scholarly journals 1-Algebra of a Locally Compact Groupoid

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Massoud Amini ◽  
Alireza Medghalchi ◽  
Ahmad Shirinkalam
Keyword(s):  
Invariant Measure ◽  
Banach Algebra ◽  
Haar System ◽  
Locally Compact ◽  
Radon Measures ◽  
Integrable Functions

For a locally compact groupoid with a fixed Haar system and quasi-invariant measure , we introduce the notion of -measurability and construct the space 1(, , ) of absolutely integrable functions on and show that it is a Banach -algebra and a two-sided ideal in the algebra () of complex Radon measures on . We find correspondences between representations of on Hilbert bundles and certain class of nondegenerate representations of 1(, , ).

scholarly journals On continuity of derivations and epimorphisms on some vector-valued group algebras

1997 ◽  
Vol 56 (2) ◽  
pp. 209-215
Author(s):  
Ramesh V. Garimella
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Group Algebras ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Zero Divisors ◽  
Integrable Functions ◽  
Vector Valued ◽  
Nontrivial Zero

For a locally compact Abelian group G and a commutative Banach algebra B, let L1(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L1(G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L1(G, B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L1(G, B) and every epimorphism from a commutative Banach algebra onto L1(G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.

scholarly journals Tensor products of commutative Banach algebras

1982 ◽  
Vol 5 (3) ◽  
pp. 503-512
Author(s):  
U. B. Tewari ◽  
M. Dutta ◽  
Shobha Madan
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Group Algebra ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Integrable Functions ◽  
Commutative Banach Algebras

LetA1,A2be commutative semisimple Banach algebras andA1⊗∂A2be their projective tensor product. We prove that, ifA1⊗∂A2is a group algebra (measure algebra) of a locally compact abelian group, then so areA1andA2. As a consequence, we prove that, ifGis a locally compact abelian group andAis a comutative semi-simple Banach algebra, then the Banach algebraL1(G,A)ofA-valued Bochner integrable functions onGis a group algebra if and only ifAis a group algebra. Furthermore, ifAhas the Radon-Nikodym property, then the Banach algebraM(G,A)ofA-valued regular Borel measures of bounded variation onGis a measure algebra only ifAis a measure algebra.

scholarly journals On Relatively Invariant Measures

1960 ◽  
Vol 12 ◽  
pp. 367-373
Author(s):  
Mark Mahowald
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Locally Compact Groups ◽  
Major Portion ◽  
Compact Groups ◽  
Locally Compact ◽  
Multiplier Function ◽  
Baire Measure

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.

scholarly journals Tensor Products of Banach Algebras

1959 ◽  
Vol 11 ◽  
pp. 297-310 ◽  
Author(s):  
Bernard R. Gelbaum
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Integrable Functions ◽  
Group A

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

scholarly journals Topological Amenability Is a Borel Property

2015 ◽  
Vol 117 (1) ◽  
pp. 5 ◽  
Author(s):  
Jean Renault
Keyword(s):  
Haar System ◽  
Locally Compact

We establish that a $\sigma$-compact locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.

scholarly journals Induction for Banach Algebras, Groupoids and KKban

2009 ◽  
Vol 4 (3) ◽  
pp. 405-468 ◽  
Author(s):  
Walther Paravicini
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
The Other ◽  
Locally Compact ◽  
Morita Equivalent

AbstractGiven two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).

Multipliers of certain convolution algebras over locally compact semigroups

1980 ◽  
Vol 87 (1) ◽  
pp. 51-59 ◽  
Author(s):  
D. G. Todd
Keyword(s):  
Ordered Semigroup ◽  
Real Numbers ◽  
Locally Compact ◽  
Multiplier Algebra ◽  
Similar Work ◽  
Convolution Algebras ◽  
Compact Semigroups ◽  
Integrable Functions ◽  
Locally Compact Semigroups

In this paper we extend a result of Johnson and Lahr(3), which characterizes the multiplier algebra of L1(a, b) (the algebra of Lebesgue integrable functions on the interval of real numbers from a to b, under order convolution) to the L1 algebra of a general totally ordered semigroup. Similar work has been done in (l), but under more restrictive conditions.

Measure in Semigroups

1952 ◽  
Vol 4 ◽  
pp. 396-406 ◽  
Author(s):  
B. R. Gelbaum ◽  
G. K. Kalisch
Keyword(s):  
Topological Group ◽  
Invariant Measure ◽  
Commutative Semigroup ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Counter Example ◽  
Compact Semigroups ◽  
Topological Measures ◽  
Weakened Form ◽  
Locally Compact Semigroups

The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

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