7.1 Noncommutative Geometry

We use the standard multi-index notation \(\alpha =(\alpha _,\dots , \alpha _)\) with \(\alpha _\in \\) and \(|\alpha |=\sum _^n |\alpha _j|\). Let us denote the partial derivative by \(D^_f=\fracf}}^} \dots \partial _}^} }\), for \(f\in C^(}^)\). We say that \(\sigma \in C^(}^\times }^)\) is a symbol of order m if

$$\begin |D^_D^_\sigma (x,\xi )|\le C_((1+|\xi |)^) \end$$

for any multi-index \(\alpha \) and \(\beta \) and for any \(x,\xi \in }^\). Note that \( C_\) depends on the choice of \(\alpha \) and \(\beta \) and is independent of x and \(\xi \). A pseudo differential operator \(\Psi \) of order m on \(}}^n\) with symbol \(\sigma _\) is defined by

$$\begin \Psi (f) (x)\hspace= \hspace\int _}^} e^ \sigma _(x,\xi ) (}f)(\xi ) d\xi \hspace, \end$$

where \(}\) is the Fourier transformation and \(\sigma _\) is a symbol of order m. Let (M, g) be a Riemannian manifold of dimension n. Then \(\Psi :C^(M)\rightarrow C^(M)\) is said to be a pseudo-differential operator of order m if this is true for every local chart. (See e.g. [66]) The most prominent example of pseudo-differential operators is the Laplacian operator \(\Delta =-\left( \frac}}^}+\dots +\frac}}^}\right) \) on \(}^\) with the symbol \(\sigma _(x,\xi )=4\pi ^|\xi |^\). Consequently, the Laplacian power \(\Psi _=(1+\Delta )^\) is a pseudo-differential operator of order m with the symbol \(\sigma _}(x,\xi )=(1+4\pi ^|\xi |^)^\). In addition \(\Psi _\) is a classical pseudo-differential operator with the asymptotic expansion

$$\begin \sigma _} (x,\xi )\sim \sum _^ } |2\pi \xi |^\hspace. \end$$

Let us denote the Laplacian power of order n by

$$\begin d=(1+\Delta )^\hspace.\end$$

(7.1)

Recall that a pseudo-differential operator \(\Psi \) is said to be

(1)

compactly supported if there exist compactly supported \(\psi _,\psi _\in C^(M)\) such that \(\Psi =}_}\Psi \,}_}\);

(2)

compactly based if there exists compactly supported \(\psi \in C^(M)\) such that \(\Psi =}_\Psi \).

For a classical, compactly supported, pseudo differential operator \(\Psi \) of order \(-n\) , the Wodzicki residue \(\text _}(\Psi )\) is the integral of the principal symbol \(\sigma _\) over the co-sphere bundle \(S^M=(}^M\backslash \)/}_\)

$$\begin \text _}(\Psi )=\frac\int _M} \sigma _(\text )dvol\hspace, \end$$

where dvol is the volume form of \(S^M\). The Wodzicki residue is also well-defined if we replace compactly supported with compactly based. Recall that \(\text _}([\Psi ,\Phi ])\) vanishes for classical compactly based pseudo differential operators \(\Psi \) of order \(k_\) and \(\Phi \) of order \(k_\) with \(k_+k_=-n.\)

For any \(f\in L_()\), the left multiplication \(}_f}: L_()\rightarrow L_()\) is a bounded linear operator,

$$\begin }_f(F)(x)}:=f(x)F(x)\hspace. \end$$

Indeed, we obtain the inclusion \(L_(M)\subset }(L_2(M))\) by this left regular representation. For any \(p\in }\), then \(}}_}=}}_^\). Moreover, there exists a constant \(c(n)=\frac(S^)}}\) such that

$$\begin \text _}(}_}d^)=c(n) \int _ f(x) dvol(x), \forall f\in L_(M)\hspace. \end$$

(7.2)

Recall that the Macaev ideal \(}_\) given by compact operators \(T\in }(L_2(M))\) such that

$$\begin \Vert T\Vert _}_} \hspace= \hspace\sup _}}} \frac\sum _^n u_j(T) \hspace<\hspace\infty \hspace,\end$$

where \(\(T)\}\) is the decreasing sequence of singular values of T. The Laplacian power d has continuous extension in \(}_\), still denoted as d.

