Completely bounded kernels
Tóm tắt
We introduce completely bounded kernels taking values in
$$\mathcal{L(A,B)}$$
where
$$\mathcal{A}$$
and
$$\mathcal{B}$$
are C*-algebras. We show that if
$$\mathcal{B}$$
is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and that this is automatic when the index set is countable.
Tài liệu tham khảo
Daniel Alpay, Some remarks on reproducing kernel Kreĭn spaces, Rocky Mountain J. Math., 21 (1991), 1189–1205.
Daniel Alpay, Aad Dijksma, James Rovnyak and Hendrik De Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications 96, Birkhäuser Verlag, Basel, 1997.
Stephen D. Barreto, B.V. Rajarama Bhat, Volkmar Liebscher and Michael Skeide, Type I product systems of Hilbert Modules, J. Funct. Anal., 212 (2003), 121–181.
Charles O. Christenson and William L. Voxman, Aspects of topology, Pure and applied Mathematics, Vol. 39, Marcel Dekker Inc., New York, 1977.
Tiberius Constantinescu and Aurelian Gheondea, On L. Schwartzés boundedness condition for kernels, Positivity, 10 (2006), 65–86.
Uffe Haagerup, Injectivity and decomposition of completely bounded maps, Operators algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, 170–222.
Jaeseong Heo, Completely multi-positive linear maps and representations on Hilbert C*-modules, J. Operator Theory, 41 (1999), 3–22.
Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics No. 24, Springer-Verlag, New York, 1975.
Gerard J. Murphy, Positive definite kernels and Hilbert C*-modules, Proc. Edinburgh Math. Soc. (2), 40 (1997), 367–374.
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002.
Laurent Schwartz, Sous-espaces hilbertiens déespaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math., 13 (1964), 115–256.
Michael Skeide, Hilbert modules - square roots of positive maps, Quantum Probability and Related Topics - Proceedings of the XXXth. Conference (R. Rebolledo and M. Orszag, eds.), Quantum Probability and White Noise Analysis, no. XXVII, World Scientific, 2011, 296–322.
R. R. Smith and D. P. Williams, The decomposition property for C*-algebras, J. Operator Theory, 16 (1986), 51–74.