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$$A$$-valued semicircular systems. (English) Zbl 0951.46035
Let $$A$$ 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.

##### MSC:
 46L09 Free products of $$C^*$$-algebras 46L10 General theory of von Neumann algebras
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