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1. CMB 2011 (vol 54 pp. 654)
Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm |
Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 |
2. CMB 2010 (vol 54 pp. 207)
A Bilinear Fractional Integral on Compact Lie Groups As an analog of a well-known theorem on the bilinear
fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein,
we establish the similar boundedness
property for a bilinear fractional integral on a compact Lie group. Our
result is also a generalization of our recent theorem
about the
bilinear fractional integral on torus.
Keywords:bilinear fractional integral, $L^p$ spaces, Heat kernel Categories:43A22, 43A32, 43B25 |
3. CMB 2010 (vol 53 pp. 447)
Injective Convolution Operators on l^{∞}(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.
Categories:43A20, 46L05, 43A22 |
4. CMB 1997 (vol 40 pp. 183)
The range of group algebra homomorphisms A characterisation of the range of a homomorphism between two
commutative group algebras is presented which implies, among other
things, that this range is closed. The work relies mainly on the
characterisation of such homomorphisms achieved by P.~J.~Cohen.
Categories:43A22, 22B10, 46J99 |