Mathematics – Operator Algebras
Scientific paper
2012-01-19
Mathematics
Operator Algebras
Scientific paper
We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\cM_V$ of the multiplier algebra $\cM$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball. The related algebras of continuous multipliers are also considered. We find that $\cM_V$ is completely isometrically isomorphic to $\cM_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. A similar condition characterizes when there exists a unital completely contractive homomorphism from $\cM_V$ to $\cM_W$. If one of the varieties is a homogeneous algebraic variety, then isometric isomorphism is shown to imply completely isometric isomorphism of the algebras. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. It is shown that if there is an isomorphism between $\cM_V$ and $\cM_W$, then there is a biholomorphism (with multiplier coordinates) between the varieties. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold---particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.
Davidson Kenneth R.
Ramsey Christopher
Shalit Orr Moshe
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