Bounded trace \(C^*\)-algebras and integrable actions
Section 1 of this paper relates the ``strength of convergence'' in orbit spaces to the properties of transformation groups (actions) and their \(C^*\)-algebras. By Green's theorem, given a free action of an l.c.\ group \(G\) on a l.c.\ Hausdorff space \(X\), the crossed product \(C_0(X)\times G\mathrel{=:}{\mathcal C}\) has continuous trace iff \(G\) acts properly. Then the upper multiplicity \(M_{U}\) of irreducible representations of \({\mathcal C}\) cannot exceed 1. For a more general integrable action introduced by M.~A.\ Rieffel, A.~an Huef proved that \({\mathcal C}\) has bounded trace (BT). A \(C^*\)-algebra \({\mathcal A}\) has BT if and only if \(M_{U}(\pi )<\infty\) for any \(\pi \in\hat A\), see \textit{R.~J.\ Archbold, D.~W.~B.\ Somerset} and \textit{J.~S.\ Spielberg} [J. Funct. Anal. 146, 430--463 (1997; Zbl 0909.46049)]. The paper under review treats similar problems in the setting of (noncommutative) \(C^*\)-dynamical systems \((A,G,\alpha )\) with the arising action of \(G\) on the spectrum \(\hat A\). Section 2 estimates \(M_U\) of a certain representation \(\mathop{\text{Ind}}\pi \) of the crossed product \(A\times {}_\alpha G\) induced from a representation \(\pi \) of \(A\). For a separable system (\(A\) is separable, \(G\)-second countable), \(\mathop{\text{Ind}}\pi \) is irreducible for each \(\pi \in\hat A\) if \(A\) is type I, \(G\) acts freely on \(\hat A\) and any orbit \(G(\pi )\) is locally closed (i.e., open in its closure). Then \(M_{U}(\mathop{\text{ Ind}}\pi)\geq k\in{\mathbb N}\) if there is a sequence \(\pi _n\) in \(\hat A\) and \(k\) sequences \(t_{j,n}\) in \(G\), \(1\leq j\leq k\), so that \(t_{j,n}^{-1}\to\pi \) \ \(\forall\,j\), and, for \(i<j\), \ \(t_{j,n}^{-1}t_{i,n}\to\infty\) as \(n\to\infty\). This entails a part of Green's theorem and has other useful corollaries. The principal Section 3 generalizes \textit{A.~an Huef}'s Theorem 4.8 of [Indiana Univ. Math. J. 51, 1197--1233 (2002; Zbl 1032.46078)], viz., if \((A,G,\alpha )\) is separable, \(A\) has BT and \(G\) acts freely on \(\hat A\) with locally closed orbits, then \(A\times {}_\alpha G\) has BT if and only if \(G\) acts integrably on \(\hat A\) (Corollary 3.9). Section 4 provides remarks. In particular, in some proofs of Section 3, the freeness can be relaxed to the almost free actions defined by \textit{C.--W. Leung} and \textit{C.--K. Ng} [Math. Z. 243, 409--421 (2003; Zbl 1024.46017)].
scientific article; zbMATH DE number 2173668
scientific article