Every dilation invariant extended limit \(\omega \) on \(\ell _(}})\) defines a weight on \((}_})_\)

$$\begin \text _(T) \hspace= \hspace\omega \left( \left\\sum _^n u_j(T)\right\} _n\right) \hspace.\end$$

This weight can be extended, by linearity, to all of \(}_\) and still remains tracial, and its extension on \(}_\) is called a Dixmier trace on \(}_\). The Dixmier trace \(\text _\) is non-normal and vanishes on the Schatten 1-class \(S_\). An operator \(T\in }_\) is said to be Dixmier measurable if the value \(\text _(T)\) is independent of the choice of the dilation invariant extended limit \(\omega \). Let \(\tau \) be the trace on \(}}(L_(M))\) defined as

$$\begin \tau (T)=\sum _\langle x_, T x_ \rangle \hspace, \end$$

(7.3)

where \(\\}\) an orthonormal basis of \(L_(M)\). Let \(T\in (}_)_\), then

$$\begin \lim _ \frac\sum _^\mu _(T)<\infty \end$$

and

$$\begin \text _(T)=\lim _ \frac\tau (T^)=\lim _}(q-1)\tau (T^)\end$$

(7.4)

for any dilation invariant extended limit \(\omega \). We refer to [66, Theorem 9.3.1] for the proof and [66, Theorem 10.1.2] for a summary of formulas for Dixmier traces.

The coincidence between the geometric quantity-Wodzicki residue and the operator algebraic quantity-Dixmier trace was first discovered and proven by Alain Connes, known as Connes’ trace theorem. Here we give a simple version of Connes’ trace theorem. See [66] for a complete statement and proof.

Theorem 7.1

Let \(\Psi \) be a classical compactly supported pseudo-differential operator of order \(-n\). Then \(\Psi \) extends continuously to a Dixmier measurable operator in \(}_\). Let \(\omega \) be any dilation invariant extended limit, then

$$\begin \text _(\Psi )=\text _}(\Psi )\hspace. \end$$

In particular for any \(f\in L_(M)\), we have

$$\begin \text _(}_}d^) \hspace= \hspace\text _}(}_}d^) \hspace= \hspacec(n)\int _f(x)dvol(x)\hspace. \end$$

Actually it is sufficient to assume that \(\Psi \) is compactly based. Now are ready to prove the main result.

7.2 \(\text _}\text \) and \(\text ^+\)

To develop the anti-transference principle, we work with a slightly different version of logarithmic Sobolev inequalities, which are usually referred to as the Beckner inequality in the probability theory. Now we give a brief introduction of complete p-Sobolev type inequalities and refer to [62] (and references therein) for details and generalized results. In the sequel, we assume that \(p\in (1,2)\). For any \(\rho ,\sigma \in }_\), the quantum p-relative entropy is defined as

$$\begin d^(\rho \Vert \sigma ) \le \ \tau (\rho ^-\sigma ^)-p\tau ((\rho -\sigma )\sigma ^),& \text supp(\rho )\supset supp(\sigma )\hspace; \\ +\infty , & \text \end\right. } \end$$

It follows from Klein’s inequality that \(d^(\rho \Vert \sigma )\ge 0\), with the equality if and only if \(\rho =\sigma \). The p-relative entropy with respect to the von Neumann subalgebra \(}\subset }\) is defined by

$$\begin d^_}}(\rho )=d^(\rho \Vert E_}}(\rho ))=\inf _}}d^(\rho \Vert \sigma )\hspace. \end$$

Lemma 2.7 remains true for the p-relative entropy, see [62] for the proof.

Lemma 7.2

Let \(E_1\),...,\(E_m\) be pairwise commuting conditional expectations on \(}}\), then

$$\begin d^\left( \rho \Vert \left( \prod _^ E_j\right) (\rho ) \right) \hspace\le \hspace\sum _^m d^ (\rho \Vert E_j(\rho )) \hspace. \end$$

In [62], we defined the p-Fisher information \(I_^\) of the generator A of a semigroup \(e^\) by

$$\begin I_^(\rho )=p\tau (A(\rho )\rho ^), (})} \end$$

and the p-information \(I_^\) of \(\delta \) by

$$\begin I_^(\rho )=p\tau \left( \delta (\rho )}^_)}^}(\delta (\rho ))\right) ,\quad (})\hspace.} \end$$

Note that

$$\begin I_(\rho )=\lim _}\frac^(\rho )} \end$$

and

See [62] for the nonnegativity and convexity of the quantum p-Fisher information.

