Gauge Theory on Noncommutative Riemannian Principal Bundles

Abstract

We present a new, general approach to gauge theory on principal G-spectral triples, where G is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for G-$C^*$-algebras and prove that the resulting noncommutative orbitwise family of Kostant’s cubic Dirac operators defines a natural unbounded $KK^G$-cycle in the case of a principal G-action. Then, we introduce a notion of principal G-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded $KK^G$-theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal G-bundles and are compatible with $\theta$-deformation; in particular, they cover the $\theta$-deformed quaternionic Hopf fibration $C^\infty (S^7_\theta ) \hookleftarrow C^\infty (S^4_\theta )$ as a noncommutative principal ${{\,\mathrm{SU}\,}}(2)$-bundle.

Appendices

Appendix A: Peter–Weyl Theory and G-Hilbert Modules

In this appendix, we provide a sketch of Peter–Weyl theory for continuous representations of compact connected Lie groups on Fréchet spaces in general and for actions on $C^*$-algebras and Hilbert $C^*$-modules in particular. A detailed account of the general picture can be found in [45, Chapter 4]; specifics related to $C^*$-algebras and Hilbert $C^*$-modules can be found, for instance, in  [16, Sect. VIII.20; 44, Sect. 2].

For the moment, let E be a ${\mathbf {Z}}_2$-graded Fréchet space topologised by a countable family of seminorms $\{ \Vert \cdot \Vert _{E;i} \}_{i\in {\mathbf {N}}}$, and $\rho : G \rightarrow {{\,\mathrm{GL}\,}}(E)$ a strongly continuous representation of G on E by even isometries.

Definition A.1

For every $\pi \in \widehat{G}$, the $\pi$-isotypical component of E is the closed subspace

$$\begin{aligned} E_\pi :=E_{\pi }^\rho :=\{ T(v): T \in {{\,\mathrm{Hom}\,}}_G(V_\pi ,E), \, v \in V_\pi \}, \end{aligned}$$

which is the range of the idempotent $P_\pi \in {\mathbf {L}}(E)$ defined by

$$\begin{aligned} \forall e \in E, \quad P_\pi (e) :=d_\pi \int _G \overline{\chi _\pi (g)} \rho _g(e)\,\mathrm {d}{g}. \end{aligned}$$

In particular, the isotypical component of the trivial representation ${\mathbf {1}} \in \widehat{G}$ is the subspace $E^G$ of fixed points, while the corresponding projection $P_{{\mathbf {1}}}$ simply averages with respect to the Haar measure.

Proposition A.2

The family of idempotents $\{ P_\pi \}_{\pi \in \widehat{G}}$ defines an orthogonal resolution of the identity in the sense that

$$\begin{aligned} \forall \pi _1,\pi _2 \in \widehat{G}, \quad P_{\pi _1}P_{\pi _2} = {\left\{ \begin{array}{ll} P_{\pi _1} &{}\text {if }\pi _1=\pi _2,\\ 0 &{}\text {else,} \end{array}\right. } \end{aligned}$$

while $\sum _{\pi \in \widehat{G}} P_\pi = {{\,\mathrm{id}\,}}_E$ pointwise on the dense subspace

$$\begin{aligned} E^\mathrm {alg}:=E^{\mathrm {alg};\rho } :=\bigoplus _{\pi \in \widehat{G}}^{\mathrm {alg}} E_\pi ; \end{aligned}$$

(A.1)

if E is a Hilbert space and $\rho$ is unitary, then $\sum _{\pi \in \widehat{G}} P_\pi = {{\,\mathrm{id}\,}}_E$ strongly in ${\mathbf {L}}(E)$.

Now, for each $k \in {\mathbf {N}}$, define the subspace of $C^k$ vectors by

$$\begin{aligned} E^k :=E^{k;\rho } :=\{ e \in E \vert (g \mapsto \rho _g(e)) \in C^k(G,E) \}. \end{aligned}$$

(A.2)

