The centre of the second dual of a commutative semigroup algebra

1984 ◽  
Vol 95 (1) ◽  
pp. 71-92 ◽  
Author(s):  
D. J. Parsons
Keyword(s):  
Banach Algebra ◽  
Commutative Semigroup ◽  
Topological Semigroup ◽  
Continuous Functions ◽  
Semigroup Algebra ◽  
Measure Algebra ◽  
Semitopological Semigroup ◽  
Borel Measures ◽  
Second Dual ◽  
Right Topological Semigroup

If S is an infinite, discrete, commutative semigroup then the semigroup algebra l1(S) is a commutative Banach algebra. Its dual is l∞(S), which is isometrically iso-morphic to C(βS), the space of continuous functions on the Stone-Čech compactification of S. This fact enables us to identify the second dual of l1(S) with M(βS), the space of bounded regular Borel measures on βS. Endowed with the Arens product the second dual is also a Banach algebra, so it is natural to ask whether a product may be defined in M(βS) without reference to l1(S). In §4 this is shown to be possible even when S is a non-discrete semitopological semigroup, provided that the operation in S may be extended to make βS into a left-topological semigroup in the manner of, for example, [2] where further references may be found. (Note, however, that the construction there is of a right-topological semigroup.) Having done this we may use results on βS to provide information about the measure algebra.

scholarly journals A convolution semigroup of modular functions

1966 ◽  
Vol 6 (2) ◽  
pp. 251-255
Author(s):  
Y.-F. Lin
Keyword(s):  
Topological Semigroup ◽  
Present Note ◽  
Continuous Functions ◽  
Convolution Semigroup ◽  
Modular Functions ◽  
Star Topology ◽  
Borel Measures ◽  
Weak Star ◽  
Compact Topological Semigroup ◽  
Complex Valued

Let S be a compact topological semigroup, and let be the collection of all normalized non-negative Borel measures on S. It is well-known that , under convolution and the topology induced by the weak-star topology on the dual of the Benach space C(S) of all complex valued continuous functions on S, forms a compact topological semigroup which is known as the convolution semigroup of measures (see for instance, Glicksberg [3], Collins [1], Schwarz [5] and the author [4]). [1], Schwarz [5] and the author [4]). Professor A. D. Wallace asked if the process of forming the convolution semigroup of measures might be generalized to a more general class of set functions, the so-called “modular functions.” The purpose of the present note is to settle this question in the affirmative under a slight restriction. Before we are able to state the Wallace problem precisely, some preliminaries are necessary.

scholarly journals On well-bounded operators of type (B)

1972 ◽  
Vol 18 (1) ◽  
pp. 35-48 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Algebra ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Compact Interval ◽  
Algebra Structure ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Second Dual

The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.

scholarly journals Quasiminimal distal function space and its semigroup compactification

1995 ◽  
Vol 18 (3) ◽  
pp. 497-500
Author(s):  
R. D. Pandian
Keyword(s):  
Function Space ◽  
Topological Semigroup ◽  
Mapping Property ◽  
Semitopological Semigroup ◽  
Semigroup Compactification ◽  
Right Topological Semigroup

Quasiminimal distal function on a semitopological semigroup is introduced. The concept of right topological semigroup compactification is employed to study the algebra of quasiminimal distal functions. The universal mapping property of the quasiminimal distal compactification is obtained.

Convergence Factors and Compactness in Weighted Convolution Algebras

2002 ◽  
Vol 54 (2) ◽  
pp. 303-323 ◽  
Author(s):  
Fereidoun Ghahramani ◽  
Sandy Grabiner
Keyword(s):  
Banach Algebra ◽  
Compact Operator ◽  
Dual Space ◽  
Weighted Space ◽  
Continuous Functions ◽  
Norm Convergence ◽  
Convergence Factor ◽  
Space Of Continuous Functions ◽  
Related Space ◽  
Convolution Algebras

AbstractWe study convergence in weighted convolution algebras L1(ω) on R+, with the weights chosen such that the corresponding weighted space M(ω) of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor ɳ for which weak*-convergence of {λn} to λ in M(ω) implies norm convergence of λn * f to λ * f in L1(ωɳ). We find necessary and sufficent conditions which depend on ω and f and also find necessary and sufficent conditions for ɳ to be a convergence factor for all L1(ω) and all f in L1(ω). We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that ɳ is a convergence factor for ω and f if and only if convolution by f is a compact operator from M(ω) (or L1(ω)) to L1(ωɳ).

Johnson pseudo-contractibility of second duals of certain Banach algebras

2019 ◽  
pp. 2150012
Author(s):  
A. Sahami ◽  
E. Ghaderi ◽  
S. M. Kazemi Torbaghan ◽  
B. Olfatian Gillan
Keyword(s):  
Tensor Product ◽  
Matrix Algebra ◽  
Banach Algebras ◽  
Amenable Group ◽  
Semigroup Algebra ◽  
Archimedean Semigroup ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
The Matrix ◽  
Second Dual

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is a finite amenable group, where [Formula: see text] is an archimedean semigroup. We also show that the matrix algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

scholarly journals An application of a theorem of Singer

1974 ◽  
Vol 19 (2) ◽  
pp. 119-123 ◽  
Author(s):  
D. M. Connolly ◽  
J. H. Williamson
Keyword(s):  
Measure Zero ◽  
General Context ◽  
Measure Algebra ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Perfect Set ◽  
Convolution Power ◽  
Recent Theorem ◽  
Vector Sum ◽  
Positive Results

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect of) a multiplicity theory for the first.

scholarly journals Conditional Fourier-Feynman Transforms with Drift on a Function Space

2019 ◽  
Vol 2019 ◽  
pp. 1-16
Author(s):  
Dong Hyun Cho ◽  
Suk Bong Park
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Fourier Transforms ◽  
Convolution Product ◽  
Borel Measures ◽  
Conditional Expectations ◽  
Change Of Scale ◽  
Convolution Products ◽  
Complex Borel Measures ◽  
Generalized Cylinder

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on L2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish various changes of scale formulas for the conditional transforms and the conditional convolution products.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

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