Definition 7.3

The semigroup \(T_=e^\) or the generator A with the fixed-point algebra \(}_}\) is said to satisfy:

(1)

the modified p-Sobolev inequality \(\lambda \)-\(\text \) (with respect to the trace \(\tau \)) if there exists a constant \(\lambda >0\) such that

$$\begin \lambda d_}_}}^(\rho )\le I_^(\rho ),\quad \forall \rho \in (})}}\hspace; \end$$

(7.5)

(2)

the complete p-Sobolev inequality \(\lambda \)-\(\text _}\text \) (with respect to the trace \(\tau \)) if \(T_ \otimes \textrm_}}\) satisfies \(\lambda \)-\(\text \) for any finite von Neumann algebra \(}\).

(3)

the enhanced complete logarithmic Sobolev inequality \(\lambda \)-\(\text ^\) if A satisfies \(\lambda \)-\(\text _}\text \) for \(p\in (1,2)\).

Let \(\text _}\text (A,\tau )\) (\(\text ^(A,\tau )\)) be the supremum of \(\lambda \) such that A satisfies \(\lambda \)-\(\text _}\text \)(\(\text ^\)), or denoted by \(\text _}\text (A)\) (\(\text ^(A)\)) if there is no ambiguity. The derivation \(\delta \) is said to satisfy \(\lambda \)-\(\text \) (\(\lambda \)-\(\text _}\text \)) if \(\delta ^}\) satisfies \(\lambda \)-\(\text \) (\(\lambda \)-\(\text _}\text \)). Similarly we define \(\text _}\text (\delta ,\tau )\), \(\text _}\text (\delta )\), \(\text ^(\delta , \tau )\), and \(\text ^(\delta )\) for the derivation \(\delta \).

Theorem 7.4

Let \(E_}}\) be the conditional expectation onto the von Neumann subalgebra \(}\subset }\), then we have

$$\begin \text _}\text (I-E_}})\ge p\hspace. \end$$

Proof It is obvious that the fixed point algebra \(}_}\) of \(T_=e^}})}\) is \(}\). By Jensen’s inequality, we have \((E_}}(\rho ))^\ge E_}}(\rho ^)\). Thus

$$\begin d_}_}}^(\rho )&=\tau (\rho ^- E_}}(\rho )(E_}}(\rho ))^ )\\&\le \tau (\rho ^-E_}}(\rho ) E_}}(\rho ^))\\&=\fracI_}}}^(\rho )\hspace.\quad \end$$

\(\square \)

Here we list important properties of \(\text _}\text \) and refer to [62] for proofs.

Theorem 7.5

Let \(\lambda >0\) and \(T_=e^:}\rightarrow }\), then

(1)

\(I_^(e^(\rho ))=-\fracd_}_}}^(e^(\rho ));\)

(2)

the exponential decay of the Fisher information \(I_^(T_(\rho ))\le e^ I_^(\rho )\) implies \(\text (A)\ge \lambda \);

(3)

the inequality (7.5) equivalent to \(d_}_}}^(T_(\rho ))\le e^ d_}_}}^(\rho ),\, \forall \rho \in (})}};\)

(4)

\(\text (A)\ge \text ^(A)\);

(5)

\(\text _}\text \) and \(\text ^\) are stable under tensorization;

(6)

\(\text _}\text \) and \(\text ^\) are stable under change of measure.

Similar results remain true for \(\delta \).

Theorem 3.4 remains true for CpSI and \(\text ^\), and we refer to [62] for the proof.

Theorem 7.6

Let \((}\!\!\subset \! \!},\tau ,\delta )\) be a derivation triple with \(\text\ge \lambda >0\). Then for any \(p\in (1,2)\):

$$\begin \text _}\text (}\!\!\subset \! \!},\tau ,\delta )\ge 2\lambda \hspace. \end$$

Furthermore \(\text ^(}\!\!\subset \! \!},\tau ,\delta )\ge 2\lambda \).