The infinitesimal representation $\mathrm {d}{\rho } : {\mathfrak {g}}\rightarrow {{\,\mathrm{Hom}\,}}_{\mathbf {C}}(E^1,E)$ then permits us to topologise $E^k$ as a Fréchet space with the family of seminorms $\{ \Vert \cdot \Vert _{E;j,\varvec{m}} \}_{(j,\varvec{m}) \in {\mathbf {N}}\times {\mathbf {N}}^k}$ defined by

$$\begin{aligned} \forall j \in {\mathbf {N}}, \, \forall \varvec{m} \in {\mathbf {N}}^k, \, \forall e \in E^k, \quad \Vert e \Vert _{E;j,\varvec{m}} :=\Vert \left( \mathrm {d}\rho (\epsilon _{m_1}) \circ \cdots \circ \mathrm {d}\rho (\epsilon _{m_k})\right) (e)_{E;j} \Vert . \end{aligned}$$

As a result, the subspace of smooth vectors

$$\begin{aligned} E^\infty :=E^{\infty ;\rho } :=\bigcap _{k = 1}^\infty E^{k,\rho } \supset E^{\mathrm {alg};\rho } \end{aligned}$$

also defines a Fréchet space, and $\mathrm {d}{\rho }$ extends to a representation $\mathrm {d}{\rho } : {\mathcal {U}}({\mathfrak {g}}) \rightarrow {\mathbf {L}}(E)$ of the universal enveloping algebra ${\mathcal {U}}({\mathfrak {g}})$ of ${\mathfrak {g}}$; in particular, it follows that

$$\begin{aligned} \forall e \in E^\infty , \quad e = \sum _{\pi \in \widehat{G}} P_\pi (e), \end{aligned}$$

with absolute convergence in $E^\infty$. Finally, in this regard, note that if $\rho$ is strongly smooth, then $E = E^\infty$ as vector spaces and ${{\,\mathrm{id}\,}}_E : E^\infty \rightarrow E$ is a continuous bijection.

We now specialise to strongly continuous actions on $C^*$-algebras and Hilbert modules.

Definition A.3

A G-$C^*$-algebra is a ${\mathbf {Z}}_2$-graded $C^*$-algebra A together with a strongly continuous action $\alpha : G \rightarrow {{\,\mathrm{Aut}\,}}^+(A)$ of G on A by even $*$-automorphisms. The fixed point algebra is the $C^{*}$-subalgebra $A^{G}:=\{ a\in A: \alpha _{g}(a)=a,\,\forall g\in G \}$ of G-fixed vectors.

Suppose that $(A,\alpha )$ is a G-$C^*$-algebra. The map ${\mathbf {E}}_A :=P_{{\mathbf {1}}} : A \twoheadrightarrow A^G$ is a faithful conditional expectation onto $A^G$. More generally, for every $\pi \in \widehat{G}$, the $\pi$-isotypical component $A_\pi$ defines Hilbert $(A^G,A^G)$-bimodule with respect to left and right multiplication by $A^G$ and the Hermitian metric defined by

$$\begin{aligned} \forall a_1,a_2 \in A, \quad \left( a_1, a_2\right) _{A^G} :={\mathbf {E}}_A(a_1^*a_2) = \int _G \alpha _g(a_1^*a_2)\,\mathrm {d}{g}. \end{aligned}$$

(A.3)

In fact, for every $\pi \in \widehat{G}$, it follows that $(A_\pi )^*= A_{\pi ^*}$, where $\pi ^*$ is the contragredient of $\pi$, so that $A^{\mathrm {alg}}$ defines a dense $*$-subalgebra of A; moreover, $A^\infty$ defines a dense Fréchet pre-$C^*$-algebra closed under the holomorphic functional calculus.

Definition A.4

Let $(A,\alpha )$ and $(B,\beta )$ are G-$C^*$-algebras. Then a Hilbert G-(A, B)-bimodule is a ${\mathbf {Z}}_2$-graded Hilbert (A, B)-bimodule E together with a strongly continuous representation $U : G \rightarrow {{\,\mathrm{GL}\,}}^+(E)$ of G on E by even Banach space automorphisms, such that

$$\begin{aligned} \forall g \in G, \, \forall a \in A, \, \forall e \in E, \, \forall b \in B, \quad U_g(aeb)= & {} \alpha _g(a)U_g(e)\beta _g(b);\\ \forall g \in G, \, \forall e_1,e_2 \in E, \quad \left( U_g(e_1), U_g(e_2)\right) _B= & {} \beta _g(\left( e_1, e_2\right) _B). \end{aligned}$$

