51. 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 

52. 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 

53. CJM 2012 (vol 65 pp. 481)
54. CJM 2012 (vol 65 pp. 559)
 Helemskii, A. Ya.

Extreme Version of Projectivity for Normed Modules Over Sequence Algebras
We define and study the socalled extreme version of the notion of a
projective normed module. The relevant definition takes into account
the exact value of the norm of the module in question, in contrast
with the standard known definition that is formulated in terms of norm
topology.
After the discussion of the case where our normed algebra $A$ is just
$\mathbb{C}$, we concentrate on the case of the next degree of complication,
where $A$ is a sequence algebra, satisfying some natural conditions.
The main results give a full characterization of extremely projective
objects within the subcategory of the category of nondegenerate
normed $A$modules, consisting of the socalled homogeneous modules.
We consider two cases, `noncomplete' and `complete', and the
respective answers turn out to be essentially different.
In particular, all Banach nondegenerate homogeneous modules,
consisting of sequences, are extremely projective within the category
of Banach nondegenerate homogeneous modules. However, neither of
them, provided it is infinitedimensional, is extremely projective
within the category of all normed nondegenerate homogeneous modules.
On the other hand, submodules of these modules, consisting of finite
sequences, are extremely projective within the latter category.
Keywords:extremely projective module, sequence algebra, homogeneous module Category:46H25 

55. CJM 2012 (vol 65 pp. 331)
 Kadets, Vladimir; Martín, Miguel; Merí, Javier; Werner, Dirk

Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces
We show that for spaces with 1unconditional bases
lushness, the alternative Daugavet property and numerical
index 1 are equivalent. In the class of rearrangement
invariant (r.i.) sequence spaces the only examples of spaces with
these properties are $c_0$, $\ell_1$ and $\ell_\infty$.
The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$;
the same space is the only r.i. separable function space on $[0,1]$
with the Daugavet property over the reals.
Keywords:lush space, numerical index, Daugavet property, KÃ¶the space, rearrangement invariant space Categories:46B04, 46E30 

56. CJM 2012 (vol 65 pp. 52)
57. CJM 2012 (vol 65 pp. 485)
 Bice, Tristan Matthew

Filters in C*Algebras
In this paper we analyze states on C*algebras and
their relationship to filterlike structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the KadisonSinger conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for qpoints in fact also holds for rapid
ppoints, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
Keywords:C*algebras, states, KadinsonSinger conjecture, ultrafilters, towers Categories:46L03, 03E35 

58. CJM 2011 (vol 64 pp. 755)
 Brown, Lawrence G.; Lee, Hyun Ho

Homotopy Classification of Projections in the Corona Algebra of a Nonsimple $C^*$algebra
We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murrayvon
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.
Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra Categories:46L05, 46L80 

59. CJM 2011 (vol 64 pp. 705)
60. CJM 2011 (vol 64 pp. 544)
61. CJM 2011 (vol 64 pp. 805)
 Chapon, François; Defosseux, Manon

Quantum Random Walks and Minors of Hermitian Brownian Motion
Considering quantum random walks, we construct discretetime
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 

62. CJM 2011 (vol 63 pp. 1161)
 Neuwirth, Stefan; Ricard, Éric

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
We inspect the relationship between relative Fourier
multipliers on noncommutative LebesgueOrlicz spaces of a discrete
group $\varGamma$ and relative ToeplitzSchur multipliers on
SchattenvonNeumannOrlicz classes. Four applications are given:
lacunary sets, unconditional Schauder bases for the subspace of a
Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the
norm of the Hilbert transform and the Riesz projection on
SchattenvonNeumann classes with exponent a power of 2, and the norm of
Toeplitz Schur multipliers on SchattenvonNeumann classes with
exponent less than 1.
Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection Categories:47B49, 43A22, 43A46, 46B28 

63. CJM 2011 (vol 64 pp. 573)
 Nawata, Norio

Fundamental Group of Simple $C^*$algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$unital $C^*$algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$algebras.
We show that there exist separable stably projectionless $C^*$algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 

64. CJM 2011 (vol 64 pp. 455)
 Sherman, David

On Cardinal Invariants and Generators for von Neumann Algebras
We demonstrate how most common cardinal invariants associated with a von
Neumann algebra $\mathcal M$ can be computed from the decomposability number,
$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating
set, $\operatorname{gen}(\mathcal M)$.
Applications include the equivalence of the wellknown generator
problem, ``Is every separablyacting von Neumann algebra
singlygenerated?", with the formally stronger questions, ``Is every
countablygenerated von Neumann algebra singlygenerated?" and ``Is
the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we
determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M),
\operatorname{dens}(\mathcal M) \bigr)$,
which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq
\mathfrak C^{\operatorname{gen}(\mathcal M)}$.
Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density Category:46L10 

