Abstract
Let \({H^{\infty}(E)}\) be a non commutative Hardy algebra, associated with a \({W^*}\)-correspondence E. In this paper we construct factorizations of inner-outer type of the elements of \({H^{\infty}(E)}\) represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of \({H^\infty(\mathbb{D})}\). Our results also generalize some results that were obtained by several authors in some special cases.
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References
Arias A., Popescu G.: Factorization and reflexivity on Fock space. Integral Equ. Oper. Theory 23, 268–286 (1995)
Beurling A.: On two problems concerning linear transformations in Hilbert space. Acta. Math. 81, 239–255 (1949)
Davidson K.R., Pitts D.R.: Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. (3) 78, 401–430 (1999)
Kribs D.W., Power S.C.: Free semigroupoid algebras. J. Ramanujan Math. Soc. 19(2), 117–159 (2004)
Lance, E.C.: Hilbert \({C^*}\)-modules : a toolkit for operator algebraist, London, Math. Soc., Lecture Note Series, vol. 210. Cambridge Univ. Press (1995)
Manuilov, V., Troitsky, E.: Hilbert \({C^*}\)-modules, Translations of Mathematical Monographs, vol. 226. Amer. Math. Soc., Providence (2005)
Meyer, J.R.: Noncommutative Hardy algebras, multipliers, and quotients, Ph.D. Thesis, University of Iowa (2010)
Muhly P., Solel B.: Tensor algebras, Induced Representations and the Wold Decomposition. Can. J. Math. 51(4), 850–880 (1999)
Muhly P., Solel B.: Tensor algebras over \({C^*}\)-correspondence: representations, dilations, and \({C^*}\)-envelopes. J. Funct. Anal. 158, 398–477 (1998)
Muhly P., Solel B.: Hardy algebras, \({W^*}\)-correspondences and interpolation theory. Math. Ann. 330, 353–415 (2004)
Muhly P., Solel B.: Representations of the Hardy algebras: absolutely continuity, intertwiners and superharmonic operators. Integr. Equ. Oper. Theory 70, 151–203 (2011)
Nikolskii, N.: Operators, functions, and systems: an easy reading. vol. I&II, Parts A, B, C. Matheamical Surveys and Monographs, p. 92. AMS (2002)
Nagy B., Foiaş C.: Harmonic analysis of operators on Hilbert spaces. North-Holland Akad, Kiado (1970)
Paschke W.L.: Inner product Modules over \({B^*}\)-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)
Pimsner, M.: A class of \({C^*}\)-algebras generalizing both Cuntz-Krieger algebras and crossed products by \({\mathbb{Z}}\), In: Voiculescu, D., (ed.) Free probability theory, Fields Instit. Comm., vol. 12, pp. 189–212. Amer. Math. Soc., Providence (1997)
Popescu G.: Von Nemann inequality for \({(B(H)^n)_1}\). Math. Scand. 68, 292–304 (1991)
Popescu G.: Multy-analytic operators and some factorization theorems. Indiana Univ. Math. J. 38, 693–710 (1989)
Popescu G.: Multy-analytic operators. J. Oper. Theory 22, 51–71 (1989)
Popescu G.: Multy-analytic operators on Fock spaces. Math. Ann. 303, 31–46 (1995)
Rieffel M.: Induced representations of \({C^*}\)-algebras. Adv. Math. 13, 176–257 (1974)
Sakai S.: A characterization of \({W^*}\)-algebras. Pacific J. Math. 6, 763–773 (1956)
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This work was completed with the support of the Technion during the Ph.D. study of the author.
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Helmer, L. Generalized Inner-Outer Factorizations in Non Commutative Hardy Algebras. Integr. Equ. Oper. Theory 84, 555–575 (2016). https://doi.org/10.1007/s00020-015-2277-7
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DOI: https://doi.org/10.1007/s00020-015-2277-7