Abstract
Muhly and Solel developed a notion of Morita equivalence for \(C^{*}\)-correspondences, which they used to show that if two \(C^{*}\)-correspondences E and F are Morita equivalent then their tensor algebras \({\mathcal {T}}_{+}(E)\) and \({\mathcal {T}}_{+}(F)\) are (strongly) Morita equivalent operator algebras. We give the weak\(^{*}\) version of this result by considering (weak) Morita equivalence of \(W^{*}\)-correspondences and employing Blecher and Kashyap’s notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of \(W^{*}\)-correspondences E and F implies weak Morita equivalence of their Hardy algebras \(H^{\infty }(E)\) and \(H^{\infty }(F)\). We give special attention to \(W^{*}\)-graph correspondences and show a number of results related to their Morita equivalence.
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References
Blecher, D.P., Kashyap, U.: Morita equivalence of dual operator algebras. J. Pure Appl. Algebra 212(11), 2401–2412 (2008)
Blecher, D.P., Le Merdy, C.: Operator Algebras and Their Modules—An Operator Space Approach, London Mathematical Society Monographs. New Series. Oxford University Press, Oxford (2004)
Blecher, D.P.: On selfdual Hilbert Modules, Operator Algebras and Their Applications (Waterloo, ON, 1994/1995), Fields Institute Communications, pp. 65–80. American Mathematical Society, Providence, RI (1997)
Blecher, D.P., Muhly, P.S., Paulsen, V.I.: Categories of operator modules (Morita equivalence and projective modules). Mem. Am. Math. Soc. 143(681), viii+94 (2000)
Davidson, K.R., Pitts, D.R.: The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311(2), 275–303 (1998)
Echterhoff, S., Kaliszewski, S., Quigg, J., Raeburn, I.: A categorical approach to imprimitivity theorems for \(C^*\)-dynamical systems. Mem. Am. Math. Soc. 180(850), viii+169 (2006)
Eleftherakis, G.K.: A morita type equivalence for dual operator algebras. J. Pure Appl. Algebra 212(5), 1060–1071 (2008)
Eleftherakis, G.K., Paulsen, V.I.: Stably isomorphic dual operator algebras. Math. Ann. 341(1), 99–112 (2008)
Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6, 83–142 (1958)
Muhly, P.S., Solel, B.: Tensor algebras, induced representations, and the Wold decomposition. Canad. J. Math. 51(4), 850–880 (1999)
Muhly, P.S., Solel, B.: On the Morita equivalence of tensor algebras. Proc. Lond. Math. Soc. 81(1), 113–168 (2000)
Muhly, P.S., Solel, B.: Hardy algebras, \(W^\ast \)-correspondences and interpolation theory. Math. Ann. 330(2), 353–415 (2004)
Muhly, P.S., Solel, B.: Schur class operator functions and automorphisms of Hardy algebras. Doc. Math. 13, 365–411 (2008)
Muhly, P.S., Solel, B.: The Poisson kernel for Hardy algebras. Complex Anal. Oper. Theory 3(1), 221–242 (2009)
Paschke, W.L.: Inner product modules over \(B^{\ast } \)-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)
Popescu, G.: von Neumann inequality for \((B({\cal{H}})^n)_1\). Math. Scand. 68(2), 292–304 (1991)
Rieffel, M.A.: Induced representations of \(C^{\ast } \)-algebras. Adv. Math. 13, 176–257 (1974)
Rieffel, M.A.: Morita equivalence for \(C^{\ast } \)-algebras and \(W^{\ast } \)-algebras. J. Pure Appl. Algebra 5, 51–96 (1974)
Raeburn, I., Williams, D.P.: Morita equivalence and continuous-trace \(C^*\)-algebras, Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence, RI (1998)
Solel, B.: You can see the arrows in a quiver operator algebra. J. Aust. Math. Soc. 77(1), 111–122 (2004)
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Communicated by Ilan Hirshberg.
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Ardila, R. Morita Equivalence of \(W^{*}\)-Correspondences and Their Hardy Algebras. Complex Anal. Oper. Theory 13, 2411–2441 (2019). https://doi.org/10.1007/s11785-019-00906-1
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DOI: https://doi.org/10.1007/s11785-019-00906-1