1. CJM 2017 (vol 69 pp. 1064)
 | Crann, Jason
 |
Amenability and Covariant Injectivity of Locally Compact Quantum Groups II
Building on our previous work, we study the non-relative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1-injectivity of
$L^{\infty}(\widehat{\mathbb{G}})$
as an operator
$L^1(\widehat{\mathbb{G}})$-module.
In particular, a locally compact group $G$ is amenable if and
only if its group von Neumann algebra
$VN(G)$
is 1-injective as
an operator module over the Fourier algebra $A(G)$. As an application,
we provide a decomposability result for completely bounded
$L^1(\widehat{\mathbb{G}})$-module
maps on
$L^{\infty}(\widehat{\mathbb{G}})$,
and give a simplified proof that amenable discrete
quantum groups have co-amenable compact duals which avoids the
use of modular theory and the Powers--Størmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and co-amenability.
Keywords:locally compact quantum group, amenability, injective module Categories:22D35, 46M10, 46L89 |
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2. 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 |
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3. CJM 2007 (vol 59 pp. 828)
 | Ortner, Ronald; Woess, Wolfgang
 |
Non-Backtracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Non-backtracking random walk moves at each step with equal
probability to one of the ``forward'' neighbours of the actual state,
\emph{i.e.,} it does not go back along
the preceding edge to the preceding
state. This is not a Markov chain, but can be turned into a Markov
chain whose state space is the set of oriented edges of $X$. Thus we
obtain for infinite $X$ that the $n$-step non-backtracking transition
probabilities tend to zero, and we can also compute their limit when
$X$ is finite. This provides a short proof of old results concerning
cogrowth of groups, and makes the extension of that result to
arbitrary regular graphs rigorous. Even when $X$ is non-regular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
non-amenable if and only if the non-backtracking $n$-step transition
probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which
comprises the original cogrowth criterion for finitely generated
groups of Grigorchuk and Cohen.
Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability Categories:05C75, 60G50, 20F69 |
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4. CJM 2005 (vol 57 pp. 17)
 | Bédos, Erik; Conti, Roberto; Tuset, Lars
 |
On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations
We introduce and study several notions of amenability for unitary
corepresentations and $*$-representations of algebraic quantum groups,
which may be used to characterize amenability and co-amenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$-categories.
Keywords:quantum group, amenability Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 |
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