65. CJM 2011 (vol 63 pp. 798)
 Daws, Matthew

Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ 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 noncommutative $L^p$ 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 noncommutative $L^p$ 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 FigaTalamancaHerz
algebra built out of these noncommutative $L^p$ 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, noncommutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 

66. CJM 2011 (vol 63 pp. 1188)
 Śliwa, Wiesław; Ziemkowska, Agnieszka

On Complemented Subspaces of NonArchimedean Power Series Spaces
The nonarchimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the
best known and most important examples of nonarchimedean nuclear FrÃ©chet spaces.
We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$
has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all
bounded limit points of the double sequence
$(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It
follows that every complemented subspace of a power series space $A_p(a)$ has a
Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded.
Keywords:nonarchimedean KÃ¶the space, range of a continuous linear map, Schauder basis Categories:46S10, 47S10, 46A35 

67. CJM 2011 (vol 63 pp. 551)
 Hadwin, Don; Li, Qihui; Shen, Junhao

Topological Free Entropy Dimensions in Nuclear C$^*$algebras and in Full Free Products of Unital C$^*$algebras
In the paper, we introduce a new concept,
topological orbit dimension of an $n$tuple of elements in a unital
C$^{\ast}$algebra. Using this concept, we conclude that Voiculescu's
topological free
entropy dimension of every finite family of selfadjoint generators of a
nuclear C$^{\ast}$algebra is less than or equal to $1$. We also show that the
Voiculescu's topological free entropy dimension is additive in the full free
product of some unital C$^{\ast}$algebras. We show that the unital full free
product of Blackadar and Kirchberg's unital MF
algebras is also an MF algebra. As an application, we obtain that
$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.
Keywords: topological free entropy dimension, unital C$^{*}$algebra Categories:46L10, 46L54 

68. CJM 2011 (vol 63 pp. 460)
 Pavlíček, Libor

Monotonically Controlled Mappings
We study classes of mappings between finite and infinite dimensional
Banach spaces that are monotone and mappings which are differences
of monotone mappings (DM). We prove a RadÃ³Reichelderfer estimate
for monotone mappings in finite dimensional spaces that remains
valid for DM mappings. This provides an alternative proof of the
FrÃ©chet differentiability a.e. of DM mappings. We establish a
Morreytype estimate for the distributional derivative of monotone
mappings. We prove that a locally DM mapping between finite
dimensional spaces is also globally DM. We introduce and study a new
class of the socalled UDM mappings between Banach spaces, which
generalizes the concept of curves of finite variation.
Keywords: monotone mapping, DM mapping, RadÃ³Reichelderfer property, UDM mapping, differentiability Categories:26B05, 46G05 

69. CJM 2011 (vol 63 pp. 381)
 Ji, Kui ; Jiang, Chunlan

A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed twosided ideal of $A$ is generated by
the projections inside the ideal, as a closed twosided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.
Keywords:AI algebras, Kgroup, tracial state, ideal property, classification Categories:46L35, 19K14, 46L05, 46L08 

70. CJM 2011 (vol 63 pp. 500)
 Dadarlat, Marius; Elliott, George A.; Niu, Zhuang

OneParameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita
oneparameter, continuous fields of Kirchberg algebras with torsion free
$\mathrm{K}$groups supported in one dimension, oneparameter, separable, uni
continuous fields of AFalgebras are classified by their ordered
$\mathrm{K}_0$sheaves. EffrosHandelmanShen type theorems are pr separable
unital oneparameter continuous fields of AFalgebras and Kirchberg algebras.
Keywords:continuous fields of C$^*$algebras, $\mathrm{K}_0$presheaves, EffrosHandeen type theorem Category:46L35 

71. 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 

72. 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 

73. CJM 2010 (vol 63 pp. 222)
 Wang, JiunChau

Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for
sums of infinitesimal conditionally free random variables. We show that the weak
convergence of classical convolution and that of conditionally free convolution
are equivalent for measures in an infinitesimal triangular array,
where the measures may have unbounded support. Moreover, we use these
limit theorems to study the conditionally free infinite divisibility. These results
are obtained by complex analytic methods without reference to the
combinatorics of cfree convolution.
Keywords:additive cfree convolution, limit theorems, infinitesimal arrays Categories:46L53, 60F05 

74. CJM 2010 (vol 63 pp. 3)
 Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.

Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 

75. CJM 2010 (vol 62 pp. 961)
 Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan

Multiplicative Isometries and Isometric ZeroDivisors
For some Banach spaces of analytic functions in the unit disk
(weighted Bergman spaces, Bloch space, Dirichlettype spaces), the
isometric pointwise multipliers are found to be unimodular constants.
As a consequence, it is shown that none of those spaces have isometric
zerodivisors. Isometric coefficient multipliers are also
investigated.
Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlettype spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zerodivisors Categories:30H05, 46E15 
