%
“…For an operator algebra 𝐴 (on a Hilbert space) it is pertinent to use operator modules to build a cohomology theory (for the definitions, see Section 2.2); of the numerous contributions we only mention [7], [16], [24], [32] and [34] here. In this paper, we focus on an appropriate definition of cohomological dimension and, in particular, answer a question raised by Helemskii [16] whether quantised global dimension zero is equivalent to 2 the algebra being classically semisimple; see Theorem 5.3 below.…”
Section: Introductionmentioning
confidence: 99%
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“…4.12] Helemskii proved that a C*-algebra is classically semisimple (i.e., finite dimensional) if and only if all objects in the category Ban ∞ of Banachmodules are projective with respect to the class of epimorphisms that split as morphisms in Ban ∞ . See also [2], [3] and [16].…”
Section: Introductionmentioning
confidence: 99%
“…For C*-algebras, there are good reasons to employ modules which also carry an operator space structure; the most common ones are the h-modules and the matrix normed modules. In [14], Paulsen introduced the notions of relative injectivity and relative projectivity for h-modules, and various pieces of homological algebra, including results on homological dimension, have been obtained in this setting; see, e.g., [11,Section 7] and [17].…”
Section: Introduction Let a Be A Banach Algebra A Right A-module E mentioning
confidence: 99%