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Compact elements and operators of quantum groups. (English) Zbl 1370.46048

From the introduction: “A classical result of S. Sakai [Pac. J. Math. 14, 659–664 (1964; Zbl 0135.35803)] states that a locally compact group \(G\) is compact if and only if it admits (weakly) compact convolution operators. Dually, \(G\) is discrete if and only if it admits (weakly) compact multiplication operators on \(L^2(G)\). In this paper, we investigate these connections in the case of locally compact quantum groups.”
Various specific quantum variants of Sakai’s results are proven. For example, with \(G\) a locally compact quantum group: cm
(1)
\(G\) is discrete if and only if \(L^\infty(G) \cap \mathbb{K}(L^2(G)) \neq 0 \);
(2)
\(G\) is compact if and only if the measure algebra \(M(G)\) contains an element \(\mu\) such that convolution by \(\mu\) is a finite rank operator on \(L^1(G)\).
The latter statement is expected to hold with (weakly) compact in place of finite rank, but for this the authors need a supplementary condition: the image of convolution by \(\mu\) contains an invertible element.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L51 Noncommutative measure and integration
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.

Citations:

Zbl 0135.35803
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Full Text: DOI

References:

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