On the complexity of the description of *-algebra representations by unbounded operators

Turowska, Lyudmila

doi:10.1090/S0002-9939-02-06525-5

On the complexity of the description of $*$-algebra representations by unbounded operators
HTML articles powered by AMS MathViewer

by Lyudmila Turowska PDF

Proc. Amer. Math. Soc. 130 (2002), 3051-3065 Request permission

Abstract:

We study the complexity of the problem to describe, up to unitary equivalence, representations of $*$-algebras by unbounded operators on a Hilbert space. A number of examples are developed in detail.

References

H. Bart, T. Ehrhardt, and B. Silbermann, Zero sums of idempotents in Banach algebras, Integral Equations Operator Theory 19 (1994), no. 2, 125–134. MR 1274554, DOI 10.1007/BF01206409
Yu.L. Daletskii, Continual integrals related to operator evolution equations, Uspekhi Mat. Nauk XVII (1962), no.5, 3-115.
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Peter Donovan and Mary Ruth Freislich, The representation theory of finite graphs and associated algebras, Carleton Mathematical Lecture Notes, No. 5, Carleton University, Ottawa, Ont., 1973. MR 0357233
P. E. T. Jorgensen, L. M. Schmitt, and R. F. Werner, Positive representations of general commutation relations allowing Wick ordering, J. Funct. Anal. 134 (1995), no. 1, 33–99. MR 1359923, DOI 10.1006/jfan.1995.1139
Stanislav Krugliak and Yuriǐ Samoǐlenko, On complexity of description of representations of $\ast$-algebras generated by idempotents, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1655–1664. MR 1636978, DOI 10.1090/S0002-9939-00-05100-5
L. P. Nizhnik and L. B. Turowska, Representations of double commutator by matrix-differential operators, Methods Funct. Anal. Topology 3 (1997), no. 3, 75–80. MR 1768166
V. Ostrovskiĭ, Yu. Samoĭlenko, Introduction to the Theory Representation of Finitely Presented $*-$algebras. $1$. Representations by bounded operators. Rev. Math. Math. Phys. 11 (1999), part 1, 1–261.
Vladimir V. Sergeichuk, Unitary and Euclidean representations of a quiver, Linear Algebra Appl. 278 (1998), no. 1-3, 37–62. MR 1637319, DOI 10.1016/S0024-3795(98)00006-8
Alan L. T. Paterson, The class of locally compact groups $G$ for which $C^*(G)$ is amenable, Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 226–237. MR 974319, DOI 10.1007/BFb0086603
G. K. Pedersen, $C^*$-algebras and their Automorphism Groups, Academic Press, London, 1979.
Yu. S. Samoĭlenko and L. B. Turowska, On representations of $*$-algebras by unbounded operators, Funkt. Anal. Prilozh. 31 (1997), no. 4, 80–83.
Yu. S. Samoĭlenko and L. B. Turowska, On bounded and unbounded idempotents whose sum is a multiple of the identity, Preprint 2001-44, Chalmers University of Technology.
K. Schmüdgen, Unbounded operator algebras and representation theory, Akademie-Verlag, Berlin, 1990.
S. L. Woronowicz, Unbounded elements affiliated with $C^*$-algebras and noncompact quantum groups, Comm. Math. Phys. 136 (1991), no. 2, 399–432. MR 1096123, DOI 10.1007/BF02100032
S. L. Woronowicz and K. Napiórkowski, Operator theory in the $C^\ast$-algebra framework, Rep. Math. Phys. 31 (1992), no. 3, 353–371. MR 1232646, DOI 10.1016/0034-4877(92)90025-V
S. L. Woronowicz, $C^*$-algebras generated by unbounded elements, Rev. Math. Phys. 7 (1995), no. 3, 481–521. MR 1326143, DOI 10.1142/S0129055X95000207