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\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \newtheorem{theorem}{Theorem} \title{On representations of $*$-algebras related to extended Dynkin graphs} \author{Vasyl Ostrovskyi} \begin{document} \maketitle In several recent papers (see e.g.~\cite{sam_etal1} and references therein) algebras generated by families on idempotents satisfying a linear relation and their representations were studied. To such algebra one can put into correspondence certain simply laced graph. Of special interest are the algebras to which there correspond extended Dynkin diagrams. Representations of four-tuples of projections whose sum is equal to~$2I$ (to this algebra there corresponds~$\tilde D_4$) were studied in~\cite{os_book}, representations of the $*$-algebra generated by three partial reflections whose sum is zero (the corresponding graph is $\tilde E_6$) were discussed in~\cite{anton}. We consider representations of the $*$-algebra \begin{gather*} \mathbb C \Bigl \langle p_1,p_2,p_3,q_1,q_2,q_3,r\Bigm | p_j p_k = \delta_{jk} p_j=p_j^*, q_jq_k = \delta_{jk} q_j=q_j^*, r^2=r^*=r, \\ p_1+2p_2+3p_3+q_1+2q_2+3q_3+2r = 4e\Bigr \rangle \end{gather*} or equivalently, pairs of self-adjoint operators $A$, $B$ having their spectrum in the set $\{\pm 1/2, \pm3/2\}$ such that $(A+B)^2 = I$. According to \cite{sam_etal1} to this algebra there corresponds the $\tilde E_7$ extended Dynkin diagram. We describe the structure of irreducible $*$-representations of this algebra up to a unitary equivalence. \begin{thebibliography}{9} \bibitem{sam_etal1} M.A.Vlasenko, A.S.Mellit, Yu.Samoilenko. \emph{On algebras generated by linearly connected generators with a given spectrum}, Funct. Anal. Appl. \textbf{38} (2004), no.~2. \bibitem{os_book} V.Ostroskyi, Yu.Samoilenko. \emph{Introduction to the theory of representations of finitely presented *-algebras. Representations by buonded operators}, Rev. Math.\& Math. Phys., 1999, vol. 11, pp. 1--261, Gordon \& Breach, London 1999. \bibitem{anton} A.S.Mellit. \emph{On the case where a sum of three partial maps is equal to zero}, Ukr. Math. J., \textbf{55} (2003), no.~9, pp.~1277--1283. \end{thebibliography} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
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