Abstract
Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra \({{\mathcal{A}}}\) onto a unital *-subalgebra \({{\mathcal{B}}}\) are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras \({{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g\) for which the *-subalgebra \({{\mathcal{B}}}:={{\mathcal{A}}}_e\) is commutative. Then the canonical projection \(p:{{\mathcal{A}}}\to{{\mathcal{B}}}\) is a conditional expectation and there is a partial action of the group G on the set \({{\mathcal{B}}}p\) of all characters of \({{\mathcal{B}}}\) which are nonnegative on the cone \(\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.\) The complete Mackey theory is developed for *-representations of \({{\mathcal{A}}}\) which are induced from characters of \({{\widehat{{{\mathcal{B}}}}^+}}.\) Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras \({{\mathcal{A}}}\) is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.
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Antoine, J.-P., Inoue, A., Trapani, C.: Partial *-Algebras and their Operator Realizations. Kluwer, Dordrecht (2002)
Bursztyn, H., Waldmann, S.: Algebraic Rieffel induction, formal Morita equivalence and applications to deformation quantization. J. Geom. Phys. 37, 307–364 (2001)
Burban, I.M., Klimyk, A.U.: Representations of the quantum algebra \({{\mathcal{U}}}_q(su(1,1))\). J. Phys. A 26(9), 2139–2151 (1993)
Cimprič, J.: Formally real involutions on central simple algebras. Commun. Algebra 36(1), 165–178 (2008)
Cimprič, J., Kuhlmann, S., Scheiderer, C.: Sums of squares and moment problems in equivariant situations. Trans. Amer. Math. Soc. 361, 735–765 (2009)
Chari, V., Pressley, A.: Unitary representations of the Virasoro algebra and a conjecture of Kac. Comput. Math. 67(3), 315–342 (1988)
Dixmier, J.: Enveloping Algebras. North-Holland Publishing Co. (1976)
Effros, E.G.: Transformation groups and C *-algebras. Ann. Math. 81(2), 38–55 (1965)
Evans, D.E., Sund, T.: Spectral subspaces for compact actions. Rep. Math. Phys. 17(2), 299–308 (1980)
Exel, R.: Partial actions of groups and actions of inverse semigroups. Proc. Am. Math. Soc. 126(12), 3481–3494 (1998)
Fell, J.M.G.: A new look at Mackey’s imprimitivity theorem. In: Lecture Notes Math., vol. 266, pp. 43–58. Springer, Berlin (1972)
Fell, J.M.G., Doran, R.S.: Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles. Academic, Boston, MA (1988)
Friedan, D., Qiu, Z., Shenker, S.: Details of the nonunitarity proof for highest weight representations of the Virasoro algebra. Commun. Math. Phys. 107(4), 535–542 (1986)
Friedrich, J., Schmüdgen, K.: n-positivity of unbounded *-representations. Math. Nachr. 141, 233–250 (1989)
Gårding, L., Wightman, A.S.: Representations of anti-commutation relations. Proc. Natl. Acad. Sci. U.S.A. 10, 617–621 (1954)
Gudder, S.P., Hudson, R.L.: A noncommutative probability theory. Trans. Am. Math. Soc. 245, 1–41 (1978)
Høegh-Krohn, R., Landstad, M.B., Størmer, E.: Compact ergodic groups of automorphisms. Ann. Math. (2) 114(1), 75–86 (1981)
Inoue, A., Takakura, M., Ogi, H.: Unbounded conditional expectations for O *-algebras. In: Contemp. Math., vol. 427, pp. 225–234. Amer. Math. Soc., Providence, RI (2007)
Jorgensen, P.E.T., Schmitt, L.M., Werner, R.F.: Positive representations of general commutation relations allowing Wick ordering. J. Funct. Anal. 134(1), 33–99 (1995)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Academic, New York (1986)
Kato, T.: Perturbation Theory for Linear Operators, xix+592 pp. Springer (1966)
