View source for Operator-irreducible representation

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A representation $\pi$ of a group (algebra, ring, semi-group) $X$ on a (topological) vector space $E$ such that any (continuous) linear operator on $E$ commuting with every operator $\pi(x)$, $x\in X$, is a scalar multiple of the identity operator on $E$. If $\pi$ is a completely-irreducible representation (in particular, if $\pi$ is a finite-dimensional irreducible representation), then $\pi$ is an operator-irreducible representation; the converse is not always true. If $\pi$ is a [[Unitary representation|unitary representation]] of a group or a symmetric representation of a symmetric algebra, then $\pi$ is an operator-irreducible representation if and only if $\pi$ is an [[Irreducible representation|irreducible representation]].



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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   M.I. Graev,   N.Ya. Vilenkin,   "Generalized functions" , '''5. Integral geometry and representation theory''' , Acad. Press  (1966)  pp. 149 ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  pp. 114  (Translated from Russian)</TD></TR></table>