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\noindent{\bf Report on the paper by Yu.Savchuk and K.Schm\"udgen \\ ``Unbounded induced representations of $*$-algebras''}

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The paper under review deals with $*$-representations of $*$-algebras, that is, complex associative algebras with involution. As the authors state, the purpose of the paper is to develop the basics of a theory of \textit{unbounded} induced $*$-representations. It is fairly self-contained, however requiring basic knowledge of unbounded $*$-representations.

After the first section that sets down preliminaries, in Sections 2,3 the authors define the basic construction of induced $*$-representation. The most important notions are conditional expectation from a $*$-algebra $\cA$ onto a $*$-subalgebra $\cB$ and the induction procedure from $\cB$ to $\cA.$ The special cases of this induction include: induction of unitary representations of discrete groups, the GNS-construction for $*$-algebras and Rieffel's induction for $C^*$-algebras.

Sections 4,5,6 deal with the case of group graded $*$-algebras. If $\cA=\oplus_{g\in G}\cA_g$ is a $G$-graded $*$-algebra and $H\subseteq G$ is a subgroup, then there is an induction procedure from the $*$-subalgebra $\cA_H=\oplus_{g\in H}\cA_g$ to $\cA.$ In this context the authors define the basic notions of the Mackey analysis, such as systems of imprimitivity and prove different versions of the Imprimitivity Theorem.

A fundamental problem in unbounded representation theory is to define and characterize well-behaved representations of a general $*$-algebra. In Sections 7,8 the authors develop a new concept of well-behaved representations for group-graded $*$-algebras. Among important examples where the theory implies are $q$-deformed enveloping algebras, $q$-CCR algebras, Virasoro algebra and others. In Sections 9,10,11 a number of examples are carried out in great detail. Authors approach is indeed universal, however for some concrete examples there exist better approaches. For example, the description of irreducible representations of the algebra $\cA=\mathbb{C}\langle a,a^*\ |\ aa^*=f(a^*a)\rangle,\ f\in \mathbb{R}[t]$ is done much easier by the method developed by Ostrovskyi, V. and Samoilenko, Yu. in their monograph "Introduction to the theory of representations of finitely presented *-algebras."

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The paper is a valuable contribution to representation theory of $*$-algebras. I highly recommend it to be published in your journal.

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With best regards,

Yurii Samoilenko
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