# Quasi-equivalent representations

Two unitary representations (cf. Unitary representation) $\pi _ {1} , \pi _ {2}$ of a group $X$( or symmetric representations of a symmetric algebra $X$) in Hilbert spaces $H _ {1}$ and $H _ {2}$, respectively, satisfying one of the following four equivalent conditions: 1) there exist unitarily-equivalent representations $\rho _ {1}$ and $\rho _ {2}$ such that $\rho _ {1}$ is a multiple of $\pi _ {1}$ and $\rho _ {2}$ is a multiple of $\pi _ {2}$; 2) the non-zero subrepresentations of $\pi _ {1}$ are not disjoint from $\pi _ {2}$, and the non-zero subrepresentations of $\pi _ {2}$ are not disjoint from $\pi _ {1}$; 3) $\pi _ {2}$ is unitarily equivalent to a subrepresentation of some multiple representation $\rho _ {1}$ of $\pi _ {1}$ that has unit central support; or 4) there exists an isomorphism $\Phi$ of the von Neumann algebra generated by the set $\pi _ {1} ( X)$ onto the von Neumann algebra generated by the set $\pi _ {2} ( X)$ such that $\Phi ( \pi _ {1} ( x) ) = \pi _ {2} ( x)$ for all $x \in X$. Unitarily-equivalent representations are quasi-equivalent representations; irreducible quasi-equivalent representations (cf. Irreducible representation) are unitarily equivalent. If $\pi _ {1}$ and $\pi _ {2}$ are quasi-equivalent representations and $\pi _ {1}$ is a factor representation, then so is $\pi _ {2}$; a factor representation and a non-zero subrepresentation of it are quasi-equivalent representations; two factor representations are either disjoint or quasi-equivalent. The notion of quasi-equivalent representations leads to that of a quasi-dual object and a quasi-spectrum for locally compact groups and symmetric algebras, respectively.

#### References

 [1] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)

Two representations $\pi$ and $\pi ^ \prime$( of a group or algebra) with representation spaces $H$ and $H ^ \prime$, respectively, are said to be disjoint is there is no non-zero intertwining operator between $\pi$ and $\pi ^ \prime$. Here, an intertwining operator between $\pi$ and $\pi ^ \prime$ is a continuous linear operator $T : H \rightarrow H ^ \prime$ such that $T \pi ( x) = \pi ^ \prime T ( x)$ for all $x$.