# Free C^*-algebra

Let $A_0$ be a set of all polynomials with complex coefficients of infinitely many noncommuting (free) variables, denoted by $X_1,X_2,...,X_1^*,X_2^*,...$. We equip $A_0$ with the operation $*:A_0 \to A_0$ in such a way that this operation becomes involution. For a polynomial $p(X_1,X_2,...,X_1^*,X_2^*,...)$ we define $$\|p\|:=sup\|p(T_1,T_2,...,T_1^*,T_2^*,...)\|$$ where the supremum is taken over all possible collections of contractions $\{T_n\}_n$ in separable infinite dimensional Hilbert space (and $p(T_1,...,T_1^*,...)$ has an obvious meaning). It is not hard to see that this formula defines a seminorm satisfying all conditions for a $C^*$-norm but my question is the following: is it actually the norm? More powerful would be the following: is it possible to find an infinite collection $T_1,T_2,...$ of contractions in separable Hilbert space such that there is no "$C^*$-algebraic" relation relating them? I never saw a proof that the formula above defines a norm but I've heard that at least one author mentions this fact.

See Lemma 3.7 in my paper C*-relations, Math. Scand., 107(1), 43--72 (2010). I show that this seminorm is actually a norm. This is for the finite or infinite case. For the case of countably many generators, one can embed the "free" C*-algebra in $B(H)$ for a separable Hilbert space and conclude you last assertion.
I can't recall who told me this, nor any reference, but I recall that for any NC $*$-polynomial, there are finite matrices, contractions, not satisfying the results. In terms I am more comfortable with, the C*-algebra generated by $n$ universal contractions is RFD, so such matrices exists. (Recall projective implies RFD.)