Non-commutative disc algebras and their representations

Abstract:

It is shown that the smallest closed subalgebra \begin{equation*}Alg(I_{ \mathcal {K}} ,V_{1},\dots ,V_{n})\subset \mathcal {B} (\mathcal {K}) \qquad (n=2,3,\dots ,\infty )\end{equation*} generated by any sequence $V_{1},\dots , V_{n}$ of isometries on a Hilbert space $\mathcal {K}$ such that $V_{1}V_{1}^{*}+\cdots +V_{n}V_{n}^{*}\le I_{\mathcal {K}}$ is completely isometrically isomorphic to the non-commutative “disc” algebra $\mathcal {A} _{n}$ introduced in Math. Scand. 68 (1991), 292–304. We also prove that for $n\ne m$ the Banach algebras $\mathcal {A} _{n}$ and $\mathcal {A} _{m}$ are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the “disc” algebras $\mathcal {A} _{n} (n=2,3,\dots ,\infty )$ on a Hilbert space are characterized. In particular, we prove that a sequence of operators $A_{1},A_{2},\dots$ is simultaneously similar to a contractive sequence $T_{1},T_{2},\dots$ (i.e., $T_{1}T_{1}^{*}+\cdots +T_{n}T_{n}^{*} \le I$ ) if and only if it is completely polynomially bounded. The first cohomology group of $\mathcal {A} _{n}$ with coefficients in $\mathbb {C}$ is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras $F_{n}^{\infty }$ introduced in Math. Scand. 68 (1991), 292–304. The right joint spectrum of the left creation operators on the full Fock space is also determined.