Eberhard Kaniuth

A course in commutative Banach algebras

General Theory of Banach Algebras.- Gelfand Theory.- Functional Calculus, Shilov Boundary, and Applications.- Regularity and Related Properties.- Spectral Synthesis and Ideal Theory.

Ideals with bounded approximate identities in Fourier algebras

Abstract We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.

On -amenability of Banach algebras

Generalizing the notion of left amenability for so-called F-algebras [12], we study the concept of -amenability of a Banach algebra A, where is a homomorphism from A to . We establish several characterizations of -amenability as well as some hereditary properties. In addition, some illuminating examples are given.

ZEROS OF THE ZAK TRANSFORM ON LOCALLY COMPACT ABELIAN GROUPS

Let G be a locally compact abelian group. The notion of Zak transform on L2 (Rd) extends to L2 (G). Suppose that G is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for L2 (R), we then prove that if f E L2(G) is such that its Zak transform Zf is continuous on G x G, then Zf has a zero.

Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups

BY KARLHEINZ GROCHENIGDepartment of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.EBERHARD KANIUTHFachbereich Mathematik/Informatik, Universitdt Paderborn, D-WA790 Paderborn,GermanyAND KEITH F . TAYLO RDepartment of Mathematics, University of Saskatchewan, Saskatoon, SaskatchewanSIN OWO, Canada(Received 14 March 1991; revised 23 August 1991)1. IntroductionThe main purpos of thi es pape isr t studo y projections is self-adjoin,, that tidempotents, in L

The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras

The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M(G) of G. This has led to the study of the algebra of BSE-functions on the spectrum of an arbitrary commutative Banach algebra and of the concept of a BSE-algebra as introduced by Takahasi and Hatori. Since then BSE-algebras have been studied by several authors. In this paper we investigate BSE-algebras in the general context on the one hand and, on the other hand, we specialize to Fourier and Fourier-Stieltjes algebras of locally compact groups.

Spectral synthesis for () and subspaces of ()

Let G be a locally compact group, A(G) the Fourier algebra of G and VN(G) the von Neumann algebra generated by the left regular representation of G. We introduce the notion of X-spectral set and X-Ditkin set when X is an A(G)-invariant linear subspace of VN(G), thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of X-spectral sets and X-Ditkin sets, and an injection theorem for X-spectral sets.

Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. We study the problem of how weak8-closedness of some translation invariant subspaces of B(G) is related to the structure of G. Moreover, we prove that for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak8-continuous only when H is open in G. INTRODUCTION Let G be a locally compact group, and let B(G) be the Fourier-Stieltjes algebra of G as defined by Eymard [8]. Recall that B(G) is the linear span of all continuous positive definite functions on G and can be identified with the Banach space dual of C*(G), the group C*-algebra of G. The space B(G), with the xlorm as dual of C*(G), is a commutative Banach *-algebra with pointwise multiplication and complex conjugation. The Fourier algebra A(G) of G is the closed *-subalgebra of B(G) generated by the functions in B(G) with compact support. In particular, A(G) is contained in Co(G), the algebra of complex valued continuous functions on G vanishing at infinity. As is well known A(G) is weak*-dense in B(G) if and only if G is amenable. In [3] translation invariant *-subalgebras A of B(G) were studied, and it was shown that if such A is weak*-closed and point separating, then it must contain A(G). However, apart from this, very little seems to be known about weak*-closed subspaces of B(G). The first purpose of this paper is to investigate the relation between weak*closedness of certain interesting norm-closed translation invariant subspaces of B(G) and the structure of G. Secondly, we solve the problem of whexl, for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak*-continuous. A brief outline of the paper is as follows. In Section 2 we estsablish for almost connected locally compact groups G the relation between weclk*-closed.ness of Bo(G) = B(G) n Co(G) in E3(G) and the structure of G (Theorem 2.10). The key result is that for a connected Lie group G, Bo(G) is weak*-closed in B(G) if and only if G is a reductive Lie group with compact centre and Kazhdans property (T). Received by the editors December 1S, 1995. 199l Mathematics Subject Classification. Primary 22D10, 43A30. Work supported by NATO collaborative research grant CRG 940184. (g)1997 American Mathematical Society

A Qualitative Uncertainty Principle for Certain Locally Compact Groups

It is known that if the Supports of a function/e L (IR*) and its Fourier transform have finite measure then / = 0 almost everywhere. We study generalizations of this property for several classes of locally compact groups. These include compact extensions of groups having the above property and multiplier extensions of Tby abelian groups. Some examples of lowdimensional nilpotent Lie groups having this property are also discussed. 1980 Mathematics Subject Classification (1985 Revision): 43A30; 22D99.

The generalized Wiener Theorem for groups with finite dimensional irreducible representations

Abstract It is shown that the generalized Wiener Theorem for ideals in group algebras of locally compact Abelian groups extends to group algebras of locally compact groups with finite-dimensional irreducible representations (Moore groups).