Banach algebras which are ideals in a Banach algebra

Barnes, Bruce A.

doi:10.2140/pjm.1971.38.1

Cited by 36 publications

(31 citation statements)

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“…A Segal algebra S 1 (G) on a locally compact group G is a dense left ideal of L 1 (G) that satisfies the following conditions: Equipped with the norm || · || S and the convolution product, denoted by f, S 1 (G) is a Banach algebra. The inequality ||h f f|| S [ ||h|| 1 ||f|| S holds for all h ¥ L 1 (G) and f ¥ S 1 (G). It is shown in [12] (see also [5]) that every Segal algebra has a left approximate identity which is bounded in L 1 -norm.…”

Section: Introductionmentioning

confidence: 99%

“…, and R y f is continuous in y, where R y is the right translation operator defined by R y f(x)= f(xy Reiter's classical Segal algebra theory has been extended to a general abstract Segal algebra theory in [1][2][3]10] (see also [4,6]). Although this paper deals only with classical Segal algebras, our discussion will link up with some results established in abstract Segal algebras.…”

Section: Introductionmentioning

confidence: 99%

“…The structure of various ideals of a Segal algebra has been studied in [12,13] (see also [1,10]). For a compact group G, it is shown in [12,Sect.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Identities for Ideals of Segal Algebras on A Compact Group

Zhang¹

2002

Journal of Functional Analysis

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We show that every closed ideal of a Segal algebra on a compact group admits a central approximate identity which has the property, called condition (U), that the induced multiplication operators converge to the identity operator uniformly on compact sets of the ideal. This result extends a known one due to H. Reiter who has considered the problem under the condition that the Segal algebra is symmetric. We prove further that a closed right ideal of a Segal algebra on a compact group admits a left approximate identity satisfying condition (U) if and only if it is approximately complemented as a subspace of the Segal algebra; if in addition the Segal algebra is symmetric, then a closed left ideal admits a right approximate identity satisfying condition (U) if and only if it is approximately complemented. © 2002 Elsevier Science (USA)

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate Identities for Ideals of Segal Algebras on A Compact Group

Zhang¹

2002

Journal of Functional Analysis

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“…Main result. As in [3], we denote the spectrum of an element x in a Banach algebra B by SpÄ(x) and its spectral radius by vB(x). In an earlier manuscript [8], the authors gave a proof of Theorem 2 in the spirit of [9], [2].…”

Section: 2 Theorem 23])mentioning

confidence: 99%

Multipliers of 𝐴*-algebras

Johnson¹,

Lahr²

1979

Proc. Amer. Math. Soc.

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Abstract. Let A be an A '-algebra of the first kind with C*-algebra completion St. It is known that if A is dual then A2 is dense in A and the Banach algebras ML(A) and ML(2t) of left multipliers of A and 31 are algebra isomorphic. In this note it is proved that ML(A) and ML(%) are topologically algebra isomorphic when A is an arbitrary A '-algebra of the first kind such that A2 is dense in A. As a consequence, it follows that every left multiplier of a replete Hilbert algebra A is automatically continuous.

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“…In this case | • | is a norm on A with the B*-property by (1) and (3) We are now in a position to answer Question 2 affirmatively when a. is assumed to be a strictly pure state of A. …”

Section: Introductionmentioning

confidence: 99%