Expand all Collapse all | Results 1 - 8 of 8 |
1. CJM 2013 (vol 65 pp. 1005)
Uniformly Continuous Functionals and M-Weakly Amenable Groups Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote
the closure of $A(G)$, the Fourier algebra of $G$ in the space of
bounded (completely bounded) multipliers of $A(G)$.
We call a locally compact group M-weakly amenable if
$A_M(G)$
has a
bounded approximate identity. We will show that when $G$ is M-weakly
amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have
properties that are characteristic of the Fourier algebra of an
amenable group. Along the way we show that the sets of tolopolically
invariant means associated with these algebras have the same
cardinality as those of the Fourier algebra.
Keywords:Fourier algebra, multipliers, weakly amenable, uniformly continuous functionals Categories:43A07, 43A22, 46J10, 47L25 |
2. CJM 2011 (vol 63 pp. 798)
Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces We show that the multiplier algebra of the Fourier algebra on a
locally compact group $G$ can be isometrically represented on a direct
sum on non-commutative $L^p$ spaces associated with the right von
Neumann algebra of $G$. The resulting image is the idealiser of the
image of the Fourier algebra. If these spaces are given their
canonical operator space structure, then we get a completely isometric
representation of the completely bounded multiplier algebra. We make
a careful study of the non-commutative $L^p$ spaces we construct and
show that they are completely isometric to those considered recently
by Forrest, Lee, and Samei. We improve a result of theirs about module
homomorphisms. We suggest a definition of a Figa-Talamanca-Herz
algebra built out of these non-commutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.
Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 |
3. CJM 2010 (vol 63 pp. 123)
Strong and Extremely Strong Ditkin sets for the Banach Algebras $A_p^r(G)=A_p\cap L^r(G)$
Let $A_p(G)$ be the Figa-Talamanca,
Herz Banach Algebra on $G$; thus $A_2(G)$
is the Fourier algebra. Strong Ditkin (SD) and
Extremely Strong Ditkin (ESD) sets for the Banach algebras
$A_p^r(G)$ are investigated for abelian and nonabelian
locally compact groups $G$. It is shown that SD and ESD sets
for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$,
with strict inclusion for ESD sets. The case for the strict
inclusion of SD sets is left open.
A result on the weak sequential completeness of $A_2(F)$
for ESD sets $F$ is proved and used to show that Varopoulos,
Helson, and Sidon sets are not ESD sets for $A_2(G)$, yet they
are such for $A_2^r(G)$ for discrete groups $G$, for
any $1\le r\le 2$.
A result is given on the equivalence of the sequential and the net
definitions of SD or ESD sets for $\sigma$-compact groups.
The above results are new even if $G$ is abelian.
Keywords:Fourier algebra, Figa-Talamanca-Herz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness Categories:43A15, 43A10, 46J10, 43A45 |
4. CJM 2009 (vol 61 pp. 382)
Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra $A(G)$ Let $\mathcal{A}$ be a Banach algebra with a bounded right
approximate identity and let $\mathcal B$ be a closed ideal of
$\mathcal A$. We study the relationship between the right identities
of the double duals ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$ under
the Arens product. We show that every right identity of ${\mathcal
B}^{**}$ can be extended to a right identity of ${\mathcal A}^{**}$ in
some sense. As a consequence, we answer a question of Lau and
\"Ulger, showing that for the Fourier algebra $A(G)$ of a locally
compact group $G$, an element $\phi \in A(G)^{**}$ is in $A(G)$ if and
only if $A(G) \phi \subseteq A(G)$ and $E \phi = \phi $ for all right
identities $E $ of $A(G)^{**}$. We also prove some results about the
topological centers of ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$.
Keywords:Locally compact groups, amenable groups, Fourier algebra, identity, Arens product, topological center Category:43A07 |
5. CJM 2007 (vol 59 pp. 966)
Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finite-dimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$-multiplier
norm.
Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 |
6. CJM 2006 (vol 58 pp. 768)
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal non-zero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.
Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 |
7. CJM 2004 (vol 56 pp. 1259)
The Fourier Algebra for Locally Compact Groupoids We introduce and investigate using Hilbert modules the properties
of the {\em Fourier algebra} $A(G)$ for
a locally compact groupoid $G$. We establish a duality theorem for
such groupoids in terms of multiplicative module maps. This includes
as a special case the classical duality theorem for locally compact
groups proved by P. Eymard.
Keywords:Fourier algebra, locally compact groupoids, Hilbert modules,, positive definite functions, completely bounded maps Category:43A32 |
8. CJM 2002 (vol 54 pp. 1100)
The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
Keywords:locally compact group, Fourier algebra, operator space, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 |