Automatic continuity and representation of certain linear isomorphisms between group algebras

Abstract. A linear map T from a Banach algebra A into another B preserves zero products if T (a)T (b) = 0 whenever a, b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T : A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras.Our method involves continuous bilinear maps φ : A × A → X (for some Banach space X) with the property that φ(a, b) = 0 whenever a, b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies φ(aµ, b) = φ(a, µb) for all a, b ∈ A and for all µ from the closure with respect to the strong operator topology of the subalgebra of M(A) (the multiplier algebra of A) generated by doubly power-bounded elements of M(A). This method is also shown to be useful for characterizing derivations through the zero products.Introduction. S. Banach [4] was the first to describe isometries on L p ([0, 1]) with p = 2. Although Banach did not give the full proof for this case (this was provided by J. Lamperti [26]), he made the key observation that isometries on L p ([0, 1]) must take functions with disjoint support into functions with disjoint support. This property arises in a variety of situations and has been considered by several authors. For example, in the theory of Banach lattices there is an extensive literature about linear maps T : X → Y , where X and Y are Banach lattices, with the property that |T (x)| ∧ |T (y)| = 0 whenever x, y ∈ X are such that |x| ∧ |y| = 0. Such maps are usually called disjointness preserving operators or d-homomorphisms. The reader interested in this setting is referred to the monograph [1]. The concept of a disjointness preserving operator was exported to function algebras by E. Beckenstein and L. Narici (see [5] for general information).

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“…J. Font and S. Hernández proved in [17] that if G 1 and G 2 are locally compact abelian groups, then every bijective linear map T :…”

Section: (Ab)t (C) = T (A)t (Bc) (A B C ∈ A)mentioning

confidence: 99%

Maps preserving zero products

et al. 2009

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“…[1], [2] or [6]), on spaces of continuous functions (see e.g. [14], [3], [7], [15] or [12]), on group algebras of locally compact Abelian groups ( [8]), on Fourier algebras ( [10] and [20]) and on some others (see e.g. [16], [17] or [5]).…”

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confidence: 99%

“…In [8] we proved that the existence of a disjointness separating bijection between the group algebras of two locally compact Abelian groups implies that these algebras are algebraically isomorphic. A similar result was obtained in [10] (resp.…”

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confidence: 99%

Disjointness preserving mappings on BSE Ditkin algebras

Font¹

2011

Publ. Math. Debrecen

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Let A and B be regular Banach function algebras. A linear map T defined from A into B is said to be disjointness preserving or sepa-We prove that if there exists a disjointness preserving bijection between two BSE Ditkin algebras with a BAI, then they are isomorphic as algebras. As a corollary we can deduce that two of these algebras are algebraically isomorphic if there exists a surjective isometry between them for the supremum norm.

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“…cit.) in [16], [10] and [11] are extended to a wider class of regular Banach function algebras that includes, for instance, Segal algebras [21] or the Banach sequence algebras / P (N), p e (0, °°), in [9]. It is, however, important to remark that a separating map need not be continuous; indeed, K. Jarosz proved [16] that, given two compact spaces X (infinite) and Y, there, always exists a discontinuous separating map defined from C(X) into C(Y); see also [8].…”

Section: Juan J Font Theorem ([11]) a Separating Bijection T:l 1 (mentioning

confidence: 98%

Automatic continuity of certain isomorphisms between regular Banach function algebras

Font¹

1997

Glasgow Math. J.

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Abstract. Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if/.g = 0 implies Tf.Tg = 0, for all f,g e A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.Finally, we show that linear isometries between regular uniform algebras are separating. As corollaries, a classical theorem of Nagasawa ([19]) and the Banach-Stone theorem (both for regular uniform algebras) are easily inferred. Introduction.The basic problem in the theory of automatic continuity of linear operators consists of giving algebraic conditions on two Banach algebras si and 38 which ensure that every algebra homomorphism !T:s4-*2ft is necessarily continuous. The most important positive result in this context is due to B. E. Johnson ([15]). Every algebra homomorphism of a Banach algebra onto a semisimple Banach algebra is continuous.In this paper we shall study the automatic continuity of a special type of linear mapping which extends the concept of algebra homomorphism: Let A and B be two independently, Koldunov [17] have shown that the inverse of a disjointness preserving bijection is disjointness preserving under certain general conditions. Disjointness preserving maps were also considered in [8] for spaces of real or complex-valued continuous functions denned on a compact Hausdorff space with the name of separating maps. The main goal in this context is to prove automatic continuity results for separating maps between different kinds of function spaces of the following type.

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Automatic continuity and representation of certain linear isomorphisms between group algebras

Cited by 20 publications

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Maps preserving zero products

Maps preserving zero products

Disjointness preserving mappings on BSE Ditkin algebras

Automatic continuity of certain isomorphisms between regular Banach function algebras

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