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**Symmetries of some reduced free product \(C^ *\)-algebras.**
*(English)*
Zbl 0618.46048

Operator algebras and their connections with topology and ergodic theory, Proc. Conf., Buşteni/Rom. 1983, Lect. Notes Math. 1132, 556-588 (1985).

[For the entire collection see Zbl 0562.00005.]

Reduced free products of \(C^*\)-algebras with specified states are constructed and the symmetries of two types of \(C^*\)-algebras arising as free products are studied. The first type arises from the constructions on the Fock-space for Boltzman statistics, which gives an action of the orthogonal group of an n-dimensional real Hilbert space on the factor of the free group on n generators, which is applied to obtain a non commutative central limit theorem with appropriate definition of independent random variables. The other type deals with certain extensions of the Cuntz-algebras on the Fock-space for Boltzman statistics, whose automorphism groups are U(n,1). Also is given a representation of U(n,1) along with its decomposition into irreducible ones and its relation to the Lie superalgebra \(\ell (n,1)\).

Reduced free products of \(C^*\)-algebras with specified states are constructed and the symmetries of two types of \(C^*\)-algebras arising as free products are studied. The first type arises from the constructions on the Fock-space for Boltzman statistics, which gives an action of the orthogonal group of an n-dimensional real Hilbert space on the factor of the free group on n generators, which is applied to obtain a non commutative central limit theorem with appropriate definition of independent random variables. The other type deals with certain extensions of the Cuntz-algebras on the Fock-space for Boltzman statistics, whose automorphism groups are U(n,1). Also is given a representation of U(n,1) along with its decomposition into irreducible ones and its relation to the Lie superalgebra \(\ell (n,1)\).

Reviewer: T.V.Panchapagesan

### MSC:

46L05 | General theory of \(C^*\)-algebras |

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

82B10 | Quantum equilibrium statistical mechanics (general) |

60F05 | Central limit and other weak theorems |