Kirillov, A.A.: Elements of the Theory of Representations. Springer (1976)
Klimek, S., Lesniewski, A.: A two-parameter quantum deformation of the unit disc. J. Funct. Anal. 115(1), 1–23 (1993)
Klimyk, A.U., Schmüdgen, K.: Quantum Groups and Their Representations. Springer, Berlin (1997)
Kulish, P.P., Reshetikhin, N.Yu.: Quatum linear problem for the sone-Gordon equation and higher representations. Zap. Nauc. Semin. LOMI 101, 101–110 (1981)
Marcus, A.: Representation Theory of Group Graded Algebras. Nova, Commack, NY (1999)
Năstăsescu, C., Van Oystaeyen, F.: Methods of Graded Rings. Lecture Notes in Math., vol. 1836. Springer (2004)
Ostrovskyĭ, V., Samoĭlenko, Yu.: Introduction to the Theory of Representations of Finitely Presented *-Algebras. I. Gordon and Breach, London (1999)
Podleś, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987)
Powers, R.T.: Self-adjoint algebras of unbounded operators. Commun. Math. Phys. 21, 85–124 (1971)
Proskurin, D.: Homogeneous ideals in Wick *-algebras. Proc. Am. Math. Soc. 126(11), 3371–3376 (1998)
Proskurin, D., Savchuk, Yu., Turowska, L.: On \(C\sp*\)-algebras generated by some deformations of CAR relations. In: Noncommutative Geometry and Representation Theory in Mathematical Physics. Contemp. Math., vol. 391, pp. 297–312. Amer. Math. Soc., Providence, RI (2005)
Pusz, W.: Twisted canonical anti-commutation relations. Rep. Math. Phys. 27, 349–360 (1989)
Pusz, W., Woronowicz, S.L.: Twisted second quantization. Rep. Math. Phys. 27, 231–257 (1989)
Rieffel, M.A.: Induced representations of C *-algebras. Adv. Math. 13, 176–257 (1974)
Rudin, W.: Functional Analysis. McGraw-Hill Book Co. (1973)
Samoilenko, Yu., Spectral Theory of Families of Self-Adjoint Operators. Kluwer, Dordrecht (1991)
Samoĭlenko, Yu., Turovskaya, L.: On representations of *-algebras by unbounded operators. Funct. Anal. Appl. 31(4), 289–291 (1997)
Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser Verlag, Basel (1990)
Schmüdgen, K.: On well-behaved unbounded representations of *-algebras. J. Oper. Theory 48, 487–502 (2002)
Schmüdgen, K.: A note on commuting unbounded selfadjoint operators affiliated to properly infinite von Neumann algebras. II. Bull. Lond. Math. Soc. 18(3), 287–292 (1986)
Schmüdgen, K.: Noncommutative real algebraic geometry—some basic concepts and first ideas. In: Emerging applications of algebraic geometry, pp. 325–350, IMA Vol. Math. Appl., 149. Springer, New York (2009)
Vaksman, L.L., Soibelman, Ya.S.: An algebra of functions on the quantum group SU(2). Funct. Anal. Appl. 22(3), 170–181 (1988)
Vershik, A.M.: Gelfand–Tsetlin algebras, expectations, inverse limits, Fourier analysis. In: The Unity of Mathematics, Progr. Math., vol. 244, pp. 619–631. Birkhäuser Boston, Boston, MA (2006)
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Dedicated to the memory of A.U. Klimyk (14.04.1939–22.07.2008).
The first author was supported by the International Max Planck Research School for Mathematics in the Sciences (Leipzig).
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Savchuk, Y., Schmüdgen, K. Unbounded Induced Representations of ∗-Algebras. Algebr Represent Theor 16, 309–376 (2013). https://doi.org/10.1007/s10468-011-9310-6
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DOI: https://doi.org/10.1007/s10468-011-9310-6
Keywords
- Induced representations
- Group graded algebras
- Well-behaved representations
- Conditional expectation
- Partial action of a group
- Imprimitivity Theorem
- Mackey analysis