Remark 7.7

Results and examples based on the two theorems above automatically apply to \(\text ^\). In particular, Theorem 1.1, Theorem 1.2, Theorem 1.4 and Theorem 1.5 remain true for \(\text ^\).

Remark 7.8

Definition 2.12 and Definition 7.3 can be generalized to a semi-finite von Neumann algebra \((}},\tau )\), where \(\tau \) is a normal semi-finite faithful trace on \(}}\). We need to modify (2.9) and (7.5) as follows:

$$\begin \lambda D_}}_}}(\rho )\le I_(\rho ),\quad \forall \rho \in L_(}})_+ \end$$

and

$$\begin \lambda d_}_}}^(\rho )\le I_^(\rho ),\quad \forall \rho \in L_(}})_+ \hspace,\end$$

respectively. Here we interpret

$$\begin I_A(\rho ) \hspace= \hspace\sup _ \int _^} \tau (\delta (\rho )^*(\rho +r)^ \delta (\rho ) (\rho +r)^) dr \hspace. \end$$

Then one can use an approximation procedure leading to (2.10). CLSI and CpSI are still defined as in Definition 2.12 and Definition 7.3, respectively. Then \(\text (A)\ge \text ^(A)\) remains true. In full generality executing this approximation procedure is beyond the scope of this paper. However, if in addition there is a sequence of projections \(e_\) of finite trace converging strongly to 1 such that \(A(e_xe_) \hspace= \hspacee_ A(x) e_\) it is enough to work with finite traces and \(S_C(e_Ne_)\) is still meaningful.

7.3 Main Result and Proof

In the following, let M be a compact Riemannian manifold of dimension n and \(}}\) be a finite von Neumann algebra equipped with a normal faithful trace \(\tau _}}}\). Let \(\text :=\tau \otimes \tau _}}}\), where \(\tau \) is defined in (7.3) For \(a\in L_(M, L_ (}}}})),\) define \(\text _}}}}}(a)=\int _ \tau _}}}}}(a(x))dvol(x).\) Now we ready to state the main result of this section.

Theorem 7.9

Let \(T_=e^}}: }}(L_(M))\rightarrow }}(L_(M))\) be a semigroup of self-adjoint completely positive trace preserving maps, where \(}}(L_(M))\) is equipped with trace \(\tau \). Let \(E:}}(L_(M))\rightarrow }}_}\) be the normal conditional expectation onto the fixed point algebra \(}}_}\) of \(T_\). We assume \(T_\) satisfies:

(1)

\(}\) maps \(C^(M)\) to \(C^(M)\), where we use the inclusion \(L_(M)\subset }}(L_(M))\) by left regular representation;

(2)

The domain \(}(}|_(M)})\) of \(}\) restricted to \(C^(M)\)is \(w^\)-dense in \(L_(M)\);

(3)

\(}(d^ad^)=d^}(a)d^\) for any 0-th order pseudo differential operator a, where d is the Laplacian power defined in (7.1).

Then \(\text ^(L_(M),})\ge \text ^(}}(L_(M)), })\).

It is noteworthy that we use the generalization of \(\text ^(}}(L_(M)))\) explained in Remark 7.8 since \(\tau \) is not a finite trace. For \(a\in C^(M)\), we can identify \(}(}_)\) and \(E(}_)\) with \(}_\) and \(E(}_g)\), respectively, for some \(f,g\in C^\), i.e., \(}_=}(}_)\) and \(}_=E(}_)\). For simplicity, we denote f and g by \(}(a)\) and E(a), respectively. Let \(}}_,M}\subset L_(M)\) be the fixed point algebra of \(T_\) restricted to \(L_(M)\). The following diagrams commute:

where \(}\) denotes the inclusion defined via left regular representation.

