## Annals of Functional Analysis

### Hyperrigid operator systems and Hilbert modules

#### Abstract

It is shown that, for an operator algebra $A$, the operator system $S=A+A^{*}$ in the $C^{*}$-algebra $C^{*}(S)$, and any representation $\rho$ of $C^{*}(S)$ on a Hilbert space $\mathcal{H}$, the restriction $\rho_{|_{S}}$ has a unique extension property if and only if the Hilbert module $\mathcal{H}$ over $A$ is both orthogonally projective and orthogonally injective. As a corollary we deduce that, when $S$ is separable, the hyperrigidity of $S$ is equivalent to the Hilbert modules over $A$ being both orthogonally projective and orthogonally injective.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 1 (2017), 133-141.

Dates
Received: 17 February 2016
Accepted: 1 August 2016
First available in Project Euclid: 12 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1478919627

Digital Object Identifier
doi:10.1215/20088752-3773182

Mathematical Reviews number (MathSciNet)
MR3572336

Zentralblatt MATH identifier
1369.46051

#### Citation

Shankar, P.; Vijayarajan, A. K. Hyperrigid operator systems and Hilbert modules. Ann. Funct. Anal. 8 (2017), no. 1, 133--141. doi:10.1215/20088752-3773182. https://projecteuclid.org/euclid.afa/1478919627

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