Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

Abstract

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's L p -spaces.

Gradient forms and strong solidity of free quantum groups

“…We need a slightly more general version going beyond the case of tracial states of their construction. Note that we do not prove the existence of a square root in the non-tracial setting however (which is one of the main results of [21]; the question is also asked for in [60]).…”

“…their generators) to obtain strong solidity for all free orthogonal and unitary quantum groups. Dirichlet forms have been studied extensively [18,[20][21][22]29,35,58,60]. In particular in [21] it was shown that in the tracial case a Dirichlet form always leads to a derivation as a square root.…”

Generating functionals for locally compact quantum groups

“…Only the relevant parts are stated; for the rest, see [SkV]. We take the opportunity to fix a mistake in the statement of [SkV,Theorem 0.1]: the words 'modulo multiplication of forms by a positive number' should have been 'modulo subtracting a positive multiple of the quadratic form • 2 ', see Remark 1.5.…”

“…(2,ϕ) µt for all t ≥ 0 ((a)⇔(b)) and the general correspondence between selfadjoint completely Markov semigroups and completely Dirichlet forms [SkV,Corollary A.8] ((b)⇔(c)); the latter means that (S t ) t≥0 = (e −tA ) t≥0 , where A is the positive selfadjoint operator on L 2 (G) such that Q = A 1/2 • 2 . Remark 1.5.…”

“…Remark 1.5. A w * -continuous convolution semigroup of contractive positive functionals can be normalised to form one consisting of states [SkV,Remark 3.3]. To restate the 1 − 1 correspondence of Theorem 1.4 (a)⇔(c) for states one has to 'normalise' the completely Dirichlet form as well; the following text should therefore be added to the statement of (c): modulo subtracting from Q a positive multiple of the quadratic form • 2 .…”