J. J. Grobler

Spectral properties of elements in different Banach algebras

by J. J. GROBLER and H. RAUBENHEIMER(Received 7 June, 1989)1. Introduction. Let A be a Banach algebra with unit 1 and let B be a Banachalgebra which is a subalgebra of A and which contains 1. In [5] Barnes gave sufficientconditions for B to be inverse closed in A. In this paper we consider single elements andstudy the question of how the spectrum relative to B of an element in B relates to thespectrum of the element relative to A.This question has been studied by a number of authors (see [14, 2, 3, 4, 12, 13, 15])under various conditions, usually requiring the norm on B to be finer than the norm on Aand in some cases requiring B to be a Banach algebra of operators on a Banach lattice.We shall merely assume the algebra B to be a subalgebra of A and we shall not assumeany relationship between the norms of A and B. Of course, if B is semi-simple and closedin A, the embedding of B into A is continuous (see [7, Theorem 25.9]). We do not intendto enter into a discussion of the question whether or not this is always true—this is a deepquestion which seems also to be related to the model of set theory used (see [8]). Thepoint we want to emphasize is that the main results are independent of the continuity ofthe embedding.If A is an algebra we denote the spectrum of an element a eA by o(a, A) and itsspectral radius by

A note on the theorems of Jentzsch-Perron and Frobenius

Abstract In this note we prove extensions of the theorems mentioned in the title. In the existing versions of these theorems it is assumed that the Banach lattice is Dedekind complete and that the operator which occurs is an (abstract) kernel operator. We show that the theorems hold in arbitrary Banach lattices for operators which are only assumed to be σ-order continuous. Only for some details the Banach lattice is assumed to be Dedekind σ-complete.

DOUBLY ABEL BOUNDED OPERATORS WITH SINGLE SPECTRUM

Abstract In a unitary Banach algebra conditions are sought for an element, the spectrum of which is the singleton {1}, to be the identity of the algebra. Our main result is that this is true if the element is Abel bounded and doubly (N)-uniformly Abel bounded. This condition is fulfilled if the element is doubly power bounded in which case the theorem was proved by I. Gelfand [4] (1941). If the element is doubly Cesaro bounded, the result is due to M. Mbekhta and J. Zemanek [7, Theorem 2]. We present an example of a matrix which is Abel bounded and doubly (2)-Abel bounded, but not Cesaro bounded.

AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Abstract We construct the Riesz tensor product of Archimedean Riesz spaces and derive its properties using functional calculus and f-algebras. We improve results on the approximation of elements in the Riesz tensor product by means of elements in the vector space tensor product in such a way that the order density property is a consequence of the improved approximation result.

ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS

ABSTRACT We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.

Carleman operators in Riesz spaces

Abstract A linear operator T from a normed space G into a Riesz space F is called a Carleman operator if the image of the unit ball in G is an order bounded subset of the universal completion of F . This abstract formulation is adequate to generalize the classical results concerning Carleman operators to the setting where no measure space is involved. This includes the characterizations of these operators, their relationship with abstract kernel operators and some compactness-type properties. In the special case where G and/or F are ideals of measurable functions, we regain the results of V.B. Korotkov and the recent results of A.R. Schep (Proc. Kon. Ned. Akad. Wetensch. A83 (Indag. Math. 42), 49–59 (1980)). We also show how some of the results of N.E. Gretsky and J.J. Uhf (Acta Sci. Math. (Szeged) 43, 207–218, 1981) fit into this general framework.

The zero-two law in Banach lattice algebras

LetA be a unital Banach lattice algebra and leta εA+ satisfy ‖a ‖≦1. Then either ‖an+1 −an ‖=2 for alln≧0 or else ‖an+1 −an ‖ → 0 asn → ∞. Cyclicity of the peripheral spectrum ofa is also established.

A CHARACTERIZATION OF THE BAND OF KERNEL OPERATORS

ABSTRACT In this note we present a characterization of the band of kernel operators in the abstract setting of Riesz spaces. Under the assumptions that E is an Archimedean Riesz space and F a Dedekind complete Riesz space separated by its ex= tended order continuo88 dual, we obtain a characterization of the band (Eoo ⊗ F)dd in terms of (sequentially) star or= der continuous operators.

Disjointness Preserving Operators on Complex Riesz Spaces

AbstractIt is proven that ifE