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\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amsfonts} \makeatletter \renewcommand{\@listI}{\setlength{\itemsep}{0pt}} \renewcommand{\@listi}{\setlength{\itemsep}{0pt}} \makeatother \begin{document} \begin{center}{\bf APPLICATION FOR SCIENTIFIC COOPERATION\\[4pt] between German and Ukrainian groups in the period 2003\,--\,2006 in the framework of the Agreement on Scientific Cooperation\\ between the Deutsche Forschungsgemeinschaft and\\ the Ukrainian National Academy of Sciences} \end{center} {\large\bf\noindent 1. General statement} \smallskip\noindent The project proposal is new. \medskip\noindent {\bf 1.1. Applicants.} \smallskip\noindent {\bf In Germany:}\ \ Universit\"at Bonn, Institut f\"ur Angewandte Mathematik, Abteilung Sto\-chas\-tik, Wegelerstra\ss{}e 6, D-53115 Bonn. Tel.: +049 (0228) 733418. FAX: +049 (0228) 739537. E-mail: \texttt{albeverio@uni-bonn.de} \smallskip\noindent {\bf Project responsible:}\ \ Prof.\ Dr.\ Sergio Albeverio \begin{enumerate} \item 17.01.1939 \item Dr.\ Rer.\ Nat., Professor \item 1977 C3-Professor, 1979 C4-Professor \item Director of the Institute for Stochastics, Faculty of Applied Mathematics, Bonn University; Director of BiBoS \item Number of publications: ca.\ 430 \item Address: \begin{enumerate} \item[6.1] Universit\"at Bonn, Institut f\"ur Angewandte Mathematik, Abteilung Sto\-chas\-tik, Wegelerstra\ss e 6, D-53115 Bonn \item[6.2] Private address: Auf dem Aspei 55, D-44801 Bochum. Tel.\ +049 (0234) 704214 \end{enumerate} \end{enumerate} \smallskip\noindent {\bf In Ukraine:}\ \ Institute of Mathematics of National Academy of Sciences of Ukraine, Tereschenkivska st. 3, 01601 Kyiv, Ukraine. Tel.: 380 44 2246153 FAX: 380 44 2352010. E-mail: yurii\_sam@imath.kiev.ua \smallskip\noindent {\bf Project responsible:}\ \ Prof.\ Dr.\ Yurii Samoilenko \begin{enumerate} \item 17.09.1943 \item Doctor in Sciences, Professor \item 1989 Doctor in Sciences, 1992 Professor \item Head of Department of Functional Analysis, Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska st. 3, 10601, Kyiv, Ukraine. \item Number of publications: 120 \item Address: \begin{enumerate} \item[6.1.] Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska st. 3, 10601, Kyiv, Ukraine. \item[6.2.] Private address: Apt. 40, Stroitelei st. 9, 02105 Kyiv, Ukraine, Tel: 380 44 5598316 \end{enumerate} \end{enumerate} \smallskip\noindent {\bf 1.2. Title.}\ \ Algebraic sructures in Operator Theory and Mathematical Physics \smallskip\noindent {\bf 1.3. Key words.}\ \ $*$-algebras of bounded and unbounded operators, Jacobi fields, sums of projections, traces on $W^*$-algebras, differential forms on configuration spaces, $q$-CCR fields, Poisson and Gamma-analysis, Fermion, Boson and anion models. \smallskip\noindent {\bf 1.4. Area of applications.}\ \ Families of operators obeying relations and their applications to models of Mathematical Physics. \smallskip\noindent {\bf 1.5. Period of work.}\ \ Commencement of work: 01.01.2003\\ \hphantom{{\bf 1.5. Period of work:}}\ \ End of work: 31.12.2006 \smallskip\noindent {\bf 1.6. Duration.}\ \ 36 months \smallskip\noindent {\bf 1.7. For a new agreement.}\ \ Desirable commencement of work: \,01.01.2003 \smallskip\noindent {\bf 1.8. Resume.} Families of operators obeying certain relations (in particular, representations of finitely generated $*$-algebras), their structure, properties, examples etc. are widely exploited in Theoretical Physics. In the project, we plan to joint results of the Ukrainian team in this area and significant experience of the German team in applications of such results to Mathematical Physics. The aims of the project are to study classes of $*$-algebras and their representations and apply the obtained results to problems of modern analysis, operator theory, mathematical physics etc. In particular, we plan to study classes of $*$-algebras for which faithful (``good'') representation by unbounded operators in rigged spaces exist which generalizes the