Remark 7.10

For the reader worried about the existence of \(E_\) for normal semigroup on N, we recall that the ergodic averages \(M_t(\rho ) \hspace= \hspace\int _0^t T_s^*(\rho ) \frac\) converge weakly to a projection \(E_*\). Using density of \(L_1\cap L_2\), and the fact that \(T_t=T_t^*\) maps \(L_1\cap L_2\) to itself, it turns out that \(E_*(L_1(N))\subset L_1(N)\). Then \(E=E_*\) maps onto the fixed points of \(N_\). Moreover, if \(\rho =E_*(\rho )\) is a faithful fixed point state, we deduce that \(N_\) is an algebra. In any case we may use the same argument for states with the trace, if we restrict \(E=E_*\) to \(N\cap L_1(N)\) and find that \(N_\cap L_1(N)\) is an algebra. Since \(E=E_*^*\) is normal this shows that also \(N_\) is an algebra. In our case the existence of \(E:N\rightarrow N_\) is guaranteed by transference principle.

We postpone the proof of the main result and obtain the following result as a corollary of Theorem 7.1.

Corollary 7.11

Let \(1\le p<\infty \) and \(f\in C^(M)\). Then

$$\begin c(n) \Vert f\Vert _p^p \hspace= \hspace\text _(|d^}}_}d^}|^p) \hspace= \hspace\lim _} (q-1) \Vert d^}}_}d^}\Vert _^ \hspace. \end$$

Proof

Noting that \(}}_^d^\) and \(|d^}}_} d^} |^\) have the same principal symbol of order \(-n\) [83], we infer that

$$\begin \text _}(}}_^pd^)=\text _}(|d^}}_}d^} |^)\hspace. \end$$

Together with (7.2), we obtain that

$$\begin c_\Vert f\Vert _^\hspace= \hspace\text _}(}}_^d^) \hspace= \hspace\text _}(|d^}}_}d^} |^)\hspace. \end$$

(7.6)

For any even integer p, the assertion follows from Theorem 7.1 and (7.4).

Then we prove the upper estimate for all p by interpolation. We may assume that \(c(n)\Vert f\Vert _^<1\). Let \(\frac=\frac+\frac\) and \(p_0\) be an even integer. Let \(\alpha (z)=\frac+\frac\) and consider the analytic function

$$\begin F(z) \hspace= \hspaceu }}_}^d^\hspace,\end$$

where \(}_}=u}}_\) is given by the polar decomposition. Thus \(F(\theta )=u}}_d^=}_}d^\). Note that for different values of \(z=it\) the functions F(it) and F(0) only differ by left and right multiplications by unitaries, and

$$\begin |F(it)|=|F(0)|=}}_}}d^}\hspace. \end$$

Applying the limit \(q\rightarrow 1^\) applies uniformly to t and together with (7.6), we infer that

$$\begin&\lim _}(q-1)\sup _\Vert F(it)\Vert _}^}\\&\quad \lim _} (q-1)\Vert d^)}}}_}^} d^)} \Vert _}^} =c(n)\Vert f\Vert _^\le 1\hspace. \end$$

Similarly we obtain that \(\lim _} (q-1) \sup _ \Vert F(1+it)\Vert ^_ \hspace\le \hspace1 \hspace.\) By the three line lemma (see [15]), we deduce that

$$\begin \lim _} (q-1) \Vert }_}d^\Vert _^ \hspace= \hspace\lim _} (q-1)\Vert F(\theta )\Vert _^ \hspace\le \hspace1 \hspace.\end$$

Homogeneity implies

$$\begin \lim _} (q-1) \Vert }_}d^\Vert _^ \hspace\le \hspacec(n) \Vert f\Vert _p^p \end$$

and

$$\begin \lim _} (q-1) \Vert d^}_}^\Vert _^ \hspace\le \hspacec(n) \Vert f\Vert _p^p \hspace. \end$$

For the lower estimate, we assume that \(\Vert f\Vert _p^p=1\). For any \(\epsilon >0\), there exists \(g\in C^(M)\) such that \(\int |g^*f| d\textrm(x)\hspace\ge \hspace(1-\varepsilon )\) and \(\Vert g\Vert _\le 1\) with \(\frac+\frac=1\). Note that \(d^}}_d^\) is a pseudo-differential operator of order \(-n\), then

$$\begin c(n) \int |g^*f| d\textrm(x) \hspace= \hspace\lim _} (q-1) \Vert d^}}_d^\Vert _q^q \hspace. \end$$

