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Register a new userCompletely bounded bimodule maps and spectral synthesis
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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 $G$.
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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.
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 examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $sum{overline{X_i}}otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescus theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.
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.
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