Gelfand-Naimark theorem, to study representations by bounded and unbounded operators of Wick algebras depending on parameters, investigate depending on parameters families of $*$-algebras generated by projections and idempotents and their representations, families of algebras related to dynamical systems, their enveloping $C^*$-algebras, investigate their dependence on the parameters, including improvement of results on stability of Cuntz algebras, spectral properties of Jacobi fields, moment problem etc. Apply these results to the study of symmetric differential forms on configuration spaces, models of boson, fermion and quon fields, models of statistical mechanics constructed using the corresponnding systems of orthogonal projections. \bigskip\noindent {\large\bf 2. State of scientific investigation and directions of the research. %Background of the\\[2pt] %\hphantom{2. }participants} } \medskip\noindent {\bf 2.1. State of scientific investigation.}\ \ Problems of families of operators obeying relations, operator algebras and representations theory are closely related to modern physics, in particular, quantum mechanics, statistical mechanics, quantum field theory etc. Due to numeral applications in analysis and mathematical physics, theory of $*$-algebras and their representations by bounded and unbounded operators in a Hilbert space is being developed actively. We list some branches of this theory related to the scope of the project. {\bf I. Unbounded representations of $*$-algebras and Gelfand-Naimark type theorems} For $C^*$-algebras, the Gelfand-Naimark theorem (1943) states that any $C^*$-algebra is isomorphic to a closed $C^*$-subalgebra of $L(H)$. In other words, for any $C^*$\nobreakdash-algebra there exists a faithful $*$-representation. One of our goals is to study similar problem for more general classes of $*$-algebras, considering their representations by generally speaking unbounded operators. Unbounded repesentations and algebras of unbounded operators were studied by numerous authors. We mention papers by E.~Nelson (1959), R.~Powers (1971, 1974), S.~L.~Woronowicz (1991, 1995), books by P.~E.~T.~Jorgensen (1988), A.~Inoue (1998), K.~Schm\"udgen (1990), Yu.~Samo\u{\i}lenko (1991) and others. For representations of commutative $*$-algebras and locally compact groups techniques of rigged spaces and decomposition on generalized eigenvectors of families of self-adjoint operators, developed by Yu.~Berezansky, I.~Gelfand, A.~Kostyuchenko, G.~Katz (1960--1970) and others, enables one to answer many questions regarding representations of such algebras, and provide variants of such theorems. We will consider classes of $*$-algebras for which this technique can be used to study ``good'' unbounded representations of $*$-algebras, and in particular, prove analogues of the Gelfand-Naimark theorem, i.e., the existence of faithful $*$-representaion (by unbounded operators) in rigged spaces. We plan to use rigged spaces approach to study infinite-dimensional and non-commutative moment problem (for contribution of the project teams to this subject see \cite{1,29,5,dal,dalsam,4} and others), enveloping $C^*$-algebra or topological $*$-algebra (\cite{sam_str_mfat2002,28,14,vin_pop} and others), its deformations \cite{11,13a,12,28,14,faa,7,26}, identities in PI-algebras \cite{22,30} etc. {\bf II. Representations of $*$-algebras generated by projections and idempotents} We plan to study $*$-algebras generated by projections and idempotents obeying certain linear conditions, their representations and relation with other known classes of algebras generated by projections (group algebras of Coxeter groups, Temperley-Lieb algebras etc.) (see \cite{27,faa,6,24,25} for project teams contribution). Notice that several old problems on eigenvalues, invariant factors, Schubert calculus etc. can be reduced to the study of representations of such algebras. Our goal is to study $*$-algebras generated by projections, whose linear combinations are multiple of the identity. Such algebras depend on