Abstract
In this chapter, we start by giving an overview of quantum (locally) compact metric spaces. Then, we show that we can associate quantum compact metric spaces to some Markov semigroups of Fourier multipliers satisfying additional conditions: an injectivity and a gap condition on the cocycle which represents the semigroup, and the finite dimensionality (with explicit control on p) of the cocycle Hilbert space. We show a similar result for semigroups of Schur multipliers and obtain a quantum locally compact metric space. We further explore the connections of our gap condition between Fourier multipliers and Schur multipliers with some examples. In the sequel, we introduce spectral triples (= noncommutative manifolds) associated to Markov semigroups of Fourier multipliers or Schur multipliers satisfying again some technical conditions, and in all we investigate four different settings. Along the way, we introduce a Banach space variant of the notion of spectral triple suitable for our context. Finally, we investigate the bisectoriality and the functional calculus of some Hodge-Dirac operators which are crucial in the noncommutative geometries which we introduce here.
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Notes
- 1.
Recall that D is an unbounded operator acting on the Hilbert space of L2-spinors and that the functions of C(M) act on the same Hilbert space by multiplication operators.
- 2.
Indeed, \( \operatorname {{\mathrm {Lip}}}(X)\) contains the constant functions. Moreover, \( \operatorname {{\mathrm {Lip}}}(X)\) separates points in X. If x 0, y 0 ∈ X with x 0 ≠ y 0, we can use the lipschitz function \(f \colon X\to \mathbb {R}\), x↦dist(x, y 0) since we have f(x 0) > 0 = f(y 0).
- 3.
- 4.
It is not clear if the support must be in addition open in [143] since the indicator 1A of a subset A is continuous if and only if A is both open and closed.
- 5.
Recall that a family (x s)s ∈ I vanishes at infinity means that for any ε > 0, there exists a finite subset J of I such that for any s ∈ I − J we have |x s| < ε.
- 6.
Write \(x=A^{-\frac {1}{2}}A_p^{\frac {1}{2}}x\).
- 7.
More generally, if 1 < p < ∞ the operator space \(\mathrm {L}^p(\mathbb {T}^2)\) has CCAP by Junge and Ruan [121, Proposition 3.5] since an abelian group is weakly amenable with Cowling-Haagerup constant equal to 1.
- 8.
If \(\eta \in \big ( \xi _1 + \frac {\sqrt {\mathrm {Gap}_\alpha }}{2} \mathrm {B}_n \big ) \cap \big ( \xi _2 + \frac {\sqrt {\mathrm {Gap}_\alpha }}{2} \mathrm {B}_n \big )\) then \(|\xi _1-\xi _2| \leqslant |\xi _1-\eta |+|\eta -\xi _2| < \frac {\sqrt {\mathrm {Gap}_\alpha }}{2}+\frac {\sqrt {\mathrm {Gap}_\alpha }}{2}=\sqrt {\mathrm {Gap}_\alpha }\) which is impossible.
- 9.
Since I is finite, we have \( \operatorname {\mathrm {dom}} A_p^{\frac {1}{2}}=S^p_I\). Write \(x=A^{-\frac {1}{2}}A_p^{\frac {1}{2}}x\).
- 10.
Since I is finite, we have \( \operatorname {\mathrm {dom}} A_{p_0}^{\frac {1}{2}}=S^{p_0}_I\).
- 11.
For any \(n,m \in \mathbb {Z}\), we have the cocycle law
$$\displaystyle \begin{aligned} \begin{array}{rcl} b(n)& +&\displaystyle \pi_n(b_m) =\big(\mathrm{e}^{2 \pi \mathrm{i} \alpha n},\mathrm{e}^{2 \pi \mathrm{i} \beta n}\big) - (1,1) +\pi_n\big[\big(\mathrm{e}^{2 \pi \mathrm{i} \alpha m},\mathrm{e}^{2 \pi \mathrm{i} \beta m}\big) - (1,1) \big] \\ & =&\displaystyle \big(\mathrm{e}^{2 \pi \mathrm{i} \alpha n},\mathrm{e}^{2 \pi \mathrm{i} \beta n}\big) - (1,1) + \big(\mathrm{e}^{2 \pi \mathrm{i} \alpha (n+m)},\mathrm{e}^{2 \pi \mathrm{i} \beta (n+m)}\big) -\big(\mathrm{e}^{2 \pi \mathrm{i} \alpha n},\mathrm{e}^{2 \pi \mathrm{i} \beta n}\big) \\ & =&\displaystyle \big(\mathrm{e}^{2 \pi \mathrm{i} \alpha (n+m)},\mathrm{e}^{2 \pi \mathrm{i} \beta (n+m)}\big) - (1,1) =b(n+m). \end{array} \end{aligned} $$ - 12.
