Compact Homomorphisms of C ∗ -Algebras

Ghahramani, F.

doi:10.2307/2047161

Cited by 6 publications

(6 citation statements)

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“…Our results build on previous work in the area, which has concentrated on compact endomorphisms of commutative Banach algebras [11,12,13], compact homomorphisms between C*-algebras [9], and compact and weakly compact homomorphisms between uniform algebras [18]. These references also provide a fund of interesting examples of weakly compact homomorphisms, most of them actually compact (see §4).…”

Section: Introductionsupporting

confidence: 60%

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Weakly compact homomorphisms

Galé¹,

Ransford²,

White³

1992

Trans. Amer. Math. Soc.

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We study the structure of weakly compact homomorphisms between Banach algebras. In particular, it is shown that between many pairs of algebras, the only weakly compact homomorphisms are those of finite rank. Problem 1.1. Let A and B be Banach algebras. If 6 : A-> B is a weakly compact homomorphism, then do there exist a reflexive Banach algebra C and continuous homomorphisms cp : A-> C and i// : C-> B such that 9 = y/ o cp ? The corresponding statement for linear maps between Banach spaces is known to be true (see e.g. [5; 15, Theorem 2.g.l 1]), but its proof fails to adapt to our situation because the interpolation method used does not respect Banach algebras.

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Section: Introductionsupporting

confidence: 60%

“…In [9], Ghahramani showed that a compact homomorphism between C*algebras has a finite-dimensional, semisimple range. The principal theorem of this section is a generalization of his result.…”

Section: Homomorphisms From C* -Algebrasmentioning

confidence: 99%

Weakly compact homomorphisms

Galé¹,

Ransford²,

White³

1992

Trans. Amer. Math. Soc.

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“…In other words, the identity mapping on a C * -algebra A is weakly compact if and only if A is finite dimensional. Actually, an algebraic homomorphism from a C * -algebra to a normed algebra is weakly compact if and only if it has finite dimensional range (see [17], [30]). …”

Section: Weakly Compact Triple Homomorphismsmentioning

confidence: 99%

Weakly compact orthogonality preservers on C*-algebras

2010

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We establish a complete description of weakly compact orthogonality preserving operators from a C * -algebra A (or a JB * -algebra) to a JB * -triple E. We prove that any such operator is the sum of a series of mutually orthogonal triple homomorphisms from A * * to a certain Peirce-2 subspace of E * * multiplied by a mutually orthogonal norm-null sequence in E which enjoy certain operator commutativity relations.

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“…This line of research originates in a result of Ghahramani [6,Theorem 1], who proved that a compact homomorphism defined on a C * -algebra is a finite rank operator. Mathieu [11] generalised this result by proving that a weakly compact homomorphism defined on a C * -algebra with range in a normed algebra is a finite rank operator.…”

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confidence: 99%