parameters, and preliminary results obtained by Ukrainian team show that the set of parameters for which representations exist, has non-trivial structure \cite{faa,7,26}. Notice that similar phenomenon arise for representations of Temperley-Lieb algebras. $*$-Algebras generated by idempotents admit unbounded representations. Taking into account that both the German and Ukrainian team leaders are experts in applications of representations by unbounded operators, we intend to study "integrable" representations of classes of $*$-algebras generated by idempotents and their applications (for results obtained by the project teams in this direction, see e.g. \cite{15,25} and others). For the study of families of $*$- and $C^*$-algebras which depend on parameters it is helpful to study $*$- and $C^*$-algebras arising from the action of a dynamical system. The construction of crossed product of commutative $C^*$-algebra by $\mathbb{Z}$ enables one to describe the $C^*$-algebra associated with the bijective mapping on the real line. We plan to study similar objects for the non-bijective case. The simplest example is a $*$-algebra associated with the quadratic mapping (quantum anharmonic oscillator algebra). Our approach is the study of dependence of representations and enveloping objects on the properties of the corresponding non-bijective dynamical system \cite{26,14,bon_pop}. Also, representations of Cuntz algebras and their generalizations will be studied from this viewpoint (for results by project teams here, see \cite{18,10,28,14} and others). We also plan to study new classes of PI-algebras, constuct for them invertibility symbols, which can be used for solving singular integral operators, investigate spectral properties of Jacobi fields, apply our results to exactly solvable models of quantum statistical mechanics etc. {\bf III. Families of operators on configuration spaces} Configuration spaces on $\mathbb R^d$, operators and differential forms on functions defined on configurations are directly related to physical models (see \cite{alb_etal}). We plan to study comumutative families of operators on such spaces, e.g. Jacobi fields. The concept of a commutative Jacobi field in the Fock space was introduced in \cite{29} in connection with a generalization of the white noise analysis to the case when coupling measure is more general then the Gaussian measure. The Fourier transform related to the expansion in the generalized joint eigenvectiors of Jacobi field turned out to be a generalization of the Wiener-Ito-Segal transform in the white noise theory. Moreover, the rigging of the Fock space is transferred to some rigging of the space $L^2$ with respect to the spectral measure of Jacobi field, and the latter rigging generates the corresponding theory of generalized funcion on the spectrum. We plan to construct and investigate Jacobi fields in the case when coupling measure is Gamma-measure or generalized Poissonian measure and obtain the corresponding chaotic decompositions using the spectral theory of such fields. We also plan to get new results on traces on operator $W^*$-algebras, in particular, on $W^*$-algebras of operators acting in spaces of symmetric (or anti-symmetric) differential forms on configuration spaces (see \cite{dal_lit}) and various examples of operator families arising from representations of the $Diff \mathbb R^d$ (A.M.Vershik, I.M.Gelfand, M.I.Graev, G.A.Goldin, S.Albeverio \cite{alb_book} etc.) {\bf IV. Wick algebras and their applications} Class of Wick algebras was introduced by P.E.T.Jorgensen, L.M.Schmitt and R.F.Werner (1995) as a generalization of some $*$-algebras arising in the Mathematical Physics. The most known examples are deformations of canonical commutation and anicommutation relations such as $q$-CCR (O.W.Greenberg, A.J.Macfarlane, L.C.Biedenharn) and more general $q_{ij}$-CCR (M.Bozeiko, R.Speicher) and twisted canonical commutation relations (W.Pusz, S.L.Woronowicz). In the recent time the theory of representations of $*$-algebras and their $W^*$ and $C^*$-algebras became of special interest due to the applications in mathematical physics. One of the directions of these applications is a construction of models of particles with generalized statistics (anyones, generalized quons etc.) The Fock realization of the deformed commutation relations is one of the central points in the studying of the corresponding physical sysytem. The existence of positive Fock representation of Wick algebras is in focus of our investgations. Note, that in general the Fock representation is not unique irreducible one, it is a fact even for CCR with infinitely many generators. We plan to study some classes of irreducible representations of deformed CCR (or equivalently pure states on the $*$-algebras) and clarify their physical nature. When Fock representation is bounded it is interesting to compare the norm of elements of enveloping $C^*$-algebra and their images in the Fock representation (for the generalized quons algebra and twisted CCR these norm coinside). An interesting detail is that the $C^*$-isomorphism classes of enveloping $C^*$-algebras are stable under the deformation parameters (it is a case for the $q$-CCR and TCCR) \cite{8,11,13a,10,23,12,9,13}. We plan to study the problem of stability for some non-linear deformations of CCR. We are going also to study a representations of higher-dimensional and, especially infinite-dimensional noncommutative tori whose appear naturally as a quotients of the algebra of generalized qouns by the kernel of Fock representation. \smallskip\noindent {\bf 2.2. Background of the participants.}\ \ Team members are experienced in the topics planned in the proposal, which is approved by the list of publications below: Professor Yu.~S.~Samoilenko (born in 1943) is an expert in spectral theory of operator families, representations of $*$-algebras by bounded and unbounded operators and applications, see \cite{5,18,27,8,11,6,24,22,9,13,30,7,26,25,4} and many others) Member of Ukrainian Academy of Sciences professor Yu.~M.~Berezansky (born in 1925) is a founder of rigged spaces techniques, well-known expert in the spectral theory of self-adjoint operator families, Jacobi fields and many other domains, see \cite{1,29,2} and many other papers. A.~Daletskii (born in ????) is an exellent expert in .... V.~Ostrovskyi (born in 1961, Ph.D.) is an expert in representations of $*$-algebras related to dynamical systems, Cuntz algebras and their representations etc., see \cite{5,18,16,10,22} and other papers. D.~Proskurin (born in 1973, Ph.D) is an expert in Wick algebras and their representations, quantum tori etc., see \cite{8,11,10,23,12,9,13} and other papers. S.~Popovych (born in 1975, Ph.D) is experienced in $*$-algebras arising from dynamical systems, in particular, non-bijective, and their enveloping $C^*$-algebras, see \cite{15,28,14} and other papers. V.~Rabanovich (born in 1973, Ph.D) )is experienced in algebras generated by projections, idempotents, their representations and applications, see \cite{27,30,7,25} and other papers. A. Strelets (born in 1977) is a talanted Ph.D. student who successfully study theory of PI-algebras, algebras generated by idempotents and who already obtained some interesting result in this domain, see \cite{25,sam_str_mfat2002} and other papers. \par\medskip\noindent\textbf{References.} The selected references to follow are a selection of papers by team members. They form the basis for the research which is proposed between the two teams of researchers. \vskip-1cm \ \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \renewcommand{\refname}{} \begin{thebibliography}{10} \bibitem{alb_book} S.Albeverio (book) \bibitem{alb_etal} S.Albeverio et al. \bibitem{Berez_Toda} Yu. M. Berezansky, {\sl On the direct and inverse spectral problems for Jacobi fields}, Algebra and Analysis {\bf 9} (1998), no. 6, 1053 -- 1071. \bibitem{1} Yu.M.Berezansky. \emph{Expansion on eigenfunctions of self-adjoint operators}. AMS, Providence, RI, 1968. Trans. from Russian edn., Nakova Dumka, 1965. \bibitem{29} Yu.M.Berezansky. \emph{Commutative Jacobi fields in Fock space}, Int. Equat. Oper. Theory, \textbf{30} (1998), no.