By Kennison L. S.

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Was partially supported by CONICET, Agencia C´ordoba Ciencia, ANPCyT-FONCyT, TWAS (Trieste) and Secyt (UNC). A. thanks Sonia Natale and Blas Torrecillas for interesting conversations. S. was partially supported by CSIC-Udelar and Dinacyt-MEC, Uruguay. S. thanks Ignacio L´ opez for many exchanges concerning category theory. References [BK] [Be] [CPS] [ENO] [EO] [F] [FR] [FK] [H] [M] [NT] [O1] [O2] B. Bakalov and A. , Lectures on tensor categories and modular functors, University Lecture Series 21, Amer.

Xn ) in the free group generated by x1 , . . , xn such that w(u1 , . . , un ) = 1 for all u1 , . . , un ∈ S ∩ U(A). Since results about symmetric units in rings with involution seem to be difficult to obtain, one way to begin the study of these units is to try to mimic known results for the symmetric elements. A fundamental result of this kind is the following theorem of Amitsur [1]. 2. [1] If R is an algebra with involution whose symmetric elements S satisfy a polynomial identity, then R itself satisfies a polynomial identity.

Several papers have dealt with questions of how various algebraic properties of the set R∗ affect the structure of the whole ring. Similar question may be posed by making assumptions about the symmetric units or subgroup they generate. 1. Let A be an R-algebra and S ⊂ A be a subset. We say that S satisfies a polynomial identity (PI for short) if there exists a nonzero 43 © 2006 by Taylor & Francis Group, LLC 44 O. Broche Cristo and M. Ruiz Mar´ın polynomial f (z1 , . . , zn ) in the polynomial ring R z1 , .