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Register a new userCompletely bounded norms of right module maps
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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.
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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 $G$.
A linear map $Phi :mathbb{M}_n to mathbb{M}_k$ is called completely copositive if the resulting matrix $[Phi (A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $Phi (X)=(mathrm{tr} X)I+X$ and $Psi (X)= (mathrm{tr} X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3times 3$ block matrices are included.
We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If $mathbb{G}$ is a locally compact quantum group, we characterise the completely bounded $L^{infty}(mathbb{G})$-bimodule maps that send $C_0(hat{mathbb{G}})$ into $L^{infty}(hat{mathbb{G}})$ in terms of the properties of the corresponding elements of the normal Haagerup tensor product $L^{infty}(mathbb{G}) otimes_{sigma{rm h}} L^{infty}(mathbb{G})$. As a consequence, we obtain an intrinsic characterisation of the normal completely bounded $L^{infty}(mathbb{G})$-bimodule maps that leave $L^{infty}(hat{mathbb{G}})$ invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.
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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