By Hölder’s inequality, then

$$\begin c(n) \int |g^*f| d\textrm(x)&\le \left( \limsup _} (q-1)^ \Vert d^}}_\Vert _^q\right) \\&\quad \limsup _} (q-1)^ \Vert }_}d^\Vert _^ \\&\le c(n)^ \limsup _} (q-1)^ \Vert }_}d^\Vert _^\hspace. \end$$

Together with \(\int |g^*f| d\textrm(x)\hspace\ge \hspace(1-\varepsilon )\), we have

$$\begin (1-\varepsilon )\le c(n)^ \limsup _} (q-1)^ \Vert }_}d^\Vert _^\hspace. \end$$

Taking the p-th power we deduce that

$$\begin (1-\varepsilon )^p c(n) \hspace\le \hspace\limsup _} (q-1) \Vert }_}d^\Vert _^ =\lim _} (q-1)\Vert }_}d^\Vert _^ \hspace. \end$$

(7.7)

Thus the upper and lower estimates yield the equality. \(\square \)

Remark 7.12

We could also use the pseudo-differential calculus developed by Connes ([31]) to obtain the result without interpolation.

Let us recall the definition of vector-valued mixed (p, q)-spaces:

$$\begin L_(}}},L_q(}}}})) \hspace= \hspace[}}}}}}}},L_(}}},L_1(}}}}))]_ \end$$

obtained by complex interpolation; see [54, 79]. The two expressions for the norm of any \(}}}\otimes }}}}\) coincide:

$$\begin \Vert f\Vert _(}}},L_1(}}}}))}&= \sup _ \Vert (a\otimes 1)f(b\otimes 1)\Vert _}}}\otimes }}}})}\\&= \inf _ \Vert \textrm\otimes \tau _}}}(f_1f_1^*)\Vert _}}}}^ \Vert \textrm\otimes \tau _}}}(^*)\Vert _}}}}^\hspace; \end$$

again see [54, 79]. Thus by interpolation, we deduce an isometric inclusion

$$\begin L_(}}}_1,L_q(}}}}))\subset L_(}}}_2,L_q(}}}})) \end$$

for every inclusion of von Neumann algebras \(}}}_1\subset }}}_2\). This is in particular true for the inclusion \(L_(M)\subset }(L_2(M))\) given by the left regular representation,

$$\begin L_(M,L_(}}}}))\subset L_(}}(L_(M)), L_(}}}}))\hspace. \end$$

(7.8)

Also recall Pisier’s interpolation theorem [54, 79] for vector-valued \(L_\) spaces that

$$\begin L_(L_(M)} }}}}) \hspace= \hspaceL_(M,L_(}}}}))\hspace. \end$$

(7.9)

Corollary 7.13

Let \(f\in L_(M, L_(}}}}))\), then

$$\begin\lim _} (q-1) \Vert (d^}\otimes 1)}_}(d^}\otimes 1)\Vert _^ \hspace= \hspacec(n) \Vert f\Vert _(M)} }}}})}^\hspace. \end$$

Proof

It is sufficient to consider the vector valued function \(f\in L_(M, }}}})\). Indeed, we may extend this result to all of \(L_p(L_(M)} }}}})\) using the Banach space \(\prod _}}L_\left( }(L_2(M) }}}}})\right) \).

Now let \(f\in L_(M,}}}})\), then \(f\in L_(L_(M)} }}}})\subset (L_(M)} }}}})}\). For any \(\varepsilon >0\), there exists \(p_>p\) such

$$\begin \Vert f\Vert _(M)} }}}})}\hspace\le \hspace\Vert f\Vert _}(L_(M)} }}}})}\hspace\le \hspace(1+\varepsilon )\Vert f\Vert _(M)} }}}})}\hspace. \end$$

(7.10)

Thus there exist \(f_1,f_2\in L_}(M)\) and \(F\in L_(M,L_}(}}}}))\) such that

$$\begin f= & (f_1\otimes 1)F(f_2\otimes 1), \\ & \max \}\Vert f_2\Vert _}\}\le \Vert f\Vert _}(L_(M)} }}}})}^, \end$$

and \(\Vert F\Vert _}\le 1\), where

$$\begin \Vert F\Vert _}=\int _ \tau _}}}}}( |F(x)|^} )^}}dvol(x) \hspace. \end$$

