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Countably generated Hilbert modules, the Kasparov Stabilisation Theorem, and frames in Hilbert modules. (English) Zbl 1015.46034

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.

MSC:

46L08 \(C^*\)-modules
46L05 General theory of \(C^*\)-algebras
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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
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