**
Eli
Shamovich,
Department
of
Pure
Mathematics,
University
of
Waterloo**

"Fixed points of self-maps of the free ball"

In this talk I will discuss classification of quotients of the free semigroup algebra $\mathcal{L}_d$, the weak-* closed algebra generated by the $d$-shift on the full Fock space, using geometric data. Let $I \subset \mathcal{L}_d$ be a weak-* closed ideal and set $A = \mathcal{L}_d/I$. We associate to $A$ a noncommutative variety, namely the set of all unital completely contractive representations of $A$ on finite-dimensional spaces, parametrized by $d$-tuples of matrices of all sizes (the values on the generators). Let us write $\mathbb{M}_d = \sqcup_{n=1}^{\infty} M_n(\C)^{\oplus d}$, then our variety is a subset of the free ball $\mathfrak{B}_d = \left\{ X \in \M_d \mid X X^* < I \right\}$. Given two algebras as above $A_1$ and $A_2$, when are they completely isometrically isomorphic (c.i.i.)? Jointly with Salomon and Shalit we have proved that they are c.i.i. if and only if there exists a pair of nc functions $f,g \colon \mathfrak{B}_d \to \mathfrak{B}_d$, such that $(f \circ g)|_{V_2} = \operatorname{id}_{V_2}$ and $(g \circ f)|_{V_1} = \operatorname{id}_{V_1}$. In this talk, I will extend this result and prove that if the varieties contain a scalar point, then $f$ and $g$ are M\"{o}bius maps and the algebras are unitarily equivalent. The key tool in the proof is a noncommutative analog of a classical result of Rudin and Herv\'{e} on fixed points of self maps of the unit ball of $\mathbb{C}^d$.

MC 5417