Indeed, the functions \(f_1(x)=f_2(x)= \tau _}}}}}(|f(x)|^})^}} \) will do the job. The inclusion result (7.8) implies that \(}}_\in L_(}}(L_(M)), L_}(}}}}))\). Since \(p<p_\), we apply Corollary 7.11 to \(f_f_^\) and \(f_f_^\) and continuity of \(L_\) spaces, then

$$\begin&\lim _} \left( (q-1) \Vert d^}}}_d^}\Vert _^\right) ^ \lim _} \left( (q-1) \Vert d^}}}_d^}\Vert _^\right) ^ \\&\quad = c(n) \Vert f_f_^\Vert _^ \Vert f_f_^\Vert _^ \\&\quad \le c(n) \left( \Vert f_f_^\Vert _}^ \Vert f_f_^\Vert _}^ \right) ^\\&\quad \le (1+\varepsilon )^c(n)||f||_^\hspace. \end$$

By [80] we find that

$$\begin&\lim _} (q-1) \Vert (d^}\otimes 1)}_}(d^}\otimes 1)\Vert _^\\&\quad \le \limsup _} \, (q-1) \Vert d^}}}_\Vert _^ \Vert }}_\Vert _(}}(L_(M)), L_(}}}}))}^ \Vert }}_d^}\Vert _^\\&\quad \le \limsup _} \left( (q-1) \Vert d^}}}_d^}\Vert _^\right) ^ \\&\quad \quad \limsup _} \left( (q-1) \Vert d^}}}_d^}\Vert _^\right) ^\hspace. \end$$

Thus

$$\begin \lim _} (q-1) \Vert (d^}\otimes 1)}_}(d^}\otimes 1)\Vert _^\le (1+\varepsilon )^c(n)||f||_^. \end$$

Sending \(\varepsilon \) to 0 yields the upper bound. The same interpolation argument as in (7.7) also shows the lower bound by duality. \(\square \)

Lemma 7.14

Let a,\(b\in L_(M, L_(}}}}))\) for \(p\in (1,2)\). Let a be positive and b be self-adjoint. Define \(A=(d^}\otimes 1)}}_a (d^}\otimes 1)\) and \(B=(d^}\otimes 1)}}_b(d^}\otimes 1)\). If there exists \(C>0\) such that \(-C a\le b\le C a\), then

$$\begin \lim _} (q-1)\text \left( BA^ \right) \le c(n)\text _}}}}}(ba^)\hspace. \end$$

Proof

Let \(t\ge 0\) and \(tC\le 1\), then \(tb+a\le 1\). Applying Corollary 7.13 to \(a+tb\) and a implies

$$\begin \lim _} (q-1) \Vert A+tB\Vert _^&= c(n) \Vert a+tb\Vert _(L_(M)} }}}})}^\hspace, \end$$

(7.11)

$$\begin \lim _} (q-1) \Vert A\Vert _^&= c(n) \Vert a\Vert _(L_(M)} }}}})}^\hspace. \end$$

(7.12)

Noting that \(\Vert \cdot \Vert _^\) is convex for \(q\ge 1\) small enough, we obtain

$$\begin pq \text \left( BA^\right) \hspace\le \hspace\frac^-\Vert A\Vert _^} \hspace. \end$$

Together with (7.11) and (7.12), we have

$$\begin \lim _} (q-1) \text (BA^)\le \frac(L_(M)} }}}})}^-\Vert a\Vert _(L_(M)} }}}})}^ \right) }. \end$$

Using the differentiation formula for the p-norm, we observe that

$$\begin&\Vert a+tb\Vert _(L_(M)} }}}})}^-\Vert a\Vert _(L_(M)} }}}})}^\\&\quad = t\int _0^1 p\text _}}}}}(b(a+stb)^) ds \\&\quad = tp \text _}}}}(ba^) + tp\int _0^1 \text _}}}}(b((a+stb)^-a^)) ds \hspace. \end$$

Thus it suffices to show that

$$\begin \int _0^1 \text _}}}}(b((a+stb)^-a^)) ds=}(t) \end$$

(7.13)

as \(t\rightarrow 0\), then sending \(t\rightarrow 0\) implies the assertion. Indeed, we decompose \(b=b^-b^\) for positive \(b^\) and \(b^\).