~2, 163--190. \bibitem{ber1} Spectral theory of commutative Jacobi fields: direct and inverse problems. {\it Fields Institute Communications}. --- 2000. --- {\bf 25}. --- p.~211--224. \bibitem{ber2} Poisson measure as the spectral measure of Jacobi field. {\it Infinite Dim. Anal., Quant. Prob. and Related Topics}. --- 2000. --- {\bf 3}, no.~1. --- p.~121--139. \bibitem{2} Yu.M.Berezansky, A.A.Kalyuzhnyi. \emph{Harmonic analysis in hypercomplex systems}. Kluwer Acad. Publ. 1998. Transl. from Russian edn., Naukova Dumka, 1992. \bibitem{5} Yu.M.Berezansky, V.L.Ostrovskyi, Yu.S.Samoilenko. \emph{Decomposition on eigenfunctions of families of commuting operators and representations of commutation relations}, Ukr. Math. Zh. \textbf{40} (1988), no.~1, 106--109. \bibitem{bon_pop} A. Bondarenko, S.Popovych. \emph{$C^*$-algebras associated with $\mathcal F_{2^n}$ zero Schwarzian unimodal mappings}. Proc. Int. Conf. Symmetry in Nonlinear Mathematical Physics, part II. Kyiv 2002, p. 425--431. \bibitem{18} O. Bratelli, P.E.T. Jorgensen, V. Ostrovskyi. Representations theory and numerical AF-invariants. The representations and centralizers of certain states on $O_d$. E-pirnt math.OA/9907036 (to appear in Mem Amer. Math. Soc.) \bibitem{dal} A.~Daletskii. \bibitem{dal_lit} A.Daletskii, E.Lytvynov. \bibitem{dalsam} A. Daletskii, Yu. Samoilenko. \emph{A noncommutative moment problem}. Funct. Anal. Prilozh \textbf{21} (1987), no.2, p.72--73. \bibitem{27} T.~Erhardt, V.~Rabanovich, Yu.~Samoilenko, B.~Silberman. \emph{On the decomposition of the identity into a sum of idempotents}, Methods Funct. Anal. Topol. \textbf{7} (2001), no.~2. \bibitem{8} P.E.T.Jorgensen, D.P.Proskurin, Yu.S.Samoilenko. \emph{A family of $*$-algebras allowing Wick ordering: Fock representations and universal enveloping $C^*$-algebras}, Noncommutative stuctures in Mathematics and Phyics, Proc. NATO ARW, Kiev, 2000, p. 321--330 \bibitem{11} P.E.T.Jorgensen, D.P.Proskurin, Yu.S.Samoilenko. \emph{The kernel of Fock representation of Wick algebras with braided operator of coefficients}, Pacific J. Math. \textbf{198} (2001), no.~1, 109--122. \bibitem{13a} P.E.T.Jorgensen, D.P.Proskurin, Yu.S.Samoilenko. Generalized canonical commutation relations: representations and stability of universal enveloping $C^*$-algebra. Proc. Fourth Int. Conf. Symmetry in Nonlinear Mathematical Physics, Part II, Kyiv 2002, p. 456--460. \bibitem{laa} S.Kruglyak, V.Rabanovich, Yu.Samoilenko. \emph{On decomposition of a scalar matrix into sums of idempotents}. Linear Algebra and Applications (to appear). \bibitem{faa} S.Kruglyak, V.Rabanovich, Yu.Samoilenko. \emph{On sums of projections} Functional Anal. Appl. \textbf{36} (2002), no.~4 \bibitem{6} S.A.Kruglyak, Yu.S.Samoilenko. \emph{On complexity of description of representations of $*$-algebras generated by idempotents}, Proc. AMS, \textbf{128} (2000), no.~6, 1655--1664. \bibitem{24} S.A.Kruglyak, Yu.S.Samoilenko. \emph{Structure theorems for families of idempotents}, Ukr. Mat. Zhurn., \textbf{50} (1998), no.~4, 523--533. \bibitem{lyt} E.Lytvynov. \emph{Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density} \bibitem{16} V.Ostrovskyi. \emph{On commutaion relations arising from one-dimensional flow}, Methods Funct. Anal. Topology, \textbf{6} (2000), no.~2, 60--65. \bibitem{10} V.Ostrovskyi, D.Proskurin. \emph{Operator relations, dynamical systems and representations of a clas of Wick algebras}, Operator Theory, Adv. Appl. \textbf{118} (2000), pp.335--345. \bibitem{22} V. Ostrovskyi, Yu. Samoilenko.\emph{Introduction to the theory of representations of finitely presented $*$-algebras. I. Representations by bounded operators}, Rev. Math.\& Math. Phys., 1999, vol. 11, pp. 1--261, Gordon \& Breach, London 1999. \bibitem{15} S.Popovych. \emph{Unbounded idempotents}, Methods Funct. Anal. Topology, \textbf{4} (1998), no.