We study homomorphisms ```
$\rho_{V}$($\rho_{V}(f)=\left (
\begin{smallmatrix}
f(w)I_n& \sum_{i=1}^{m} \partial_if(w)V_{i} \\
0 & f(w)I_n
\end{smallmatrix}\right ), f \in \mathcal O(\Omega_\mathbf A)$
```

) defined on
```
$\mathcal
O(\Omega_\mathbf A)$
```

, where `$\Omega_\mathbf A$`

is a bounded
domain of the form:
\(\begin{eqnarray*}
\Omega_\mathbf A & := &\{(z_1 ,z_2, \ldots, z_m) :\|z_1 A_1
+\cdots + z_mA_m \|_{\rm op} < 1\}
\end{eqnarray*}\)
for some choice of a linearly independent set of `$n\times n$`

matrices `$\{A_1, \ldots, A_m\}.$`

From the work of V. Paulsen and E. Ricard, it follows that if
`$n\geq 3$`

and `$\mathbb B$`

is any ball in `$\mathbb C^m$`

, then there exists
a contractive linear map which is not complete
contractivity. It is known that contractive homomorphisms of the
disc and the bi-disc algebra are completely contractive, thanks
to the dilation theorem of B. Sz.-Nagy and Ando. However, an
example of a contractive homomorphism of the (Euclidean) ball
algebra which is not completely contractive was given by G. Misra. The
characterization of those balls in `$\mathbb C^2$`

for which
contractive linear maps which are always comletely contractive
remained open. We answer this question.

The class of homomorphism of the form `$\rho_V$`

arise from
localization of operators in the Cowen-Douglas class of $\Omega.$ The
(complete) contractivity of a homomorphism in this class
naturally produces inequalities for the curvature of the
corresponding Cowen-Douglas bundle. This connection and some of
its very interesting consequences are discussed.

- All seminars.
- Seminars for 2014

Last updated: 11 Dec 2023