Jyunji Inoue

A spectral mapping theorem for some representations of compact abelian groups

We show that if G is a compact abelian group and U is a weakly continuous representation of G by means of isometries on a Banach space X , then holds for each measure µ in reg( M ( G )), where π(µ) denotes the generalized convolution operator in B ( X ) defined by , σ the usual spectrum in B ( X ), sp ( U ) the Arveson spectrum of U , the Fourier-Stieltjes transform of µ and reg( M ( G )) the largest closed regular subalgebra of the convolution measure algebra M ( G ) of G . reg( M ( G )) contains all the absolutely continuous measures and discrete measures.

Segal algebras in commutative Banach algebras

The notion of Reiter’s Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

Finite Dimensional Solution Sets of Extremal Problems in H

For a non-zero function f in H 1, the classical Hardy space on the unit circle, put

On the zeros of solutions of an extremal problem in H 1

{{S}_{{\left| f \right|/f}}} = \left\{ {g \in {{H}^{1}}:{{{\left\| g \right\|}}_{1}} = 1,\arg f\left( {{{e}^{{it}}}} \right)a.e.t} \right\}

An example of a non-exposed extreme function in the unit ball of H 1

, then \({S_{\left| f \right|/f}}\) is the set of extremal functions of a well known linear extremal problem in H 1.

On characterizations of the image of the Gelfand transform of commutative Banach algebras

A simple proof of Albrechts result on the existence of the largest closed regular subalgebra of a semisimple commutative Banach algebra is given. Albrecht [1] proved that any semisimple commutative Banach algebra with identity has the largest closed regular subalgebra, using the theory of decomposable operators. In this short note, we give a simple proof of Albrechts result without the assumption of semisimplicity. Our proof is quite different from Albrechts, and based on the consideration of the hull-kernel topology of the carrier space of the given Banach algebra. Theorem. If A is a commutative Banach algebra with identity, then there exists a closed regular subalgebra of A which contains all closed regular subalgebras of A. Our proof is essentially based on the following Lemma. Let X be a commutative Banach algebra with identity and B a Banach subalgebra of X. If B is regular, then for any b E B the Gelfand transform of b as an element of X is continuous on the carrier space Ox of X in the hull-kernel topology. Proof. We can assume without loss of generality that B contains the identity of X. Then it is sufficient to show that the restriction map 0: Ox DB; (0 -* 0 IB is continuous in the hull-kernel topology. To do this let F be a closed subset of 1DB in the hull-kernel topology. Then { o E ODx: (0I ker F = O} = 0I(F) . Also since kerF c ker 06 I(F), it follows that hul(ker 0I(F)) C {p E (Dx: Iker F = O}. Therefore 0(F) is closed in the hull-kernel topology. In other words, 0 is continuous in this topology. El Proof of Theorem. Let reg(A) be the closed subalgebra of A generated by the class of all closed regular subalgebras of A. It is sufficient to show that reg(A) is regular. Note that reg(A) is a commutative Banach algebra with identity. If Received by the editors November 29, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H05.

Some closed subalgebras of measure algebras and a generalization of P. J. Cohen's theorem II

For a nonzero function f in H 1, the classical Hardy space on the unit disc, we put .. The intersection of Sf and the unit sphere in H 1 is a solution set of a certain extremal problem in H 1. It is known that Sf can be represented in the form Sf = SB × g 0, where B is a Blaschke product and g 0 is a function in H 1 with Sg0 = {λ g0 : λ > 0}. Also it is known that the linear span of Sf is finite dimensional if and only if B is a finite Blaschke product, and when B is a finite Blaschke product, we can describe completely the set SB and the zeros of functions in SB . In this paper we study the set of zeros of functions in SB when B is an infinite Blaschke product whose set of singularities is not the whole circle. In particular, we study the behavior of zeros of functions in SB in the sectors of the form: δ= {reiθ:0 < r ≤ 1c1 < θ < c2 } on which the zeros of B has no accumulation points, and establish a convergence order theorem for the zeros in δ of functions in SB .