1. CJM Online first
 Ghahramani, F.; Zadeh, S.

Bipositive isomorphisms between Beurling algebras and between their second dual algebras
Let $G$ be a locally compact group and let $\omega$ be a continuous
weight on $G$. We show that for each of the Banach algebras $L^1(G,\omega)$,
$M(G,\omega)$, $LUC(G,\omega^{1})^*$ and $L^1(G,\omega)^{**}$,
the order structure combined with the algebra structure determines
the weighted group.
Keywords:locally compact group, Beurling algebra, Arens product, topological group isomorphism, bipositive algebra isomorphism Categories:43A20, 43A22 

2. CJM Online first
 Crann, Jason

Amenability and covariant injectivity of locally compact quantum groups II
Building on our previous work, we study the nonrelative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1injectivity 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 1injective 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 coamenable compact duals which avoids the
use of modular theory and the PowersStÃ¸rmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and coamenability.
Keywords:locally compact quantum group, amenability, injective module Categories:22D35, 46M10, 46L89 

3. CJM Online first
 Marquis, Timothée; Neeb, KarlHermann

Isomorphisms of twisted Hilbert loop algebras
The closest infinite dimensional relatives of compact Lie algebras are HilbertLie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple HilbertLie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple HilbertLie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple HilbertLie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Keywords:locally affine Lie algebra, HilbertLie algebra, positive energy representation Categories:17B65, 17B70, 17B22, 17B10 

4. CJM 2016 (vol 68 pp. 1067)
 Runde, Volker; Viselter, Ami

On Positive Definiteness over Locally Compact Quantum Groups
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
Keywords:bicrossed product, locally compact quantum group, noncommutative $L^p$space, positivedefinite function, positivedefinite measure, separation property Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 

5. CJM Online first
 De Bernardi, Carlo Alberto; Veselý, Libor

Tilings of normed spaces
By a tiling of a topological linear space $X$ we mean a
covering of $X$ by at least two closed convex sets,
called tiles, whose nonempty interiors are
pairwise disjoint.
Study of tilings of infinitedimensional spaces initiated in
the
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
spaces,
and then apply these results to study existence of tilings of
normed and Banach spaces by tiles possessing
certain smoothness or rotundity properties. For a Banach space
$X$,
our main results are the following.
1. $X$ admits no tiling by FrÃ©chet smooth bounded tiles.
2. If $X$ is locally uniformly rotund (LUR), it does not admit
any tiling by balls.
3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$
uncountable, do admit
a tiling by pairwise disjoint LUR bounded tiles.
Keywords:tiling of normed space, FrÃ©chet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$space Categories:46B20, 52A99, 46A45 

6. CJM 2016 (vol 68 pp. 309)
 Daws, Matthew

Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$homomorphisms between universal
$C^*$algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.
Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25 

7. CJM 2014 (vol 67 pp. 184)
 McReynolds, D. B.

Geometric Spectra and Commensurability
The work of Reid, ChinburgHamiltonLongReid,
PrasadRapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.
Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds Category:20G25 

8. CJM 2013 (vol 65 pp. 1073)
 Kalantar, Mehrdad; Neufang, Matthias

From Quantum Groups to Groups
In this paper we use the recent developments in the
representation theory of locally compact quantum groups,
to assign, to each locally compact
quantum group $\mathbb{G}$, a locally compact group $\tilde {\mathbb{G}}$ which
is the quantum version of pointmasses, and is an
invariant for the latter. We show that ``quantum pointmasses"
can be identified with several other locally compact groups that can be
naturally assigned to the quantum group $\mathbb{G}$.
This assignment preserves compactness as well as
discreteness (hence also finiteness), and for large classes of quantum
groups, amenability. We calculate this invariant for some of the most
wellknown examples of
nonclassical quantum groups.
Also, we show that several structural properties of $\mathbb{G}$ are encoded
by $\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very
important information about $\mathbb{G}$.
Keywords:locally compact quantum group, locally compact group, von Neumann algebra Category:46L89 

9. CJM 2013 (vol 66 pp. 303)
 Elekes, Márton; Steprāns, Juris

Haar Null Sets and the Consistent Reflection of Nonmeagreness
A subset $X$ of a Polish group $G$ is called Haar null if there exists
a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that
$\mu(gBh)=0$ for every $g,h \in G$.
We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a
Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in
\mathbb R$.
This answers a question from David Fremlin's problem list by showing
that one cannot simplify the definition of a Haar null set by leaving out the
Borel set $B$. (The answer was already known assuming the Continuum
Hypothesis.)
This result motivates the following Baire category analogue. It is consistent
with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor
set $C \subset G$ such that for every nonmeagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively nonmeagre in $C$. This
essentially generalises results of BartoszyÅski and BurkeMiller.
Keywords:Haar null, Christensen, nonlocally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 

10. CJM 2012 (vol 66 pp. 102)
 Birth, Lidia; Glöckner, Helge

Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$functions
and convolution of compactly supported Radon measures.
Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigmacompactness, convolution, continuity, seminorm, product estimates Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 

