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Unperforated pairs of operator spaces and hyperrigidity of operator systems

  • Raphaël Clouâtre,
    Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Manitoba
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Abstract

We study restriction and extension properties for states on C$^*$-algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson's hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a C$^*$-algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient C$^*$-algebra. We prove that commuting pairs are unperforated, and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.
Keywords: operator system, state, peak point, hyperrigidity conjecture operator system, state, peak point, hyperrigidity conjecture
MSC Classifications: 46L07, 46L30, 46L52 show english descriptions Operator spaces and completely bounded maps [See also 47L25]
States
Noncommutative function spaces
46L07 - Operator spaces and completely bounded maps [See also 47L25]
46L30 - States
46L52 - Noncommutative function spaces
 

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