Representations of B(X)

Berkson, Earl; Porta, Horácio

doi:10.1016/0022-1236(69)90048-2

Cited by 45 publications

(54 citation statements)

References 17 publications

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“…By Grothendieck's completeness theorem (7, page 103) a linear functional is tr(3I", 2l')-continuous if and only if it is ff(9l", 9T)-continuous on B for each bounded set B in 31. Thus (1) implies (2).…”

Section: Lemma 1 a Continuous Linear Functional H On A Locally Convementioning

confidence: 93%

“…For any continuous function /"of compact support on G, || F|| p -»|| F||j asp->l by the dominated convergence theorem: we may therefore find m e N so large that and l l / * « K l l , m^O -Consider (g n ) e E, where g n -0 if n ^ m and g m = e K : then IK^JI^ ^ 1 + e and The operator norm of T f is thus at least (l-e) 2 (l + e)~1 | | / | | i , and therefore (since e was arbitrary) it equals ||/||i.…”

Section: Theorem 4 the Group Algebra V (G) Of Any Locally Compact Grmentioning

confidence: 99%

“…The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned-namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra 2(E) of all continuous linear operators on E where E -I" with 1 <p # 2<oo, for Berkson and Porta proved in (2) that if E, F are taken from the spaces l p with 1 <p < oo and E # F then the only continuous homomorphism from £(£) into £(F) is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space-for example the algebra of finiterank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).…”

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confidence: 99%

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Periodicity of functionals and representations of normed algebras on reflexive spaces

Young

1976

Proceedings of the Edinburgh Mathematical Society

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It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

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Section: Lemma 1 a Continuous Linear Functional H On A Locally Convementioning

confidence: 93%

Section: Theorem 4 the Group Algebra V (G) Of Any Locally Compact Grmentioning

confidence: 99%

mentioning

confidence: 99%

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Periodicity of functionals and representations of normed algebras on reflexive spaces

Young

1976

Proceedings of the Edinburgh Mathematical Society

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“…These operator ideals generalize a construction due to Porta (see [25] and [2]). In our application we shall require the following technical fact.…”

Section: {0}mentioning

confidence: 99%

Maximal Ideals in the Algebra of Operators on Certain Banach Spaces

Laustsen

2002

Proceedings of the Edinburgh Mathematical Society

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For a Banach space X, let B(X) denote the Banach algebra of all continuous linear operators on X. First, we study the lattice of closed ideals in B(Jp), where 1 < p < ∞ and Jp is the pth James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in B(Jp). Applications of this result include the following. (ii) For each natural number n and each n-tuple (m 1 , . . . , mn) in {k 2 | k ∈ N} ∪ {∞}, there is a Banach space X such that B(X) has exactly n maximal ideals, and these maximal ideals have codimensions m 1 , . . . , mn in B(X), respectively; the Banach space X is a finite direct sum of James spaces and p-spaces.Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces X such that all the maximal ideals in B(X) can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in B(X) for each hereditarily indecomposable Banach space X, and we prove that there are 2 2 ℵ 0 distinct maximal ideals in B(G), where G is the Banach space constructed by Gowers to solve Banach's hyperplane problem.

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“…This is true for &(Y), the algebra of all bounded operators on a Banach space Y, and for &( Y), the subalgebra of 38(Y) consisting of operators with finite dimensional range. The representations of 3S(Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of ^(Y) (and also of some more general algebras of operators). In both these papers, ^(Y), which is the socle of the algebras under consideration, plays an important role in the theory.…”

Section: Introductionmentioning

confidence: 99%

Representations of normed algebras with minimal left ideals

Barnes

1974

Proceedings of the Edinburgh Mathematical Society

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The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

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Representations of B(X)

Cited by 45 publications

References 17 publications

Periodicity of functionals and representations of normed algebras on reflexive spaces

Periodicity of functionals and representations of normed algebras on reflexive spaces

Maximal Ideals in the Algebra of Operators on Certain Banach Spaces

Representations of normed algebras with minimal left ideals

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