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Operator algebraic approach to inverse and stability theorems for amenable groups. (English) Zbl 1416.22008

The “inverse and stability theorems” referred to in the title are generalisations of results of W. T. Gowers and O. Hatami [Sb. Math. 208, No. 12, 1784–1817 (2017; Zbl 1427.20022); translation from Mat. Sb. 208, No. 12, 70–106 (2017)]. Gowers and Hatami studied maps \(f:G\to M_n(\mathbb{C})\), from a finite group \(G\) to the \(n\times n\) complex matrices, proving that: (1) If such a map \(f\) has a relatively large \(U^2\)-norm, then \(f\) correlates (in a suitable sense) with a unitary representation; and, as a consequence, (2) if \(f\) has image in the unitary group, and if \(f\) is almost a homomorphism in the sense that the Hilbert-Schmidt norm of \(f(gh)-f(g)f(h)\) is always small, then \(f\) is close (in a suitable sense) to a unitary representation.
In the paper under review, these results are generalised to maps \(f:G\to M\), where \(G\) is a discrete amenable group, \(M\) is a von Neumann algebra, and the Hilbert-Schmidt norm is replaced by an appropriate kind of seminorm on \(M\), important examples being the Schatten \(p\)-semi-norms associated to a normal trace.
The “operator algebraic approach” indicated in the title involves a judicious application of the Stinespring dilation theorem for completely positive maps [W. F. Stinespring, Proc. Am. Math. Soc. 6, 211–216 (1955; Zbl 0064.36703)]. This gives an affirmative answer to a question raised by Gowers and Hatami [loc. cit.], who asked whether the operator-algebraic proof of their result (2) that had been communicated privately to them by N. Ozawa could be made to work for their result (1) as well.

MSC:

22D10 Unitary representations of locally compact groups
39B82 Stability, separation, extension, and related topics for functional equations
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References:

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