Biprojective Banach algebras, their structure, cohomologies, and connection with nuclear operators

Selivanov, Yu. V.

doi:10.1007/bf01075782

Cited by 13 publications

(11 citation statements)

References 2 publications

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“…(iv) the algebra A is superbiprojective? Earlier, similar questions were answered for C * -algebras [34,80,81,82,40,41,89,90]. Apart from this, the latter two questions were answered for σ-C * -algebras [67].…”

Section: Introductionmentioning

confidence: 85%

Structure theory of homologically trivial and annihilator locally C^*-algebras

Pirkovskii

Selivanov

2010

Banach Center Publ.

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We study the structure of certain classes of homologically trivial locally C * -algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C * -algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C * -algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically modular annihilator) topological algebra. We characterize all annihilator σ-C * -algebras and describe the structure of biprojective locally C * -algebras. Also, we present an example of a biprojective locally C * -algebra that is not topologically isomorphic to a Cartesian product of biprojective C * -algebras. Finally, we show that every superbiprojective locally C * -algebra is topologically * -isomorphic to a Cartesian product of full matrix algebras.

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Section: Introductionmentioning

confidence: 85%

Structure theory of homologically trivial and annihilator locally C^*-algebras

Pirkovskii

Selivanov

2010

Banach Center Publ.

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“…This approach gives a new proof of the well-known result that any biprojective C * -algebra is the direct sum of C * -algebras of the type M n (C) (see [34]). This approach gives a new proof of the well-known result that any biprojective C * -algebra is the direct sum of C * -algebras of the type M n (C) (see [34]).…”

Section: Commutative C * -Subalgebras Of Biprojective C * -Algebrasmentioning

confidence: 94%

“…Note that, in view of Theorem 5·1, Corollary 5·2 and the remark about biprojectivity of K(H), we have a new proof of the following theorem (see [34]). …”

Section: Commutative C * -Subalgebras Of Biprojective C * -Algebrasmentioning

confidence: 96%

Relations between the homologies of C-algebras and their commutative C-subalgebras

Lykova¹

2002

Math. Proc. Camb. Phil. Soc.

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The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A [otimes ]minA is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space.

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“…Recall Selivanov's result [25] that any biprojective C * -algebra is the direct sum of C * -algebras of the type M n (C). Another proof of this result is given in [20,Theorem 5.4].…”

Section: Biprojectivity Of Banach Algebras Of Continuous Fieldsmentioning

confidence: 99%

Projectivity of Banach and C*-Algebras of Continuous Fields

Cushing¹,

Lykova²

2012

The Quarterly Journal of Mathematics

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We give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective C * -algebras A defined by locally trivial continuous fields U = {Ω, (A t ) t∈Ω , Θ} such that each C * -algebra A t has a strictly positive element. For a commutative C * -algebra D contained in B(H), where H is a separable Hilbert space, we show that the condition of left projectivity of D is equivalent to the existence of a strictly positive element in D and so to the spectrum of D being a Lindelöf space.

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Biprojective Banach algebras, their structure, cohomologies, and connection with nuclear operators

Cited by 13 publications

References 2 publications

Structure theory of homologically trivial and annihilator locally C^*-algebras

Structure theory of homologically trivial and annihilator locally C^*-algebras

Relations between the homologies of C-algebras and their commutative C-subalgebras

Projectivity of Banach and C*-Algebras of Continuous Fields

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Biprojective Banach algebras, their structure, cohomologies, and connection with nuclear operators

Cited by 13 publications

References 2 publications

Structure theory of homologically trivial and annihilator locally C*-algebras

Structure theory of homologically trivial and annihilator locally C*-algebras

Relations between the homologies of C*-algebras and their commutative C*-subalgebras

Projectivity of Banach and C*-Algebras of Continuous Fields

Contact Info

Product

Resources

About

Structure theory of homologically trivial and annihilator locally C^*-algebras

Structure theory of homologically trivial and annihilator locally C^*-algebras

Relations between the homologies of C-algebras and their commutative C-subalgebras