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Register a new userCompletely bounded homomorphisms of the Fourier algebra revisited
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Authors
Matthew Daws
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Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that continuous piecewise affine maps $alpha: Y subseteq Hrightarrow G$ induce completely bounded homomorphisms $Phi:A(G)rightarrow B(H)$, and that when $G$ is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure theoretic ideas, following more closely the original ideas of Cohen.
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We prove that if $rho: A(H) to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $alpha : G to H$, then $rho$ extends to a w*-w* continuous map between the corresponding $L^infty$ algebras if and only if $alpha$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(alphatimesalpha)^{-1}$ preserves sets of compact operator synthesis, and conversely when $alpha$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.
It is well-known that if T is a D_m-D_n bimodule map on the m by n complex matrices, then T is a Schur multiplier and $|T|_{cb}=|T|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $|T|_{cb}=|T|$. However, this property fails if m>1 and n>2. For m>1 and n=3,4 or $ngeq m^2$, we give examples of maps T attaining the supremum C(m,n)=sup |T|_{cb} taken over the contractive, right D_n-module maps on M_{m,n}, we show that C(m,m^2)=sqrt{m} and succeed in finding sharp results for C(m,n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on the compact operators K(H) which is not completely bounded.
We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)otimes_{rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{sharp} = {(s,t) : stin E}$ and show that if $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)$-bimodule maps with the dual of $A(G)otimes_{rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.
In this paper, we introduce the concept of uniformly bounded fibred coarse embeddability of metric spaces, generalizing the notion of fibred coarse embeddability defined by X. Chen, Q. Wang and G. Yu. Moreover, we show its relationship with uniformly bounded a-T-menability of groups. Finally, we give some examples to illustrate the differences between uniformly bounded fibred coarse embeddability and fibred coarse embeddability.
We let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for all x,y in G. We show that this algebra admits bounded point derivations whenever G contains a non-abelian closed connected subgroup. Conversely when G is virtually abelian, then ZA(G) is amenable. Furthermore, for virtually abelian G, we establish which closed ideals admit bounded approximate identities. We also show that if ZA(G) is weakly amenable, even hyper-Tauberian, exactly when G admits no non-abelian connected subgroup. We also study the amenability constant of ZA(G) for finite G and exhibit totally disconnected groups G for which ZA(G) is non-amenable.
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