Geodesic
Similarity problem for certain martingale uniform algebras
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 90-102

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Let $A$ be a proper uniform algebra admitting a nontrivial bounded point derivation. Then for a certain uniform algebra $A_1$ (related to $A$ much as the algebra of Hardy martingales on $\mathbb T^\infty$ is related to the disk algebra) there exists a bounded but not completely bounded homomorphism $\varphi\colon A_1\to B(H)$. 
@article{ZNSL_2000_270_a4,
     author = {S. V. Kislyakov},
     title = {Similarity problem for certain martingale uniform algebras},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {90--102},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a4/}
}
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S. V. Kislyakov. Similarity problem for certain martingale uniform algebras. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 90-102. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a4/