Constructing gradient-S_p quantum Markov semi-groups to obtain strong solidity results for von Neumann algebras

Abstract

In this thesis we will for a
quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a
trace τ , investigate the property of the semi-group being gradient-Sp for some
p ∈ [1, ∞]. This property was introduced
in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov
semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these
classes the property gradient-Sp has not been studied; in particular for groups
and their operator algebras no (non-trivial) examples were known before this
thesis. The main aim of this thesis is therefore to construct interesting
examples of quantum Markov semi-groups that possess the gradient-Sp property. The
reason why we are interested in constructing such semi-groups, is because they
can be used to obtain properties like the Akemann-Ostrand property (AO+) and
strong solidity for the underlying von Neumann algebra. Over the last decade,
these properties have become a topic of interest and have been studied for
several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In
this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain
discrete groups Γ that possess the Haagerup property. Namely, for such groups
there exists a proper, conditionally negative definite function ψ on Γ. We can
then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as
∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆
)t≥0. For this semi-group we can investigate for what p it has the gradient-Sp
property. In particular we will be considering group von Neumann algebras of
Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by
[4], and a proper conditionally negative function ψ on W is given by the
minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost
completely’ characterize for what types of Coxeter systems the semi-group
corresponding to the word length is gradient-Sp. Moreover, in the cases that we
get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and
strong solidity for L(W). Hereafter, we will also consider other quantum Markov
semi-groups on L(W). We consider word lengths that arise by putting different
weights on the generators, and consider the semi-groups associated to these
proper, conditionally negative functions. From this we obtain (AO+) and strong
solidity for L(W) for some other cases. Thereafter, we will generalize some of
our results obtained for L(W) to the Hecke algebras Nq(W), which are
q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for
general groups, we shall examine for semi-groups induced by a proper,
conditionally negative function ψ, how the gradient-Sp property of the
semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the
semi-group that is generated by the αth-root ∆α of the generator. Last, we will
also show a method that allows us, for right-angled word hyperbolic Coxeter
groups, to obtain (AO+) and strong solidity for L(W) without building a
gradient-Sp quantum Markov semi-group, but by using a slightly different
method.