OnC *-algebras having linear,polynomial and subexponential growth

Summary Answering a question of Voiculescu 16, Problem 5.9], we show thatC ^*-algebras having filtrations (A _n)_n satisfying the condition lim sup_n ln dimA _n/n=0 (in particular having subexponential growth), are nuclear.For the case of linear growth we obtain the following particular result: letX be a finite dimensional self-adjoint generating system of aC ^*-algebraA such that dim (span (X ⁿ +1))1+dim (span (X ⁿ )), then there exist a finite dimensionalC ^*-algebraC having only irreducible representations of dimension 1 and aC ^*-algebraB, which is generated by a single self-adjoint element, such thatACB.Some other results are given on linear growth and we show that there exist singly generatedC ^*-algebras such that the growth of the filtration (span (X ⁿ ))_n is polynomial, whereX={x,x ^*, 1} is a generating system, and such that in every neighbourhood ofx there exists an invertibley such thatY={1,y,y ^*} is a generating system whose associated filtration (span (Y ⁿ ))_n doesn't satisfy the previous condition of Voiculescu, and in particular does not have subexponential growth.Oblatum 19-X-1991Work partially supported by DFGAllocataire M.R.T., Université Aix-Marseille II (France)