 
Summary: MULTIPLIERS AND REPRESENTATIONS
OF NONCOMMUTATIVE DISC ALGEBRAS
Alvaro Arias
Abstract. The noncommutative disc algebra An, n 2, is the norm closure of
the nonselfadjoint algebra generated by the left regular representation of F+n , the
free semigroup on n generators. We present examples of contractive representation
of An, n 2, which are not completely contractive. This answers a question from
Po7]. We characterize the (completely) contractive Schur multipliers of An which
are indexed by subsets of F+n , and we show that for n 2, B(F+n ) M0(F+n ) and
M0(F+n )nB(F+n ) 6= . B(F+n ) is the space of coe cients of contractive representations
of F+n and M0(F+n ) is the space of completely bounded Schur multipliers of An.
0. Introduction
In this paper we study some properties of the noncommutative disc algebras
An, n 2. These nonselfadjoint algebras were introduced in 1991 by Popescu. In
a sequence of papers (see Po2], Po3], Po4], Po5], Po6], and Po7]) he established
remarkable similarities between these algebras and classical spaces appearing in
Harmonic Analysis. This line of study has been pursued recently by Davidson and
Pitts (see DP1] and DP2]). We also refer to A], APo], and DPo] for related
results.
For the moment, it su ces to say that An is an algebra generated by n isome