~1, 95--103. \bibitem{28} S.Popovych, T.Maistrenko. \emph{ $C\sp *$-algebras associated with quadratic dynamical system}, Symmetry Nonlin. Math. Phys. (1999), 364--370. \bibitem{14} S.Popovych, T.Maistrenko, \emph{On $C^*$-algebras related to simple unimodal dynamical systems}, Ukr. Mat. Zhurn. \textbf{53} (2001), no.~7 \bibitem{23} D. P. Proskurin. {Homogeneous ideals in Wick $*$-algebras}, { Proc. of Amer. Math. Soc.} (1998) V.{126}, no.~11, p. 3371--3376. \bibitem{12} D.Proskurin. \emph{Stability of special class of $q_{ij}$-CCR and extension of non-commutative tori}, Lett. Math. Phys, \textbf{52} (2000), no.~2, 165--175. \bibitem{9} D.P.Proskurin, Yu.S.Samoilenko, \emph{Representations of Wick CCR algebra}, Spectral and evolutionary problems, 1998, \textbf{8}, p. 43--45. \bibitem{13} D.Proskurin, Yu.Samoilenko. \emph{Stability of a $C^*$-algebra asociated with the twisted canonical commutation relations}, accepted for publication in Algebras and Representations Theory. \bibitem{30} V.I.Rabanovich, Yu.S.Samoilenko. \emph{On representations of $\mathcal {F}_n$-algebras}, Oper. Theory Adv. Appl. \textbf{118} (2000), 347--357. \bibitem{7} V.I.Rabanovich, Yu.S.Samoilenko. \emph{When sum of idempotents is a multiple of identity}, Funct. Anal. Prilozh., \textbf{34} (2000), no.~4, 91--93. \bibitem{26} V.I.Rabanovich, Yu.S.Samoilenko. \emph{Cases in which a scalar operator is a sum of projections}, Ukr. Mat. Zhurn., \textbf{53} (2001), no.~7. \bibitem{25} V.I.Rabanovich, Yu.S.Samoilenko, A.V.Strelets. \emph{On identities in algebras $Q_{n,\lambda}$ generated by idempotents}, Ukr. Mat. Zhurn., \textbf{53} (2001), no.~8. \bibitem{4} Yu.S.Samoilenko. \emph{Spectral theory of families of self-adjoint operators}. Kluwer Acad. Publ., 1991. Transl. from Russian edn., Naukova Dumka, 1984. \bibitem{sam_str_mfat2002} Y.Samoilenko, A.Strelets. On ``good'' vectors for a family of unbounded operators and application. Methods Funct. Anal. Appl. 2002, \bibitem{vin_pop} Vinogradov, Popovych \end{thebibliography} \bigskip\noindent {\large\bf\nopagebreak 3. Aims and work program} \medskip\noindent {\bf 3.1. Aims.}\ \ 1. We plan to select classes of $*$-algebras for which there exists a faithful representations by bounded and unbounded operators, and obtain results similar to Gelfand-Naimark theorem for a wide class of $*$-algebras. In particular, we plan detailed study of $C^*$-representability of algebras generated by words (Lance-Tapper problem, 1996, 1999), enveloping algebras of real Lie groups, PI $*$-algebras, enveloping algebras for algebras having only bounded irreducible representations etc. 2. For classes of algebras generated by projections, we plan to construct functors similar to Coxeter functor used by Gelfand-Ponomarev (1970--1980) in the study of representaions of quivers, and apply them to the study of ses $\Sigma_n=\{\alpha \in \mathbb{R}: \exists P_1,\dots,P_n, \sum P_k=\alpha I\}$, $P_k$ are orthoprojections, and representations of the corresponding algebras $P_{n,\alpha}=\mathbb{C}\langle p_1,\dots,p_n: p_k^2=p_k^*, \sum p_k=\alpha I\rangle$. Using such functors, we plan to study sets of parameters, for which representations exist, for ``discrete spectrum'' describe classes of such representations, study how these algebras are related to Temperley-Lieb algebras $TL_{n,\tau}$ and the corresponding model of statistical physics. We also plan to study representations of $*$-algebras generated by projections related to the H.Weyl problem on spectrum of a sum of operators and to the P.Halmos problem on operators which can be given as a sum of projections. 3. Concrete examples of $*$-algebras arising in Mathematical Physics are examples of so-called ``dynamical'' $*$-algebras. Their representation theory have a fruitful connection with corresponding dynamical systems. The problem here is to find topological invariants of dynamical systems responsible for isomorphism of the corresponding $C^*$-algebras. We plan to apply techniques of non-bijective dynamical systems and based on them commutative models to more detailed study of states and representations of quantum unharmonic oscillator and Cuntz algebras and their generalizations. 