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Search: MSC category 46L07 ( Operator spaces and completely bounded maps [See also 47L25] )

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1. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
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)

Samei, Ebrahim; Spronk, Nico; Stokke, Ross
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)

Dong, Z.
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)

Forrest, Brian E.; Runde, Volker; Spronk, Nico
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)

Junge, Marius
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)

Ruan, Zhong-Jin
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)

Wood, Peter J.
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

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