J. Duncan

C *-Algebras of inverse semigroups

There are various algebras which may be associated with a discrete group G . In particular we may consider the complex group ring ℂ G , the convolution Banach algebra l 1 (G), the enveloping C *-algebra C *( G ) of l 1 ( G ), and the reduced C *-algebra determined by the completion of l 1 ( G ) under the left regular representation on l 2 ( G ). There is a substantial literature on the circle of ideas associated with the embeddings

Carleman's inequality

The statement of the inequality is actually a little more precise, but we shall discuss that later. Carleman used the inequality as part of a new proof of a delightful theorem of Denjoy about quasi-analytic functions. We shall not pursue that application here since we are interested in the inequality as a theorem about infinite series. We wish to expound in detail some of the many proofs of this theorem. We remark in passing, with approval, that editors of mathematical journals in the 1920s were happy to publish many different proofs of the same theorem. There are many known results about the convergence (or divergence) of series obtained by linear transformations of a nonnegative sequence {an }. For example, if Ain is the arithmetic mean of a1, a2, ... , an, then Y An, always diverges unless {an } is the zero sequence (for the sum dominates part of the harmonic series). The arithmeticgeometric mean inequality (AGM) asserts that y, < An, with equality if and only if the sequence {an} is constant. So the AGM-inequality must be strict when E an is convergent and nonzero. It is perhaps surprising a priori that the AGM-inequality is so

Monogenic inverse semigroups and their C* -algebras

By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C* -algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroup ℑℐ t .

Dual representations of Banach algebras

The irepresentation theory for Banach algebras has three main branches that are only rather loosely connected with each other. The Gelfand representation of a commutative algebra represents the given algebra by continuous complex valued functions on a space built from the multiplicativc linear functionals on the algebra. A Banach star algebra is represented by operators on a Hilbert space, the Hilbert space being built by means of positive I.iermitian funetionals on the algebra. Finally, for general non-commutat ive Banach algebras, an extension of the Jacobson theory of representations of rings is available. In this general theory, the representations are built in terms of irreducible operator representations on Banach spaces, and, on the face of it, no par t is played by the linear functionals on the algebra. There is some evidence that the concepts involved in the general theory are not sufficiently strong to exploit to the full the Banach algebra situation. The purpose of the present paper is to develop a new unified general representation theory that is more closely related than the Jaeobson theory to the special theories for commutat ive and star algebra s. The central concept is that of a dual representation on a pair of Banach spaces in normed duality. I t is found tha t each continuous linear functional on a Banach algebra gives rise to a dual representation of the algebra, and thus the dual space of the Mgebra enters representation theory in a natural way. One may ask of a dual representation that it be irreducible on each of the pair of spaces in duality, and thus obtain a concept of irreducibility stronger than the classical one. Correspondingly one obtains a stronger concept of density. For certain pairs of spaces in duality, topological irreducibility on one of the spaces implies topological irreducibility on the other. However, we show tha t this is very far from being the case in general. We also consider a further concept of irreducibility, namely uniform strict transitivity, which is stronger than strict irreducibility. For certain pairs of spaces in duality, uniform strict transit ivity on one of the spaces implies

Some inequalities for norm unitaries in Banach algebras

Let A be a complex unital Banach algebra. An element u ∈ A is a norm unitary if (For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u ∈ A , we have Sp( u )⊃Γ, where Sp( u ) denotes the spectrum of u and Γ denotes the unit circle in C . If Sp( u )≠Γ we may suppose, by replacing e iθ u , that . Then there exists h ∈ A such that

On one-sided primitivity of Banach algebras

Let

The evaluation functionals associated with an algebra of bounded operators

S

Some extremal algebras for hermitians

be the semigroup with identity, generated by

On compact normal semigroups

x

THE EXTREMAL ALGEBRA ON TWO HERMITIANS WITH SQUARE 1

and