Raeburn, Iain; Thompson, Shaun J. Countably generated Hilbert modules, the Kasparov Stabilisation Theorem, and frames in Hilbert modules. (English) Zbl 1015.46034 Proc. Am. Math. Soc. 131, No. 5, 1557-1564 (2003). Let \(H\) be a Hilbert (right) module over a \(C^*\)-algebra \(A\) [cf. E. C. Lance, Hilbert \(C^*\)-modules. A toolkit for operator algebraists, Cambridge Univ. Press (1995; Zbl 0822.46080)]. Define \(M(H)\) to be the Banach space \(L(A,H)\) of adjointable operators from \(A\) to \(H\). Then \(M(H)\) is a Hilbert \(M(A)\)-module with \((m.b)(a):=m(ba)\) and \(\langle m,n\rangle:=m^*\circ n\) for \(m,n \in M(H)\) and \(b\in M(A), a\in A\). Furthermore, \(H\) can be regarded as a closed \(M(A)\)-submodule of \(M(H)\) via \(\iota_H:H\rightarrow M(H)\) where \(\iota_H(h)(a)=ha, h\in H, a\in A\). \(M(H)\) as an \(M(A)\)-module is called the multiplier module. If there is a sequence \(\{h_i\}\subset M(H)\) such that the elements \(h_i.a\) span a dense submodule of \(H\), then \(H\) is said to be countably generated in \(M(H)\). The authors establish a version of Kasparov’s theorem for these countably generated modules, and also extend the work of M. Frank and D. R. Larson on frames in Hilbert modules [Contemp. Math. 247, 207-233 (1999; Zbl 0949.46027)] by showing that every such module admits a frame of multipliers. As a remark a version of the Banach-Green-Rieffel theorem for non-\(\sigma\)-unital \(C^*\)-algebras is given. Reviewer: Mohammad Sal Moslehian (Mashhad) Cited in 2 ReviewsCited in 25 Documents MSC: 46L08 \(C^*\)-modules 46L05 General theory of \(C^*\)-algebras Keywords:multiplier module; countably generated module; Kasparov’s theorem; frame; reconstruction formula; frame formula Citations:Zbl 0822.46080; Zbl 0949.46027 PDFBibTeX XMLCite \textit{I. Raeburn} and \textit{S. J. Thompson}, Proc. Am. Math. Soc. 131, No. 5, 1557--1564 (2003; Zbl 1015.46034) Full Text: DOI References: [1] Saad Baaj and Georges Skandalis, \?*-algèbres de Hopf et théorie de Kasparov équivariante, \?-Theory 2 (1989), no. 6, 683 – 721 (French, with English summary). · Zbl 0683.46048 · doi:10.1007/BF00538428 [2] B. Blackadar, article to appear in the Encyclopedia of Mathematics. [3] Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of \?*-algebras, Pacific J. Math. 71 (1977), no. 2, 349 – 363. · Zbl 0362.46043 [4] Siegfried Echterhoff and Iain Raeburn, Multipliers of imprimitivity bimodules and Morita equivalence of crossed products, Math. Scand. 76 (1995), no. 2, 289 – 309. · Zbl 0843.46049 · doi:10.7146/math.scand.a-12543 [5] M. Frank and D.R. Larson, Frames in Hilbert \(C^*\)-modules and \(C^*\)-algebras, J. Operator Theory, to appear. · Zbl 1029.46087 [6] Michael Frank and David R. Larson, A module frame concept for Hilbert \?*-modules, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 207 – 233. · Zbl 0949.46027 · doi:10.1090/conm/247/03803 [7] Kjeld Knudsen Jensen and Klaus Thomsen, Elements of \?\?-theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. · Zbl 1155.19300 [8] E.C. Lance, Hilbert \(C^*\)-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Series, vol. 210, Cambridge Univ. Press, 1994. [9] J. A. Mingo and W. J. Phillips, Equivariant triviality theorems for Hilbert \?*-modules, Proc. Amer. Math. Soc. 91 (1984), no. 2, 225 – 230. · Zbl 0546.46049 [10] Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace \?*-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. · Zbl 0922.46050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.