11. CJM 2012 (vol 65 pp. 1043)
 Hu, Zhiguo; Neufang, Matthias; Ruan, ZhongJin

Convolution of Trace Class Operators over Locally Compact Quantum Groups
We study locally compact quantum groups $\mathbb{G}$ through the
convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})),
\triangleright)$. We prove that the reduced quantum group
$C^*$algebra $C_0(\mathbb{G})$ can be recovered from the convolution
$\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$module
$\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is
equal to $C_0(\mathbb{G})$. On the other hand, we show that the left
$T(L_2(\mathbb{G}))$module $\langle T(L_2(\mathbb{G}))\triangleright
K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product
$C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is
a much larger $C^*$subalgebra of $B(L_2(\mathbb{G}))$.
We establish a natural isomorphism between the completely bounded
right multiplier algebras of $L_1(\mathbb{G})$ and
$(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance
problems associated with the representation theorem of
JungeNeufangRuan (2009). We characterize regularity and discreteness
of the quantum group $\mathbb{G}$ in terms of continuity properties of
the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove
that if $\mathbb{G}$ is semiregular, then the space
$\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right
$\mathbb{G}$continuous operators on $L_2(\mathbb{G})$, which was
introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$subalgebra
of $B(L_2(\mathbb{G}))$. In the representation framework formulated by
NeufangRuanSpronk (2008) and JungeNeufangRuan, we show that the
dual properties of compactness and discreteness can be characterized
simultaneously via automatic normality of quantum group bimodule maps
on $B(L_2(\mathbb{G}))$. We also characterize some commutation
relations of completely bounded multipliers of $(T(L_2(\mathbb{G})),
\triangleright)$ over $B(L_2(\mathbb{G}))$.
Keywords:locally compact quantum groups and associated Banach algebras Categories:22D15, 43A30, 46H05 

12. CJM 2012 (vol 64 pp. 1182)
 Tall, Franklin D.

PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$
Extending the work of Larson and Todorcevic,
we show there
is a model of set theory in which normal spaces are collectionwise
Hausdorff if they are either first countable or locally compact, and
yet there are no first countable $L$spaces or compact
$S$spaces. The model is one of the form PFA$(S)[S]$, where $S$
is a coherent Souslin tree.
Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinal Categories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65 

13. CJM 2010 (vol 63 pp. 436)
 Mine, Kotaro; Sakai, Katsuro

Simplicial Complexes and Open Subsets of NonSeparable LFSpaces
Let $F$ be a nonseparable LFspace homeomorphic to
the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$,
where $\aleph_0 < \tau_1 < \tau_2 < \cdots$.
It is proved that
every open subset $U$ of $F$ is homeomorphic to the product $K \times F$
for some locally finitedimensional simplicial complex $K$ such that
every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$
with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$
(and $\operatorname{card} K^{(0)} \le \tau$),
and, conversely, if $K$ is such a simplicial complex,
then the product $K \times F$ can be embedded in $F$ as an open set,
where $K$ is the polyhedron of $K$ with the metric topology.
Keywords:LFspace, open set, simplicial complex, metric topology, locally finitedimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$ Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40 

14. CJM 2010 (vol 63 pp. 123)
 Granirer, Edmond E.

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 FigaTalamanca,
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, FigaTalamancaHerz algebra, locally compact group, Ditkin sets, Helson sets, Sidon sets, weak sequential completeness Categories:43A15, 43A10, 46J10, 43A45 

15. 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 

16. CJM 2009 (vol 61 pp. 382)
 Miao, Tianxuan

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 

17. CJM 2006 (vol 58 pp. 768)
 Hu, Zhiguo; Neufang, Matthias

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 nonzero
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 

18. CJM 2006 (vol 58 pp. 64)
 Filippakis, Michael; Gasiński, Leszek; Papageorgiou, Nikolaos S.

Multiplicity Results for Nonlinear Neumann Problems
In this paper we study nonlinear elliptic problems of Neumann type driven by the
$p$Laplac\ian differential operator. We look for situations guaranteeing the existence
of multiple solutions. First we study problems which are strongly resonant at infinity
at the first (zero) eigenvalue. We prove five multiplicity results, four for problems
with nonsmooth potential and one for problems with a $C^1$potential. In the last part,
for nonsmooth problems in which the potential eventually exhibits a strict
super$p$growth under a symmetry condition, we prove the existence of infinitely
many pairs of nontrivial solutions. Our approach is variational based on the critical
point theory for nonsmooth functionals. Also we present some results concerning the first
two elements of the spectrum of the negative $p$Laplacian with Neumann boundary condition.
Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$Laplacian Categories:35J20, 35J60, 35J85 

19. CJM 2004 (vol 56 pp. 1259)
 Paterson, Alan L. T.

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 

20. CJM 2004 (vol 56 pp. 344)
21. 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 

22. CJM 2002 (vol 54 pp. 795)
 Möller, Rögnvaldur G.

Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
Willis's structure theory of totally disconnected locally compact groups
is investigated in the context of permutation actions. This leads to new
interpretations of the basic concepts in the theory and also to new proofs
of the fundamental theorems and to several new results. The treatment of
Willis's theory is selfcontained and full proofs are given of all the
fundamental results.
Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs Categories:22D05, 20B07, 20B27, 05C25 
