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Search: MSC category 43A22 ( Homomorphisms and multipliers of function spaces on groups, semigroups, etc. )

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1. CJM Online first

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, centraliserCategories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25

2. 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 functionalsCategories:43A07, 43A22, 46J10, 47L25

3. 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 projectionCategories:47B49, 43A22, 43A46, 46B28

4. 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 interpolationCategories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

5. 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, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

6. 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 amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25

7. CJM 2001 (vol 53 pp. 565)

Hare, Kathryn E.; Sato, Enji
 Spaces of Lorentz Multipliers We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 8. CJM 1997 (vol 49 pp. 1117) Hu, Zhiguo  The von Neumann algebra$\VN(G)$of a locally compact group and quotients of its subspaces Let$\VN(G)$be the von Neumann algebra of a locally compact group$G$. We denote by$\mu$the initial ordinal with$\abs{\mu}$equal to the smallest cardinality of an open basis at the unit of$G$and$X= \{\alpha; \alpha < \mu \}$. We show that if$G$is nondiscrete then there exist an isometric$*$-isomorphism$\kappa$of$l^{\infty}(X)$into$\VN(G)$and a positive linear mapping$\pi$of$\VN(G)$onto$l^{\infty}(X)$such that$\pi\circ\kappa = \id_{l^{\infty}(X)}$and$\kappa$and$\pi$have certain additional properties. Let$\UCB (\hat{G})$be the$C^{*}$-algebra generated by operators in$\VN(G)$with compact support and$F(\hat{G})$the space of all$T \in \VN(G)$such that all topologically invariant means on$\VN(G)$attain the same value at$T$. The construction of the mapping$\pi$leads to the conclusion that the quotient space$\UCB (\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$has$l^{\infty}(X)$as a continuous linear image if$G$is nondiscrete. When$G$is further assumed to be non-metrizable, it is shown that$\UCB(\hat{G})/F (\hat{G})\cap \UCB(\hat{G})$contains a linear isomorphic copy of$l^{\infty}(X)$. Similar results are also obtained for other quotient spaces. Categories:22D25, 43A22, 43A30, 22D15, 43A07, 47D35 9. CJM 1997 (vol 49 pp. 736) Fendler, Gero  Dilations of one parameter Semigroups of positive Contractions on$L^{\lowercase {p}}$spaces It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an$L^p$-space, for$1 Categories:47D03, 22D12, 43A22
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