\(A\)-valued semicircular systems.

*(English)*Zbl 0951.46035Let \(A\) be a von Neumann algebra, \(\gamma_{ij}\) \((i,j\in I)\) be a collection of linear maps from \(A\) to itself, so that the associated map \(\eta: A\to A\otimes B(\ell^2(I))\) given by \(\gamma(a)= (\eta_{i,j}(a))_{i,j\in I}\) is normal and completely positive.

It is associated a von Neumann algebra \(\Phi(A,\eta)\), which is generated by \(A\) and an \(A\)-valued semicircular system \(X_i\) \((i\in I)\), associated to \(\eta\). In the case of a single completely positive map \(\eta: A\to A\), the algebra \(\Phi(A,\eta)\) coincides with the algebra \(W^*(A,X)\), where \(X= L+L^*\), with \(\eta(a)= L^*aL\), \(a\in A\).

It is proved that there exists a conditional expectation from \(\Phi(A,\eta)\) to \(A\), which is under a certain assumption faithful; if \(A\) has a trace, under certain assumptions on \(\eta\), \(\Phi(A,\eta)\) also has a trace. It is shown that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction \(\Phi\). Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form \(\Phi(L^\infty[0, 1],\eta)\). The cores of free Araki-Woods factors are free Krieger algebras. The author studied the free Krieger algebras and as a result obtained several non-isomorphism results for free Araki-Woods factors. It is computed the \(\tau\) invariant of Connes for free products of von Neumann algebras, which generalizes earlier work on computation of \(T\), \(S\), and \(Sd\) invariants for free product algebras.

It is associated a von Neumann algebra \(\Phi(A,\eta)\), which is generated by \(A\) and an \(A\)-valued semicircular system \(X_i\) \((i\in I)\), associated to \(\eta\). In the case of a single completely positive map \(\eta: A\to A\), the algebra \(\Phi(A,\eta)\) coincides with the algebra \(W^*(A,X)\), where \(X= L+L^*\), with \(\eta(a)= L^*aL\), \(a\in A\).

It is proved that there exists a conditional expectation from \(\Phi(A,\eta)\) to \(A\), which is under a certain assumption faithful; if \(A\) has a trace, under certain assumptions on \(\eta\), \(\Phi(A,\eta)\) also has a trace. It is shown that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction \(\Phi\). Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form \(\Phi(L^\infty[0, 1],\eta)\). The cores of free Araki-Woods factors are free Krieger algebras. The author studied the free Krieger algebras and as a result obtained several non-isomorphism results for free Araki-Woods factors. It is computed the \(\tau\) invariant of Connes for free products of von Neumann algebras, which generalizes earlier work on computation of \(T\), \(S\), and \(Sd\) invariants for free product algebras.

Reviewer: Abdugafur Rakhimov (Tashkent)

##### Keywords:

free product algebras; completely positive; semicircular system; completely positive map; conditional expectation; Krieger algebras; crossed products; Krieger factors; free Araki-Woods factors; \(\tau\) invariant of Connes
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\textit{D. Shlyakhtenko}, J. Funct. Anal. 166, No. 1, 1--47 (1999; Zbl 0951.46035)

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