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1. CJM Online first

Kaniuth, Eberhard
The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products
Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta$ of $B$ is a certain Banach algebra product $A\times_\theta B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\times_\theta B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner-Schoenberg-Eberlein property and the algebra of Bochner-Schoenberg-Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\times_\theta B$.

Keywords:commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function
Categories:46J10, 46J25, 43A30, 43A45

2. CJM Online first

Chen, Fulin; Gao, Yun; Jing, Naihuan; Tan, Shaobin
Twisted Vertex Operators and Unitary Lie Algebras
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral $\mathbb Z_2$-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.

Keywords:Lie algebra, vertex operator, representation theory
Categories:17B60, 17B69

3. CJM Online first

Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ${\mathbb C}$, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ${\mathbb N}$-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$ and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the structure of these algebras with the property $L=\langle L_1 \rangle$. In this paper we study those generated by the first and $q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification
Categories:17B70, 14F45

4. CJM 2014 (vol 66 pp. 1358)

Osėkowski, Adam
Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .

Keywords:Fourier multiplier, martingale, laminate
Categories:42B15, 60G44, 42B20

5. CJM 2013 (vol 66 pp. 1250)

Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module for the group $G^a$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$, we call the closure $\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$ of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where even for fundamental weights $\omega$ the varieties $\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula.

Keywords:Lie algebras, flag varieties, symplectic groups, representations
Categories:14M15, 22E46

6. CJM 2013 (vol 65 pp. 1005)

Forrest, Brian; Miao, Tianxuan
Uniformly Continuous Functionals and M-Weakly Amenable Groups
Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote the closure of $A(G)$, the Fourier algebra of $G$ in the space of bounded (completely bounded) multipliers of $A(G)$. We call a locally compact group M-weakly amenable if $A_M(G)$ has a bounded approximate identity. We will show that when $G$ is M-weakly amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of tolopolically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

Keywords:Fourier algebra, multipliers, weakly amenable, uniformly continuous functionals
Categories:43A07, 43A22, 46J10, 47L25

7. CJM 2012 (vol 66 pp. 700)

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

8. CJM 2012 (vol 65 pp. 82)

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude
The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $ \sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$.

Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category
Categories:55P35, 55P62, , , , 17B70

9. 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, sigma-compactness, convolution, continuity, seminorm, product estimates
Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

10. CJM 2012 (vol 65 pp. 510)

Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco
Transference of vector-valued multipliers on weighted $L^p$-spaces
We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be below one. We also develop some ad-hoc methods which apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allow us to extend results to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities.

Keywords:Fourier multipliers, restriction theorems, weighted spaces
Categories:42B15, 42B35

11. CJM 2012 (vol 65 pp. 299)

Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito
On Multilinear Fourier Multipliers of Limited Smoothness
In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

Keywords:multilinear Fourier multipliers, Hörmander multiplier theorem, Hardy spaces
Categories:42B15, 42B20

12. CJM 2012 (vol 65 pp. 241)

Aguiar, Marcelo; Lauve, Aaron
Lagrange's Theorem for Hopf Monoids in Species
Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange's theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies $\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of any one of the generating series of $\mathbf h$ by the corresponding generating series of $\mathbf k$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities, and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.

Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, Poincaré-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement
Categories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35

13. CJM 2012 (vol 65 pp. 66)

Deng, Shaoqiang; Hu, Zhiguang
On Flag Curvature of Homogeneous Randers Spaces
In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groups
Categories:22E46, 53C30

14. CJM 2011 (vol 63 pp. 1416)

Shelah, Saharon
MAD Saturated Families and SANE Player
We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any two is finite) that is saturated (completely separable \emph{i.e.,} any $X \subseteq \mathbb{N}$ is included in a finite union of members of the family \emph{or} includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation: (i) if $2^{\aleph_0} \lt \aleph_\omega$, then there is such a family; (ii) if there is no such family, then some situation related to pcf holds whose consistency is large (and if ${\mathfrak a}_* \gt \aleph_1$ even unknown); (iii) if, \emph{e.g.,} there is no inner model with measurables, \emph{then} there is such a family.

Keywords:set theory, MAD families, pcf, the continuum
Categories:03E05, 03E04, 03E17

15. CJM 2011 (vol 64 pp. 481)

Chamorro, Diego
Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition.

Keywords:Sobolev inequalities, polynomial volume growth Lie groups
Category:22E30

16. 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 Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz 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 Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann 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

17. CJM 2011 (vol 63 pp. 1083)

Kaletha, Tasho
Decomposition of Splitting Invariants in Split Real Groups
For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

Keywords:endoscopy, real lie group, splitting invariant, transfer factor
Categories:11F70, 22E47, 11S37, 11F72, 17B22

18. 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

19. CJM 2010 (vol 62 pp. 1276)

El Wassouli, Fouzia
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

20. CJM 2010 (vol 62 pp. 961)

Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
Multiplicative Isometries and Isometric Zero-Divisors
For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisors
Categories:30H05, 46E15

21. CJM 2010 (vol 62 pp. 870)

Valdimarsson, Stefán Ingi
The Brascamp-Lieb Polyhedron
A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
Categories:44A35, 14M15, 26D20

22. CJM 2009 (vol 62 pp. 284)

Grbić, Jelena; Theriault, Stephen
Self-Maps of Low Rank Lie Groups at Odd Primes
Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

Keywords:Lie group, self-map, H-map
Categories:55P45, 55Q05, 57T20

23. CJM 2009 (vol 62 pp. 52)

Deng, Shaoqiang
An Algebraic Approach to Weakly Symmetric Finsler Spaces
In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions $2$ and $3$. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature
Categories:53C60, 58B20, 22E46, 22E60

24. CJM 2009 (vol 61 pp. 721)

Calin, Ovidiu; Chang, Der-Chen; Markina, Irina
SubRiemannian Geometry on the Sphere $\mathbb{S}^3$
We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group $\mathbb{S}^3$.

Keywords:noncommutative Lie group, quaternion group, subRiemannian geodesic, horizontal distribution, connectivity theorem, holonomic constraint
Categories:53C17, 53C22, 35H20

25. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
$H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions
Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers
Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22
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