A Three-Dimensional Operator System without the Smith--Ward Property

Abstract

Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith--Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span$\{1,q(D),q(K)\}$, where $D$ and $K$ are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator $D+iK$ gives a counterexample to the Smith--Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.

Disclosure

“Acknowledgement The author thanks Orr Shalit for careful proofreading, which led to an improvement in the readability of the paper. AI Disclosure: ChatGPT was used to perform literature search and to accelerate the search for the operator system S. References [1] W. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969), 141–224. [2] L. G. Brown, R. G. Douglas, and P. A. Fillmore,”

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2026-08-24T00:34:05.733Z

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