Homology and cohomology for topological algebras

Taylor, Joseph L.

doi:10.1016/0001-8708(72)90016-3

Cited by 139 publications

(138 citation statements)

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“…If 1 (S) is biprojective, it is in particular biflat, so is uniformly locally finite. For each unit e α ∈ G α , α ∈ we have e α g β = g β e α , g β ∈ G β , β ∈ , so 1 (e α S) is a unital biprojective Banach algebra possesing Grothendieck's approximation property, thus is finite dimensional ( [15]). Hence each group G α is finite.…”

Section: Applicationsmentioning

confidence: 99%

Biflatness and Biprojectivity of Banach Algebras Graded Over a Semilattice

Grønbæk

Habibian

2010

Glasgow Math. J.

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Section: Applicationsmentioning

confidence: 99%

Biflatness and Biprojectivity of Banach Algebras Graded Over a Semilattice

Grønbæk

Habibian

2010

Glasgow Math. J.

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“…Thus the admissible sequences in (A, R)-mod are those that split in R-mod. In particular, when R = C, we recover the standard definition of an admissible (or C-split) sequence of A-⊗-modules used in the homological theory of topological algebras (see [25,66] …”

Section: R-modules)mentioning

confidence: 99%

“…Arens-Michael envelopes (under a different name) were introduced in [66]. Here we adopt the terminology from [24].…”

Section: Arens-michael Envelopesmentioning

confidence: 99%

“…By definition, the Arens-Michael envelope of a C-algebra A is its completion A with respect to the system of all submultiplicative prenorms. The Arens-Michael envelopes were introduced by J. Taylor [66,67] in relation to a problem of multi-operator holomorphic functional calculus. The reason why those algebras may be relevant to noncommutative complex-analytic geometry is that the Arens-Michael envelope of the algebra O alg (V ) of regular (polynomial) functions on an affine algebraic variety V coincides with the algebra O(V ) of holomorphic functions on V endowed with the standard compact-open topology (see Example 3.6 below).…”

Section: Introductionmentioning

confidence: 99%

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Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

Pirkovskii

2008

Trans. Moscow Math. Soc.

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Abstract. We describe and investigate Arens-Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens-Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens-Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum (2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.

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“…By the requirements of his theory [13], J. L. Taylor developed in [12] important homological methods in the general framework of topological algebras. In the latter paper, among other interesting results, Proposition 5.7 was presented, which loosely speaking, has the following meaning: A contractible Arens-Michael algebra is topologically isomorphic to the direct sum of the topological cartesian product of a certain family of full matrix algebras and of some hypothetical "badly behaving" algebra, which in the commutative case is always zero.…”

Section: Introductionmentioning

confidence: 99%

Structure of contractible locally $C^*$-algebras

Fragoulopoulou¹

2001

Proc. Amer. Math. Soc.

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Abstract. A locally C * -algebra is contractible iff it is topologically isomorphic to the topological cartesian product of a certain family of full matrix algebras. IntroductionAt the beginning of the 1970's a well-known series of papers by J. L. Taylor appeared, viewing his concepts of joint spectrum and multi-operator functional calculus under the light of homology. By the requirements of his theory [13], J. L. Taylor developed in [12] important homological methods in the general framework of topological algebras. In the latter paper, among other interesting results, Proposition 5.7 was presented, which loosely speaking, has the following meaning: A contractible Arens-Michael algebra is topologically isomorphic to the direct sum of the topological cartesian product of a certain family of full matrix algebras and of some hypothetical "badly behaving" algebra, which in the commutative case is always zero. Around the 1980's A. Ya. Helemskii traced back a gap in the proof of the preceding result, which was filled by himself in 1985 in the case where the algebra under consideration is moreover commutative; for a proof of this result, see [5, Theorem IV. 5.27]. The aforementioned Taylor's result initiates a more general problem of whether an arbitrary contractible Arens-Michael algebra is topologically isomorphic to the topological cartesian product of a certain family of full matrix algebras. This, among other questions, was considered by the school of Helemskii in Moscow, with successful answers in several cases. Nevertheless, the main problem still remains open, even in the normed case (see [5, p. 195, comments after Exercise 5.28]). More precisely, in 1996, positive answers were given by Y. V. Selivanov [11] for contractible semiprime Fréchet Arens-Michael algebras having a nice geometrical property, the so-called "approximation property" of A. Grothendieck (cf. e.g.,

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Homology and cohomology for topological algebras

Cited by 139 publications

References 10 publications

Biflatness and Biprojectivity of Banach Algebras Graded Over a Semilattice

Biflatness and Biprojectivity of Banach Algebras Graded Over a Semilattice

Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

Structure of contractible locally $C^*$-algebras

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