Expand all Collapse all | Results 1 - 7 of 7 |
1. 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 |
2. CJM 2010 (vol 62 pp. 845)
Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.
Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebra Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 |
3. CJM 2009 (vol 61 pp. 1262)
On the Local Lifting Properties of Operator Spaces In this paper, we mainly study operator spaces which have the
locally lifting property (LLP). The dual of any ternary ring of operators is shown to
satisfy the strongly local reflexivity, and this is used to prove
that strongly local reflexivity holds also for operator spaces
which have the LLP. Several homological characterizations of the
LLP and weak expectation property are given. We also prove that for any operator space
$V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and
$V^{*}$ is exact.
Keywords:operator space, locally lifting property, strongly locally reflexive Category:46L07 |
4. 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 |
5. CJM 2004 (vol 56 pp. 983)
Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$-Spaces Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von
Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$,
respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in
$\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be
bounded families. We show the following equality
$$
\lim_{i,\U} \lim_{j,\U'} \Big\| \summ_{k=1}^n x_k(i)\otimes
y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U}
\Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} .
$$
For $p=1$ this Fubini type result is related to the local
reflexivity of duals of $C^*$-algebras. This fails for $p=\infty$.
Keywords:noncommutative $L_p$-spaces, ultraproducts Categories:46L52, 46B08, 46L07 |
6. CJM 2004 (vol 56 pp. 843)
Type Decomposition and the Rectangular AFD Property for $W^*$-TRO's We study the type decomposition and the rectangular AFD property for
$W^*$-TRO's. Like von Neumann algebras, every $W^*$-TRO can be
uniquely decomposed into the direct sum of $W^*$-TRO's of
type $I$, type $II$, and type $III$.
We may further consider $W^*$-TRO's of type $I_{m, n}$
with cardinal numbers $m$ and $n$, and consider $W^*$-TRO's of
type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$.
It is shown that every separable stable $W^*$-TRO
(which includes type $I_{\infty,\infty}$, type $II_{\infty,
\infty}$ and type $III$) is TRO-isomorphic to a von Neumann algebra.
We also introduce the rectangular version of the approximately finite
dimensional property for $W^*$-TRO's.
One of our major results is to show that a separable $W^*$-TRO
is injective if and only
if it is rectangularly approximately finite dimensional.
As a consequence of this result, we show that a dual operator space
is injective if and only if its operator predual is a rigid
rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular
Categories:46L07, 46L08, 46L89 |
7. 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 |