On the Existence of Minimal Ideals in a Banach Algebra

Barnes, Bruce A.

doi:10.2307/1994992

Cited by 6 publications

(11 citation statements)

References 4 publications

(7 reference statements)

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“…Also C(t) is the closed linear span of its selfadjoint idempotents by the remarks preceding the statement of the theorem. This verifies (1) and (2) of Lemma 2.6. Therefore there exists a selfadjoint idempotent e e K such that (1 -e)K(l -e) is ¿""-equivalent, e e J for some J in <€ and J is ¿""-equivalent.…”

supporting

confidence: 78%

“…Therefore F = 0. This proves (1). (2) Given / g A, / = /*, then the spectrum of/ in C(t) is real by [6,Lemma (4.8.1)(i), p. 240].…”

mentioning

confidence: 75%

“…If / e A and ? has range in 3, then / = / * by (1 ) and (3) of Proposition 2.1. Then C(t)~ is a sup norm complete *-subalgebra of Â.…”

mentioning

confidence: 89%

“…Now assume that t e As. C(t) is ¿""-equivalent, and therefore, from [1,Corollary,p. 517] and [8,Theorem 4.1,p.…”

mentioning

confidence: 95%

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Locally 𝐵*-equivalent algebras

Barnes¹

1972

Trans. Amer. Math. Soc.

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supporting

confidence: 78%

“…Therefore F = 0. This proves (1). (2) Given / g A, / = /*, then the spectrum of/ in C(t) is real by [6,Lemma (4.8.1)(i), p. 240].…”

mentioning

confidence: 75%

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Locally 𝐵*-equivalent algebras

Barnes¹

1972

Trans. Amer. Math. Soc.

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“…The theory of modular annihilator algebras was developed by Yood in [13]. In [l], [2] B arnes has extended this study to semi-simple Banach algebras.…”

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confidence: 99%

Modular annihilator algebras

Giotopoulos

Katseli

1991

Period Math Hung

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The elements of minimal left (right) ideals in a semi-prime modular annihilator algebra A completely characterized by the property of being singles not in rad A. An element s of A is called single if whenever asb = 0 for some a, b in A then at least one of as, sb is zero. B(X) of bounded operators on X containing it. In a number of operator algebras the converse is also true (see [7], [lo], [12]) making the property of being equivalent to being an operator of rank one. In an abstract semi-prime modular annihilator Banach, the property of being rank one is meaningless,'but the link between single elements and rank oneness is crucial. To be specific, we show that such Banach algebras can faithfully represented on a Banach space X in such a way that the image of an element s of A is of rank one if and only if the element is single and s 4 rad A. Also we completely characterize the minimal left (right) ideals of A by showing that such ideals are precisely the ideals of the form As(sA) where s is single, s 4 rad A. Finally, we characterize the socle of A as the set of all finite sums of single elements that are not in the radical of A.In general, the terminology and the notation used will be as in F. F. Bonsall and J. Duncan in [3]. All the algebras considered will be over the complex field.Mathematics subject classification numbers, Primary 1980/85.

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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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On the Existence of Minimal Ideals in a Banach Algebra

Cited by 6 publications

References 4 publications

Locally 𝐵*-equivalent algebras

Locally 𝐵*-equivalent algebras

Modular annihilator algebras

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

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On the Existence of Minimal Ideals in a Banach Algebra

Cited by 6 publications

References 4 publications

Locally 𝐵*-equivalent algebras

Locally 𝐵*-equivalent algebras

Modular annihilator algebras

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

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Product

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