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On the complexity of description of representations of $*$-algebras generated by idempotents

Authors: Stanislav Krugliak and Yurii Samoilenko
Journal: Proc. Amer. Math. Soc. 128 (2000), 1655-1664
MSC (2000): Primary 46K10, 46L05; Secondary 16G60
Published electronically: February 16, 2000
MathSciNet review: 1636978
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In this paper, we introduce a quasiorder $\succ$ (majorization) on $*$-algebras with respect to the complexity of description of their representations. We show that $C^*({\mathcal F}_2) \succ \mathfrak A $ for any finitely generated $*$-algebra $\mathfrak A$ (algebras $\mathfrak B$ such that $\mathfrak B\succ C^*({\mathcal F}_2)$ are called $*$-wild). We show that the $*$-algebra generated by orthogonal projections $p$, $p_1$, $p_2$, ..., $p_n$ ($p_ip_j=0$ for $i\neq j$) is $*$-wild if $n\geq 2$. We also prove that $*$-algebras generated by a pair of idempotents and an orthogonal projection, or by a pair of idempotents $q_1$, $q_2$ ( $q_1q_2=q_2 q_1=0$), etc., are $*$-wild.

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Additional Information

Stanislav Krugliak
Affiliation: Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchinkivs’ka, 3, Kiev, 252001, Ukraine

Yurii Samoilenko
Affiliation: Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchinkivs’ka, 3, Kiev, 252001, Ukraine

Keywords: Involutive algebras, idempotents, orthogonal projections, $*$-representations, irreducible representations, majorizing of representations, $*$-wildness
Received by editor(s): February 5, 1997
Received by editor(s) in revised form: May 17, 1998
Published electronically: February 16, 2000
Additional Notes: This work has been supported in part by the Ukrainian Committee for Fundamental Studies and by CRDF grant no. UM1-311
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 2000 American Mathematical Society