It may perhaps be possible to replace the reflexivity by an assumption of weak compactness, see [110, p. 361].
- 13.
In the sense of [136, p. 167].
- 14.
Recall that the term ∂ ψ,q,p(a)x is by definition equal to
. - 15.
For any x ∈ Γq(H) and any s ∈ G, we have
and
.
- 16.
For t > 0, let μ t(b) denote the generalized singular number of |b|, so that
for any continuous function \(\mathbb {R}_+ \to \mathbb {R}\) of bounded variation [227, p. 30]. If b were unbounded, then μ
t(b) →∞ as t → 0. Taking ξ = ϕ(|b|) ≠ 0 with ϕ a smoothed indicator function of an interval [x, y] with large x, it is not difficult to see that \(\left \Vert b\xi \right \Vert { }_p \geqslant x \left \Vert \xi \right \Vert { }_p\), which is the desired contradiction. - 17.
Recall that the term ∂ α,q,p(a)x is by definition equal to ∂ α,q(a)(1 ⊗ x).
- 18.
Recall that a family (x i)i ∈ I vanishes at infinity means that for any ε > 0, there exists a finite subset J of I such that for any i ∈ I − J we have \(|x_i| \leqslant \varepsilon \).
- 19.
The entries are 1 on the j-row and zero anywhere else.
- 20.
Note that since \(\Gamma _{-1}(H) \overline \otimes \mathrm {B}(\ell ^2_I)\) is then finite-dimensional, the spaces \(\mathrm {L}^p(\Gamma _{-1}(H) \overline \otimes \mathrm {B}(\ell ^2_I))\) are all isomorphic for different values of p, \(\overline { \operatorname {\mathrm {Ran}} \mathscr {D}_{\alpha ,-1,2}} = \overline { \operatorname {\mathrm {Ran}} \mathscr {D}_{\alpha ,-1,p}}\) for all 1 < p < ∞ and thus, \(|\mathscr {D}_{\alpha ,-1,2}|{ }^{-1}\) extends to a compact operator on \(\overline { \operatorname {\mathrm {Ran}} \mathscr {D}_{\alpha ,-1,p}}\), too.
- 21.
We have
$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \sum_{k,l} J_{kl} R(\lambda,\mathrm{L}_{s_q(\alpha_k -\alpha_l)})Q_{kl} \left( \lambda - \sum_{i,j} J_{ij} \mathrm{L}_{s_q(\alpha_i - \alpha_j)} Q_{ij} \right) \\ & &\displaystyle \qquad \quad = \sum_{k,l}J_{kl} \lambda R(\lambda,\mathrm{L}_{s_q(\alpha_k - \alpha_l)})Q_{kl} - \sum_{k,l,i,j} J_{kl}R(\lambda,\mathrm{L}_{s_q(\alpha_k - \alpha_l)})Q_{kl}J_{ij} \mathrm{L}_{s_q(\alpha_i - \alpha_j)}Q_{ij} \\ & &\displaystyle \qquad \quad = \sum_{k,l} J_{kl}\lambda R(\lambda,\mathrm{L}_{s_q(\alpha_k - \alpha_l)})Q_{kl}- \sum_{k,l,i,j} \delta_{k=i} \delta_{l = j} J_{kl}R(\lambda,\mathrm{L}_{s_q(\alpha_k - \alpha_l)}) \mathrm{L}_{s_q(\alpha_i - \alpha_j)} Q_{ij} \\ & &\displaystyle \qquad \quad = \sum_{k,l}J_{kl} R(\lambda,\mathrm{L}_{s_q(\alpha_k - \alpha_l)})(\lambda - \mathrm{L}_{s_q(\alpha_k-\alpha_l)}) Q_{kl} \\ & &\displaystyle \qquad \quad = \sum_{k,l}J_{kl}\mathrm{Id}_{\mathrm{L}^p(\Gamma_q(H))} Q_{kl} = \mathrm{Id}_{\mathrm{L}^p(\Gamma_q(H) \overline\otimes \mathrm{B}(\ell^2_I))}, \end{array} \end{aligned} $$and similarly the other way around.
.
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Arhancet, C., Kriegler, C. (2022). Locally Compact Quantum Metric Spaces and Spectral Triples. In: Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers. Lecture Notes in Mathematics, vol 2304. Springer, Cham. https://doi.org/10.1007/978-3-030-99011-4_5
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