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Paper IPM / M / 17353  


Abstract:  
Let ${\Bbb G}$ be a locally compact quantum group.Then the space $T(L^2({\Bbb G}))$ of trace class operators on $L^2({\Bbb G})$
is a Banach algebra with the convolution induced by the right fundamental unitary of ${\Bbb G}$. We show that properties of ${\Bbb G}$ such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra $T(L^2({\Bbb G}))$. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of $T(L^2({\Bbb G}))^{**}$ and some properties of ${\Bbb G}$.
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