Using the the monotonicity of \(x\mapsto x^\) and \(b\le C a\) we deduce

$$\begin \text _}}}}(b^+a^)\le \text _}}}}(b^+(a+stb)^)\hspace\le \hspace(1+st C )^ \text _}}}}(b^+a^)\hspace. \end$$

The same argument applies for \(b^-\) and hence

$$\begin&|\text _}}}}\left( b((a+stb)^-a^)\right) | \\&\quad \hspace\le \hspace\left( (1+stC)^-1\right) \text _}}}}}\left( |b|a^\right) \\&\quad \hspace\le \hspace(p-1)st C \text _}}}}(|b|a^)\hspace. \end$$

Integrating the inequality above yields (7.13). \(\square \)

Now we are ready to prove the main theorem.

Proof

Let \(1<p<2\) and \(p<q<2\). Let \(a:M\rightarrow }}}}\) be a smooth positive function and \(a_=a+\varepsilon 1\) for \(\epsilon >0\). Then \(}_}}=}\otimes \textrm_}}})}}(}}}})=}\otimes \textrm_}}})}}(}_}})\) since \(}\) is self-adjoint. By (1), we can identify \((}\otimes \textrm_}}})(}_)\) with an \(}}\)-valued function f defined on M via left regular representation, i.e., \(}_=(}\otimes \textrm_}}})(}_)\). For simplicity, we denote this \(}}\)-valued function f by \(}(a)\). Let \(C=\varepsilon ^\Vert }\otimes \textrm_}}})}}(a)\Vert _(L_(M)} }}}})}\), then \(-Ca_\le b\le Ca_\). Let \(A_=(d^}\otimes 1)}}_} (d^}\otimes 1)\) and \(B=(d^}\otimes 1)}}_b(d^}\otimes 1)\). It follows from Lemma 7.14 that

$$\begin \lim _} (q-1) \text (BA_^) \hspace\le \hspacec(n)\text _}}}}( ba_^)\hspace. \end$$

(7.14)

Notice that \((}\otimes \textrm_}}}}})(A_)=B\) by (3) in Theorem 7.9. Then

$$\begin&\Vert A_\Vert _^-\Vert (d^}\otimes 1)}}__}}})}}(a_)}(d^}\otimes 1)\Vert _^\\&\quad \le \frac^(}}(L_(M)), }) } \text _}}}}}\left( BA_^\right) \hspace. \end$$

Together with (7.14), we obtain

$$\begin&\lim _}(q-1)\left( \Vert A_\Vert _^-\Vert (d^}\otimes 1)}}__}}})}}(a_)}(d^}\otimes 1)\Vert _^ \right) \\&\quad \le \frac^(}}(L_(M)), }) }\text _}}}}( ba_^).\end$$

Applying Corollary 7.13 to \((d^}\otimes 1)}}_}(d^}\otimes 1)\) and \((d^}\otimes 1)}}_)}(d^}\otimes 1)\) implies

$$\begin&\Vert a_\Vert _(M)} }}}})}^p-\Vert _}}})}}(a_)\Vert _(M)} }}}})}^p\nonumber \\&\quad \le \frac^(}}(L_(M)), }) } \text _}}}}}(}\otimes \textrm_}}})}(a)a_^) \hspace. \end$$

(7.15)

The left hand side is continuous in \(\varepsilon \). By functional calculus and the dominated convergence theorem for the sequence of functions \(g_k(x)=(x+\frac)^\), we deduce that

$$\begin \lim _ \text _}}}}}\left( b\left( a+\frac\right) ^\right) \hspace= \hspace\lim _k \int _}}} g_k(x) d\mu _b(x) \hspace= \hspace\text _}}}}}(ba^) \hspace,\end$$

where we use the spectral measure \(\int f(x)d\mu _b(x)=(b_1, f(x)b_2)\) given by a decomposition \(b=b_1b_2^*\) with \(b_,b_\in L_2(}}}})\). By sending \(\varepsilon \) to 0, then

$$\begin \lim _ \text _}}}}(}\otimes \textrm_}}})}}(a)(a+\varepsilon )^) \hspace= \hspace\text _}}}}(}\otimes \textrm_}}})}}(a)a^)\hspace.\end$$

Thus (7.15) does indeed imply