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Hypercontractivity in group von Neumann algebras. (English) Zbl 1382.43002

Mem. Am. Math. Soc. 1183, xii, 88 p. (2017).
The article under review treats the problem of establishing hypercontractivity estimates for Poisson-type semigroups acting on group von Neumann algebras of discrete groups. The general setup is as follows: \(\Gamma\) is a discrete group. \(\psi: \Gamma \to \mathbb{R}_+\) is a conditionally-negative definite function and we are interested in the semigroup of Herz-Schur type multipliers \(S_{\psi,_t}:VN(\Gamma) \to VN(\Gamma)\) given by the (continuous linear extension of the) formula \[ S_{\psi,_t} (\lambda_\gamma) =\exp(-t \psi(\gamma)) \lambda_\gamma, \quad \gamma \in \Gamma. \; \] The von Neumann algebra \(VN(\Gamma)\) is equipped with a canonical trace \(\tau\) given by the evaluation at the neutral element \(e\); hence one can consider also the corresponding \(L^p\)-type norms on \(VN(\Gamma)\) for \(p \in [1, \infty)\). It is well-known that \(S_{\psi,_t}\) as above is a contraction with respect to each of these norms. We say that \((S_{\psi,_t})_{t \in \mathbb{R}_+}\) satisfies the (optimal) hypercontractivity estimate if there exists \(C >0 \) such that \[ \|S_{\psi,_t}(f)\|q \leq \|f\|_p, \quad f \in VN(\Gamma), 1< p \leq q < \infty, \quad t\geq C \log (\frac{q-1}{p-1}), \] (and \(C>0\) is the smallest number with this property). It is well-known that in this context it suffices to prove the corresponding inequality for \(p=2\), \(q=4\).
The authors develop a general combinatorial technology allowing to obtain such estimates, relating the constant \(C\) to the minimal value of \(\psi\) outside \(e\), for functions \(\psi\) satisfying natural assumptions of ‘spectral gap’ and ‘restricted growth’ and further a condition of ‘no short loops in the minimal set’. It is worth noting that establishing optimal hypercontractivity estimates is notoriously difficult even for finite groups and for example for the cyclic group of order 3 equipped with the length function the optimal bound is not yet known. The techniques presented in the reviewed article yield however optimal or close to optimal results for the free groups, some Coxeter groups and ‘large’ finite cyclic groups. In fact in large parts of the paper the authors do not assume that \(\psi\) is conditionally negative definite (so in particular one needs to consider general parameters \(p\) and \(q\)).
The article begins with a short survey of the known results and applications of hypercontractivity, and ends with appendices describing the technical parts of the proof and outlining the above mentioned argument due to Gross, allowing to deduce the general \((p,q)\)-case, possibly with a suboptimal estimate, from the \((2,4)\)-bound. Although the combinatorial arguments are rather daunting, and occasionally require computer-assisted counting, the main steps and general strategy of the proofs are always carefully explained. Several examples of ‘length functions’ \(\psi\) fitting the framework of the paper are discussed, as well as some implications of the main results for the ultracontractivity questions (i.e. the case where \(q=\infty\)).

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
47D06 One-parameter semigroups and linear evolution equations
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
46L10 General theory of von Neumann algebras
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