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Amenability and covariant injectivity of locally compact quantum groups. (English) Zbl 1330.22013
The paper is devoted to the connection between amenability of a locally compact group \(G\) and injectivity of the von Neumann algebra \(L(G)\) associated with the left regular representation. The amenability of \(G\) implies the injectivity of \(L(G)\), but the converse is not true in general, but it is true for inner amenable groups. In the present paper the authors show that the equivalence of the above properties is valid for all locally compact groups if \(L(G)\) is considered as a \(T(L_{2}(G))\)-module with respect to a natural action. They prove an appropriate version of this result for every locally compact quantum group.

MSC:
22D15 Group algebras of locally compact groups
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
81R15 Operator algebra methods applied to problems in quantum theory
43A07 Means on groups, semigroups, etc.; amenable groups
46M10 Projective and injective objects in functional analysis
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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