Isometries of Hilbert 𝐶*-modules

Solel, Baruch

doi:10.1090/S0002-9947-01-02874-4

Isometries of Hilbert $C^*$-modules
HTML articles powered by AMS MathViewer

by Baruch Solel PDF

Trans. Amer. Math. Soc. 353 (2001), 4637-4660 Request permission

Abstract:

Let $X$ and $Y$ be right, full, Hilbert $C^*$-modules over the algebras $A$ and $B$ respectively and let $T:X\to Y$ be a linear surjective isometry. Then $T$ can be extended to an isometry of the linking algebras. $T$ then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If $A$ (or $B$) is a factor von Neumann algebra, then every isometry $T:X\to Y$ is either a (bi-)module map or reverses the (bi-)module actions.

References

Jonathan Arazy and Baruch Solel, Isometries of nonselfadjoint operator algebras, J. Funct. Anal. 90 (1990), no. 2, 284–305. MR 1052336, DOI 10.1016/0022-1236(90)90085-Y
David P. Blecher, A new approach to Hilbert $C^*$-modules, Math. Ann. 307 (1997), no. 2, 253–290. MR 1428873, DOI 10.1007/s002080050033
David P. Blecher, On selfdual Hilbert modules, Operator algebras and their applications (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 65–80. MR 1424955, DOI 10.1007/bf02426790
M. Frank, A multiplier approach to the Lance-Blecher theorem, Z. Anal. Anwendungen 16 (1997), no. 3, 565–573. MR 1472718, DOI 10.4171/ZAA/778
Masamichi Hamana, Triple envelopes and Šilov boundaries of operator spaces, Math. J. Toyama Univ. 22 (1999), 77–93. MR 1744498
Lawrence A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, Proceedings on Infinite Dimensional Holomorphy (Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973) Lecture Notes in Math., Vol. 364, Springer, Berlin, 1974, pp. 13–40. MR 0407330
Günther Horn, Characterization of the predual and ideal structure of a $\textrm {JBW}^*$-triple, Math. Scand. 61 (1987), no. 1, 117–133. MR 929400, DOI 10.7146/math.scand.a-12194
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR 1468229, DOI 10.1090/gsm/015
Wilhelm Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503–529. MR 710768, DOI 10.1007/BF01173928
E. Christopher Lance, Unitary operators on Hilbert $C^*$-modules, Bull. London Math. Soc. 26 (1994), no. 4, 363–366. MR 1302069, DOI 10.1112/blms/26.4.363
E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
P. S. Muhly and B. Solel, On the Morita equivalence of tensor algebras, Proc. London Math. Soc. 81 (2000), 113-118.
William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
Gilles Pisier, The operator Hilbert space $\textrm {OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. MR 1342022, DOI 10.1090/memo/0585
Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR 1634408, DOI 10.1090/surv/060
Baruch Solel, Isometries of CSL algebras, Trans. Amer. Math. Soc. 332 (1992), no. 2, 595–606. MR 1053117, DOI 10.1090/S0002-9947-1992-1053117-5
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728