##
**Dilations for representations of triangular algebras.**
*(English)*
Zbl 0721.46034

Given a (not necessarily selfadjoint) subalgebra A of a unital \(C^*\)- algebra B and a contractive representation \(\rho\) of A on a Hilbert space \({\mathcal H}\), a dilation of \(\rho\) is a triple (\(\pi\),V,\({\mathcal K})\), where \(\pi\) is a *-representation of B on a Hilbert space \({\mathcal K}\) and V is an isometric mapping \({\mathcal H}\) into \({\mathcal K}\) such that \(\rho (a)=V^*\pi (a)V\) for all \(a\in A\). A dilation is said to be minimal if the smallest subspace of \({\mathcal K}\) that reduces \(\pi\) and contains the range of V is \({\mathcal K}\) itself. This notion of dilation was introduced by W. B. Arveson [Acta Math. 123, 141-224 (1969; Zbl 0194.157)], where he showed that a dilation exists precisely when \(\rho\) is completely contractive.

In the present note, the authors consider the situation when B is a von Neumann algebra, A is \(\sigma\)-weakly closed, and \(\rho\) is \(\sigma\)- weakly continuous.

The main result of the paper can be stated as follows. Let M be a hyperfinite von Neumann algebra and let \({\mathcal T}\) be a \(\sigma\)-weakly closed subalgebra of M such that

(a) \({\mathcal T}\) is a \(\sigma\)-Dirichlet subalgebra of M (i.e. \(A+A^*\) is \(\sigma\)-weakly dense in M);

(b) \({\mathcal T}\) contains a Cartan subalgebra of M (i.e. a subalgebra whose normalizer in M generates M and onto which there is a faithful normal expectation of M).

Then every \(\sigma\)-weakly continuous contractive representation of T has a unique (up to unitary equivalence) minimal dilation, which is a normal *-reprsentation of M.

The proof uses earlier work of J. Feldman and C. Moore [Trans. Am. Math. Soc. 234, 289-324 and 325-359 (1977; Zbl 0369.22009 and Zbl 0369.22010)], as well as work of the present authors and K.-S. Saito [Ann. of Math., II. Ser. 127, No.2, 245-278 (1988; Zbl 0649.47036)], giving a more concrete realization of M and \({\mathcal T}\). The paper ends with a result asserting that, if M and \({\mathcal T}\) are as above, \(\rho\) is a \(\sigma\)-weakly continuous representation of \({\mathcal T}\) on \({\mathcal H}\) with minimal dilation (\(\pi\),V,\({\mathcal K})\), and S is an operator on \({\mathcal H}\) commuting with \(\rho\) (\({\mathcal T})\), then there is an operator \(\tilde S\) on \({\mathcal K}\) commuting with \(\pi\) (M) such that \(S=V^*\tilde SV\) and \(\| \tilde S\| =\| S\|\).

In the present note, the authors consider the situation when B is a von Neumann algebra, A is \(\sigma\)-weakly closed, and \(\rho\) is \(\sigma\)- weakly continuous.

The main result of the paper can be stated as follows. Let M be a hyperfinite von Neumann algebra and let \({\mathcal T}\) be a \(\sigma\)-weakly closed subalgebra of M such that

(a) \({\mathcal T}\) is a \(\sigma\)-Dirichlet subalgebra of M (i.e. \(A+A^*\) is \(\sigma\)-weakly dense in M);

(b) \({\mathcal T}\) contains a Cartan subalgebra of M (i.e. a subalgebra whose normalizer in M generates M and onto which there is a faithful normal expectation of M).

Then every \(\sigma\)-weakly continuous contractive representation of T has a unique (up to unitary equivalence) minimal dilation, which is a normal *-reprsentation of M.

The proof uses earlier work of J. Feldman and C. Moore [Trans. Am. Math. Soc. 234, 289-324 and 325-359 (1977; Zbl 0369.22009 and Zbl 0369.22010)], as well as work of the present authors and K.-S. Saito [Ann. of Math., II. Ser. 127, No.2, 245-278 (1988; Zbl 0649.47036)], giving a more concrete realization of M and \({\mathcal T}\). The paper ends with a result asserting that, if M and \({\mathcal T}\) are as above, \(\rho\) is a \(\sigma\)-weakly continuous representation of \({\mathcal T}\) on \({\mathcal H}\) with minimal dilation (\(\pi\),V,\({\mathcal K})\), and S is an operator on \({\mathcal H}\) commuting with \(\rho\) (\({\mathcal T})\), then there is an operator \(\tilde S\) on \({\mathcal K}\) commuting with \(\pi\) (M) such that \(S=V^*\tilde SV\) and \(\| \tilde S\| =\| S\|\).

Reviewer: T.A.Gillespie (Edinburgh)

### MSC:

46L10 | General theory of von Neumann algebras |

47L30 | Abstract operator algebras on Hilbert spaces |

47A20 | Dilations, extensions, compressions of linear operators |

47A66 | Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators |