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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.
D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between $C^*$-algebras by D. Kretschmann, D. Schlingemann and R. F. Wern
A matrix convex set is a set of the form $mathcal{S} = cup_{ngeq 1}mathcal{S}_n$ (where each $mathcal{S}_n$ is a set of $d$-tuples of $n times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We st
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
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 propert
The generalized state space $ S_{mathcal{H}}(mathcal{mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $mathcal{A}$ taking values in the algebra $mathcal{B}(mathcal{H})$ of all bounded operators on a Hilbert space $