Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riesz transforms, Hodge-Dirac operators and functional calculus for
  multipliers I   In this work, we solve the problem explicitly stated at the end of a paper of
Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups
including all amenable groups and free groups. More precisely, we prove that
the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry
associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is
bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a
bisector which implies a positive answer to the quoted problem. Our result can
be seen as a strengthening of the dimension free estimates of Riesz transforms
of the above authors and also allows us to provide Hodge decompositions. A part
of our proof relies on a new transference argument between multipliers which is
of independent interest. Our results are even new for the Poisson semigroup on
$\mathbb{T}^n$. We also provide a similar result for Markov semigroups of Schur
multipliers and dimension free estimates for noncommutative Riesz transforms
associated with these semigroups. Along the way, we also obtain new Khintchine
type equivalences for $q$-Gaussians in $\mathrm{L}^p$-spaces associated to
crossed products. Our results allow us to introduce new spectral triples (i.e.
noncommutative manifolds) and new quantum (locally) compact metric spaces, in
connection with the carr\'e du champ, which summarize the underlying geometry
of our setting. Finally, our examples lead us to introduce a Banach space
variant of the notion of spectral triple suitable for our context.

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The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn