## 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 $$\ell (n,1)$$.
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

Zbl 0562.00005