4. We plan to investigate interesting examples of commuting families of operators on configuration spaces, in particular, Jacobi fields. We plan to study their spectral measure and properties. We will apply these results to the study of traces on $W^*$-algebras of operators on spaces of symmetric and skew-symmetric differential forms on configuration spaces etc. 5. We plan to study irreducible representations by bounded and unbounded operators of deformed CCR. These deformations include such examples as $q_{ij}$-CCR constructed by M.Bozejko and R.Speicher and twisted CCR studied by W.Pusz and S.L.Woronowicz. It was shown that for these examples the Fock representation is faithful for both $*$-algebra and universal enveloping $C^*$-algebra constructed using bounded representations. Our goal is to prove similar results for classes of Wick algebras with countable number of generators. The probem is also to determine the $C^*$-isomorphism classes for deformed CCR enveloping $C^*$-algebras. In particular we plan to prove the Jorgensen-Schmitt-Werner conjecture about stability of the Cuntz algebras in the class of $q$-CCR algebras and study the extensions of noncommutative higher-dimensional tori. We plan to apply these results to the study of anyon fields. 6. We expect to obtain better estimates for the stability of the Cuntz algebras in the class of Wick algebras and their polynomial deformations; study unharmonic deformations of one- and many-dimensional $q$-CCR. Here, endomorphic mappings arise, and the properties of the dynamical system (cycles, entropy etc.) depend essentially on the values of the deformation parameters. For instance, in the one-dimensional case, the algebras are generated by $x$, $x^*$, subject to the relation $xx^* = 1+ax^*x - b(x^*x)^2$, $a$, $b>0$. The study of the corresponding algebra involves the study of positive orbits of the mapping $t \to 1+at -bt^2$. \vfill\eject \bigskip\noindent {\large\bf 4. Expenses} \medskip {\bf Travels of foreign experts to German partner institutions:} \medskip\smallskip \begin{center} \begin{tabular}{|l|c|c|c|} \hline &1st year&2nd year &3rd year\\ \hline Number of stays&6&6&6 \\ \hline Duration of each stay&2 months &2 months &2 months\\ \hline \end{tabular} \end{center} \bigskip {\bf Travels of German experts to foreign partner institutions:} \medskip \begin{center} \begin{tabular}{|l|c|c|c|} \hline &1st year&2nd year &3rd year\\ \hline Number of stays&6& 6& 6\\ \hline Duration of each stay&$\lefteqn{\rm{\ 2\ weeks}}\hphantom{2 months}$ &\lefteqn{\rm{\ 2\ weeks}}\hphantom{2 weeks} &\lefteqn{\rm{\ 2\ weeks}}\hphantom{2 weeks}\\ \hline \end{tabular} \end{center} \medskip \bigskip\noindent {\large\bf 5. Implementation of the project} \medskip\smallskip\noindent {\bf 5.1. Work group.} \medskip %\noindent {\bf Participants from Germany} {\bf Participants from Ukraine} Yu.S.Samoilenko Yu.M.Berezansky A.Yu.Daletskii V.Ostrovskyi D.Proskurin S.Popovych V.Rabanovich A.Strelets \smallskip \noindent {\bf 5.2. Cooperation with other scientists.} A.Klimyk (Ukraine), V.Shulman (Russia) \noindent {\bf 5.3. Cooperation with foreign scientists.} P.E.T.Jorgensen (USA), K.Schm\"udgen (Germany), B.Silberman (Germany), L.Turowska (Sweden), N.Krupnik (Israel) \medskip\noindent {\bf 5.4. Equipment and consumables.} \vskip2pt\smallskip \begin{tabbing} \qquad\= 2nd year \ \=---\ \ \=8000 DM\kill \>1st year \>--- \>3000 DM\\[3pt] \>2nd year \>--- \> 3000 DM\\[3pt] \>3rd year \>--- \>3000 DM \end{tabbing} \bigskip\noindent {\large\bf 6. Power of attorney} \medskip\noindent This project will not be submitted for financial support to any other organization. In case of doing so, the Deutsche Forschungsgemeinschaft will be notified. \bigskip\bigskip\noindent {\large\bf Signatures} \bigskip\noindent From German Partner\hfill From Ukrainian Partner \vskip18mm \noindent \ \ \quad S. Albeverio\hfill Yu. Samoilenko\ \quad\mbox{} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: