Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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Eli Shamovich, Department of Pure Mathematics, University of Waterloo
"Fixed points of selfmaps 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 finitedimensional 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
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca