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Compact and “compact” operators on standard Hilbert modules over \(W^*\)-algebras. (English) Zbl 1394.46047

The authors introduce a locally convex topology on the standard Hilbert module \(l^2(A)\), where \(A\) is a unital \(W^*\)-algebra such that any “compact” operator (that is, a member of the closed linear span of operators of the form \(\theta_{y,z}(x)=z\langle y,x\rangle\), \(x,y,z\in l^2(A)\)) maps bounded sets (in the norm) into totally bounded sets in the introduced topology. In the special case where \(A=B(H)\), they show that the converse is also true. Namely, any operator \(T\) in the algebra of all adjointable operators on \(l^2(A)\) that maps bounded sets into totally bounded sets is “compact”.

MSC:

46L08 \(C^*\)-modules
47B07 Linear operators defined by compactness properties
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References:

[1] N. Bourbaki, General Topology, chapters 5-10, reprint of the 1966 edition, Elem. Math. (Berlin), Springer, Berlin, 1989.
[2] M. Frank, Self-duality and \(C^{*}\)-reflexivity of Hilbert \(C^{*}\)-moduli, Z. Anal. Anwend. 9 (1990), no. 2, 165-176.
[3] M. Frank, V. M. Manuilov, and E. V. Troitsky, Hilbert \(C^{*}\)-modules from group actions: Beyond the finite orbits case, Studia Math. 200 (2010), no. 2, 131-148. · Zbl 1210.46044
[4] J. L. Kelley, General Topology, Van Nostrand, Toronto, 1955.
[5] E. C. Lance, Hilbert \(C^{*}\)-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, Cambridge, 1995.
[6] V. M. Manuilov, Diagonalization of compact operators in Hilbert modules over finite \(W^{*}\)-algebras, Ann. Global Anal. Geom. 13 (1995), no. 3, 207-226. · Zbl 0827.46058
[7] V. M. Manuilov, Diagonalization of compact operators on Hilbert modules over \(C^{*}\)-algebras of real rank zero (in Russian), Mat. Zametki 62, no. 6 (1997), 865-870; English translation in Math. Notes 62 (1997), 726-730. · Zbl 0917.47020
[8] V. M. Manuilov and E. V. Troitsky, Hilbert \(C^{*}\)-Modules, Transl. Math. Monogr. 226, Amer. Math. Soc., Providence, 2005.
[9] W. L. Paschke, Inner product modules over \(B^{⁎}\)-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. · Zbl 0239.46062
[10] W. L. Paschke, Inner product modules arising from compact automorphism groups of von Neumann algebras, Trans. Amer. Math. Soc. 224 (1976), 87-102. · Zbl 0339.46048
[11] A. A. Pavlov and E. V. Troitsky, Quantization of branched coverings, Russ. J. Math. Phys. 18 (2011), no. 3, 338-352. · Zbl 1254.46064
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