Bhattacharyya, Tirthankar and Misra, Gadadhar (2005) Contractive and Completely Contractive Homomorphisms of Planar Algebras. In: Illinois Journal of Mathematics, 49 (4). pp. 11811201.

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Abstract
We consider contractive homomorphisms of a planar algebra $A(\Omega)$ over a nitely connected bounded domain $\Omega \subseteq C$ and ask if they are necessarily completely contractive. We show that a homomorphism $\rho : A(\Omega) \rightarrow B(H)$ for which $dim(A( \Omega )= ker \rho) = 2$ is the direct integral of homomorphisms $\rho T$ induced by operators on twodimensional Hilbert spaces via a suitable functional calculus $\rho T : f \mapsto f(T)$; $f \in A(\Omega)$. It is well known that contractive homomorphisms $\rho T$ induced by a linear transformation $T : C^2 \rightarrow C^2$ are necessarily completely contractive. Consequently, using Arveson's dilation theorem for completely contractive homomorphisms, one concludes that such a homomorphism $\rho T$ possesses a dilation. In this paper, we construct this dilation explicitly. In view of recent examples discovered by Dritschel and McCullough, we know that not all contractive homomorphisms $\rho T$ are completely contractive even if T is a linear transformation on a finitedimensional Hilbert space. We show that one may be able to produce an example of a contractive homomorphism $\rho T$ of $A(\Omega)$ which is not completely contractive if an operator space which is naturally associated with the problem is not the MAX space. Finally, within a certain special class of contractive homomorphisms $\rho T$ of the planar algebra $A(\Omega)$, we construct a dilation.
Item Type:  Journal Article 

Publication:  Illinois Journal of Mathematics 
Publisher:  University of Illinois Press 
Additional Information:  Copyright of this article belongs to University of Illinois Press. 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  21 Jul 2008 
Last Modified:  19 Sep 2010 04:47 
URI:  http://eprints.iisc.ac.in/id/eprint/15161 
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