Similarity problem for certain martingale uniform algebras (Q1405662)

Solving a major long-standing conjecture, \textit{G.~Pisier} [Taiwanese J. Math. 5, 1-17 (2001; Zbl 0999.46025)] proved that there exists a bounded, but not completely bounded, homomorphism \(\varphi:C_A\to B(H)\), where \(C_A\) is the disk algebra and \(B(H)\) the algebra of all bounded operators on a Hilbert space. This article pursues the question (asked by Pisier) of whether \(C_A\) can be replaced by other uniform algebras, and gives a very clear introduction to the problems connected with the endeavor (the conjecture whether this is possible is still open). The partial result proved in this paper requires a uniform algebra \(A\) to possess a nontrivial bounded point derivation. If this is the case, and if \(\nu\) is a point derivation measure for \(A\), then one builds the so-called martingale extension algebra \(A_1 := \text{Mart} (A,\nu)\). The author is then able to extend Pisier's argument to prove that there exists a bounded, but not completely bounded, homomorphism from \(A_1\) to \(B(H)\).