On the complexity of the description of algebra representations by unbounded operators
Author:
Lyudmila Turowska
Journal:
Proc. Amer. Math. Soc. 130 (2002), 30513065
MSC (2000):
Primary 46K10, 46L05, 47L60; Secondary 16G60
Published electronically:
March 18, 2002
MathSciNet review:
1908930
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [BES]
H.Bart, T.Ehrhardt and B.Silbermann, Zero sums of idempotents in Banach algebras, Integr. Equat. Oper. Th. 19 (1994), 125134. MR 95c:46071a
 [Dal]
Yu.L. Daletskii, Continual integrals related to operator evolution equations, Uspekhi Mat. Nauk XVII (1962), no.5, 3115.
 [D]
J. Dixmier, algebras, NorthHolland Publ. Comp., Amsterdam, 1977. MR 56:16388
 [DF]
P. Donovan, M. R. Freislich, The representation theory of finite graphs and associated algebras, Carleton Mathematical Lecture Notes, No. 5. Carleton University, Ottawa, Ont., 1973, 83 pp. MR 50:9701
 [JSW]
P.E.T. Jorgensen, L.M. Schmitt, R.F. Werner, Positive representations of general commutation relations allowing Wick ordering, J. Funct. Anal. 134 (1995), no. 1, 3399. MR 96h:81033
 [KS]
S. A. Kruglyak and Yu. S. Samolenko, On complexity of description of representation of algebras generated by idempotents, Proc. Amer. Math. Soc., 128 (2000), no. 6, 16551664. MR 2000j:46099
 [NT]
L. P. Nizhnik and L. B. Turowska, Representations of double commutator by matrixdifferential operators, Methods Funct. Anal. Topol. 3 (1997), no. 3, 7580. MR 2001e:47043
 [OS]
V. Ostrovski, Yu. Samolenko, Introduction to the Theory Representation of Finitely Presented algebras. . Representations by bounded operators. Rev. Math. Math. Phys. 11 (1999), part 1, 1261.
 [S]
V. V. Sergeichuk, Unitary and Euclidean representations of a quiver, Linear Algebra Appl. 278 (1998), no. 13, 3762. MR 99g:16020
 [Pat]
A. L. T. Paterson, The class of locally compact groups for which is amenable. Harmonic Analysis, Luxembourg, 1987, Lecture Notes in Mathematics, Vol. 1359, New York, 1988. MR 90m:22019
 [Ped]
G. K. Pedersen, algebras and their Automorphism Groups, Academic Press, London, 1979.
 [ST1]
Yu. S. Samolenko and L. B. Turowska, On representations of algebras by unbounded operators, Funkt. Anal. Prilozh. 31 (1997), no. 4, 8083.
 [ST2]
Yu. S. Samolenko and L. B. Turowska, On bounded and unbounded idempotents whose sum is a multiple of the identity, Preprint 200144, Chalmers University of Technology.
 [S]
K. Schmüdgen, Unbounded operator algebras and representation theory, AkademieVerlag, Berlin, 1990.
 [Wor1]
S. L. Woronowicz, Unbounded elements affiliated with algebras and noncompact quantum groups, Commun. Math. Phys. 136 (1991), 399432. MR 92b:46117
 [WN]
S. L. Woronowicz and K. Napiorkowski, Operator theory in the algebra framework, Rep. Math. Phys. 31 (1992), 353371. MR 94k:46123
 [Wor2]
S. L. Woronowicz, algebras generated by unbounded elements, Reviews in Mathematical Physics 7 (1995), no. 3, 481521. MR 96c:46056
 [BES]
 H.Bart, T.Ehrhardt and B.Silbermann, Zero sums of idempotents in Banach algebras, Integr. Equat. Oper. Th. 19 (1994), 125134. MR 95c:46071a
 [Dal]
 Yu.L. Daletskii, Continual integrals related to operator evolution equations, Uspekhi Mat. Nauk XVII (1962), no.5, 3115.
 [D]
 J. Dixmier, algebras, NorthHolland Publ. Comp., Amsterdam, 1977. MR 56:16388
 [DF]
 P. Donovan, M. R. Freislich, The representation theory of finite graphs and associated algebras, Carleton Mathematical Lecture Notes, No. 5. Carleton University, Ottawa, Ont., 1973, 83 pp. MR 50:9701
 [JSW]
 P.E.T. Jorgensen, L.M. Schmitt, R.F. Werner, Positive representations of general commutation relations allowing Wick ordering, J. Funct. Anal. 134 (1995), no. 1, 3399. MR 96h:81033
 [KS]
 S. A. Kruglyak and Yu. S. Samolenko, On complexity of description of representation of algebras generated by idempotents, Proc. Amer. Math. Soc., 128 (2000), no. 6, 16551664. MR 2000j:46099
 [NT]
 L. P. Nizhnik and L. B. Turowska, Representations of double commutator by matrixdifferential operators, Methods Funct. Anal. Topol. 3 (1997), no. 3, 7580. MR 2001e:47043
 [OS]
 V. Ostrovski, Yu. Samolenko, Introduction to the Theory Representation of Finitely Presented algebras. . Representations by bounded operators. Rev. Math. Math. Phys. 11 (1999), part 1, 1261.
 [S]
 V. V. Sergeichuk, Unitary and Euclidean representations of a quiver, Linear Algebra Appl. 278 (1998), no. 13, 3762. MR 99g:16020
 [Pat]
 A. L. T. Paterson, The class of locally compact groups for which is amenable. Harmonic Analysis, Luxembourg, 1987, Lecture Notes in Mathematics, Vol. 1359, New York, 1988. MR 90m:22019
 [Ped]
 G. K. Pedersen, algebras and their Automorphism Groups, Academic Press, London, 1979.
 [ST1]
 Yu. S. Samolenko and L. B. Turowska, On representations of algebras by unbounded operators, Funkt. Anal. Prilozh. 31 (1997), no. 4, 8083.
 [ST2]
 Yu. S. Samolenko and L. B. Turowska, On bounded and unbounded idempotents whose sum is a multiple of the identity, Preprint 200144, Chalmers University of Technology.
 [S]
 K. Schmüdgen, Unbounded operator algebras and representation theory, AkademieVerlag, Berlin, 1990.
 [Wor1]
 S. L. Woronowicz, Unbounded elements affiliated with algebras and noncompact quantum groups, Commun. Math. Phys. 136 (1991), 399432. MR 92b:46117
 [WN]
 S. L. Woronowicz and K. Napiorkowski, Operator theory in the algebra framework, Rep. Math. Phys. 31 (1992), 353371. MR 94k:46123
 [Wor2]
 S. L. Woronowicz, algebras generated by unbounded elements, Reviews in Mathematical Physics 7 (1995), no. 3, 481521. MR 96c:46056
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
46K10,
46L05,
47L60,
16G60
Retrieve articles in all journals
with MSC (2000):
46K10,
46L05,
47L60,
16G60
Additional Information
Lyudmila Turowska
Affiliation:
Department of Mathematics, Chalmers University of Technology, 412 96 Göteborg, Sweden
Email:
turowska@math.chalmers.se
DOI:
http://dx.doi.org/10.1090/S0002993902065255
PII:
S 00029939(02)065255
Keywords:
$*$algebras,
$*$representations,
unbounded operators,
$*$wildness.
Received by editor(s):
June 12, 2000
Received by editor(s) in revised form:
November 28, 2000, and May 31, 2001
Published electronically:
March 18, 2002
Communicated by:
David R. Larson
Article copyright:
© Copyright 2002 American Mathematical Society
