Free probability on $$W^{*}$$ W ∗ -dynamical systems determined by $$GL_{2}(\mathbb {Q} _{p})$$ G L 2 ( Q p ) : generalized Hecke algebras

Abstract

In this paper, we generalize classical Hecke algebras \(\mathcal {H}(G_{p})\) over the generalized linear groups \(G_{p} = GL_{2}(\mathbb {Q}_{p})\) induced by the p-adic number fields \(\mathbb {Q}_{p}\), for primes p. For a given group \(G_{p},\) construct a suitable semigroup \(W^{*}\)-dynamical system \((M, \sigma (G_{p}), \pi ),\) where M is a fixed von Neumann algebra, and \( \pi \) is a semigroup-action of the \(\sigma \)-algebra \(\sigma (G_{p})\) of \( G_{p}\) acting on M. By constructing the corresponding crossed product \( W^{*}\)-algebra \(M \times _{\pi } \sigma (G_{p})\) generated by \((M, \sigma (G_{p}), \pi ),\) we study free probability on the \(W^{*}\) -subalgebra \(\mathcal {H}_{M}(G_{p})\) of \(M \times _{\pi } \sigma (G_{p})\) . One can understand our von Neumann algebra \(\mathcal {H}_{M}(G_{p})\) as a generalized \(*\)-algebra over both M and a Hecke algebra \(\mathcal {H} (G_{p}),\) for a prime p.