Journal of Operator Theory

Volume 55, Issue 1, Winter 2006 pp. 3-16.

Two reformulations of Kadison's similarity problem

Authors: Donald Hadwin (1) and Vern I. Paulens (2)
Author institution: (1) Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA
(2) Department of Mathematics, University of Houston, Texas 77204-3476, USA

Summary: First, we prove that Kadison's similarity problem is equivalent to a problem about the invariant operator ranges of a single operator. We construct an operator $T$ on a separable Hilbert space such that Kadison's problem is equivalent to deciding if Dixmier's invariant operator range problem is true for each of the operators $\{ T \otimes I_{n} \},$ where $I_{n}$ denotes the identity operator on a Hilbert space of dimension $n$ with $n$ a countable cardinal. We prove that the answer to Dixmier's invariant operator range problem is affirmative when $n$ is finite. Second, using Pisier's theory of similarity and factorization degree, we prove that the answer to Kadison's problem is affirmative if and only if there exists a "universal factorization formula" of the type considered by Pisier, consisting of a particular set of scalar matrices and a set of polynomials in non-commuting variables. This formula would factor matrices over \emph{any} $C^*$-algebra into products of scalar matrices and diagonal matrices, where the entries of the diagonal matrices are determined by the non-commutative polynomials.

Keywords: completely bounded, similarity, operator range

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