scholarly journals Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

2017 ◽  
Vol 60 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Michael Christ ◽  
Marc A. Rieòel
Keyword(s):  
Finite Groups ◽  
Word Length ◽  
Polynomial Growth ◽  
Metric Spaces ◽  
Length Function ◽  
C Algebra ◽  
Pointwise Multiplication ◽  
C Algebras ◽  
Definition Of ◽  
Compact Quantum Metric Space

AbstractLet be a length function on a group G, and let M denote the operator of pointwise multiplication by on l2(G). Following Connes, M𝕃 can be used as a “Dirac” operator for the reduced group C*-algebra (G). It deûnes a Lipschitz seminorm on (G), which defines a metric on the state space of (G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators

2016 ◽  
Vol 146 (6) ◽  
pp. 1159-1166 ◽  
Author(s):  
Lucas Chaffee
Keyword(s):  
Singular Integral ◽  
Wide Class ◽  
Fractional Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Bounded Mean Oscillation ◽  
Fractional Integral Operators ◽  
Pointwise Multiplication ◽  
Bilinear Operators ◽  
Mean Oscillation

We characterize bounded mean oscillation in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón–Zygmund operators and the bilinear fractional integral operators.

On amenability of the Banach algebras l∞(S, [Ufr ])

2002 ◽  
Vol 132 (2) ◽  
pp. 319-322
Author(s):  
FÉLIX CABELLO SÁNCHEZ ◽  
RICARDO GARCÍA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Short Note ◽  
Basic Information ◽  
Arens Product ◽  
Pointwise Multiplication ◽  
Weak Amenability ◽  
Usual Norm ◽  
Bounded Functions ◽  
Second Dual

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

scholarly journals On products of Sobolev-Orlicz spaces

1990 ◽  
Vol 42 (3) ◽  
pp. 427-436 ◽  
Author(s):  
J. Appell ◽  
G. Hardy
Keyword(s):  
Bounded Operator ◽  
Orlicz Spaces ◽  
Sufficient Condition ◽  
Superposition Operator ◽  
Pointwise Multiplication

We give conditions under which pointwise multiplication is a continuous bounded operation on kth order Sobolev-Orlicz spaces. This result is used to derive a sufficient condition under which the superposition operator is a continuous bounded operator on these spaces.

scholarly journals Derivations on white noise functionals

1995 ◽  
Vol 139 ◽  
pp. 21-36 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Topological Algebra ◽  
Fundamental Question ◽  
Special Choice ◽  
Pointwise Multiplication ◽  
Infinite Dimensional ◽  
White Noise Functionals ◽  
White Noise Calculus ◽  
Gelfand Triple ◽  
Gaussian Space

The Gaussian space (E*, μ) is a natural infinite dimensional analogue of Euclidean space with Lebesgue measure and a special choice of a Gelfand triple gives a fundamental framework of white noise calculus [2] as distribution theory on Gaussian space. It is proved in Kubo-Takenaka [7] that (E) is a topological algebra under pointwise multiplication. The main purpose of this paper is to answer the fundamental question: what are the derivations on the algebra (E)?

scholarly journals Fractional integrals of generalised functions

1972 ◽  
Vol 14 (1) ◽  
pp. 30-37 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Fractional Order ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Pointwise Multiplication ◽  
Convolution Of Distributions ◽  
Independent Variable ◽  
Image Position

The concept of integrals of fractional order of a function f, defined by if Reα > 0, can be extended to generalised functions in the framework of the theory of convolution of distributions. The resulting theory [2, Chap. I §5.5] is very satisfactory for many purposes but there are circumstances in which it is not suitable. Such circumstances arise in particular if it is necessary to multiply, before or after integratrion, by non-integral powers of the variable. Pointwise multiplication by fractional powers of the independent variable does not make sense in the theory of distributions.

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