For E a G-(A, B) Hilbert module, ${\mathbf {L}}_B(E)$ denotes the $C^{*}$-algebra of adjointable endomorphisms, while ${\mathbf {K}}_B(E)$ denotes the $C^*$-subalgebra of compact endomorphisms. Although ${\mathbf {K}}_B(E)$ naturally defines a G-$C^{*}$-algebra, ${\mathbf {L}}_B(E)$ does not. This motivates the definition of the $C^{*}$-subalgebra

$$\begin{aligned} {\mathbf {L}}^{U}_B(E):=\{ T\in {\mathbf {L}}_B(E): g\mapsto U_{g}TU_{g}^{*}\in C(G,{\mathbf {L}}_B(E)) \}\subset {\mathbf {L}}_B(E), \end{aligned}$$

(A.4)

of G-continuous adjointable operators, which is a G-$C^{*}$-algebra by construction.

Example A.5

One can complete the G-equivariant ${\mathbf {Z}}_2$-graded $(A,A^G)$-bimodule A with respect to the $A^G$-valued Hermitian metric $\left( \cdot , \cdot \right) _{A^G}$ defined by Equation (A.3) to obtain a Hilbert G-$(A,A^G)$-bimodule $L^2_v(A) :=L^2_v(A;\alpha )$. In particular, $\alpha : G \rightarrow {{\,\mathrm{Aut}\,}}^+(A)$ extends to its own spatial implementation $L^2_v(\alpha ) : G \rightarrow {\mathbf {U}}_{A^G}(L^2_v(A^G))$.

Finally, suppose that $(A,\alpha )$ is a G-$C^*$-algebra and that B is a $C^*$-algebra with trivial G-action. Then, for every $\pi \in \widehat{G}$, the $\pi$-isotypical component $E_\pi$ defines a right Hilbert B-submodule of E that is G-equivariantly unitarily equivalent to $V_\pi \otimes {{\,\mathrm{Hom}\,}}_G(V_\pi ,E)$ endowed with the $A^G$-linear Hermitian metric given by

$$\begin{aligned}&\forall v_1,v_2 \in V_\pi , \, \forall T_1,T_2 \in {{\,\mathrm{Hom}\,}}_G(V_\pi ,E),\\&\quad \left( v_1 \otimes T_1, v_2 \otimes T_2\right) _B :=d_\pi ^{-1}\left( T_1(v_1), T_2(v_2)\right) _B; \end{aligned}$$

an explicit unitary equivalence $\phi _\pi : V_\pi \otimes {{\,\mathrm{Hom}\,}}_G(V_\pi ,E) \rightarrow E_\pi$ is given by

$$\begin{aligned} \forall v \in V_\pi , \, \forall T \in {{\,\mathrm{Hom}\,}}_G(V_\pi ,E), \quad \phi _\pi (v \otimes T) :=d_{\pi }^{1/2}T(v) \end{aligned}$$

with inverse $\phi _\pi ^{-1} : E_\pi \rightarrow V_\pi \otimes {{\,\mathrm{Hom}\,}}_G(V_\pi ,E)$ given by

$$\begin{aligned} \forall e \in E_\pi , \quad \phi _\pi ^{-1}(e) :=d_\pi ^{1/2}\sum _{i=1}^{d_\pi } v_i \otimes \int _G U_g (e) \otimes \langle v_i, \pi (g^{-1})(\cdot )\rangle \, \mathrm {d}{g}, \end{aligned}$$

where $\{ v_1,\dots ,v_{d_\pi } \}$ is any orthonormal basis for $V_\pi$.

Proposition A.6

(Peter-Weyl theorem for Hilbert modules). Let $(A,\alpha )$ be a G-$C^*$-algebra and B a $C^*$-algebra with trivial G-action. For every $\pi \in \widehat{G}$, the Hilbert B-submodule $E_\pi$ is complemented in E with G-invariant orthogonal projection $P_\pi \in {\mathbf {L}}_B(E)$; moreover, the map

$$\begin{aligned} E \rightarrow \bigoplus _{\pi \in \widehat{G}} E_\pi , \quad e \mapsto (P_\pi (e))_{\pi \in \widehat{G}} \end{aligned}$$

(A.5)

is an isomorphism of right Hilbert G-B-modules (i.e., Hilbert G-$({\mathbf {C}},B)$-bimodules).

In the special case of the Hilbert G-$(A,A^G)$-bimodule $(L^2_v(A),L^2_v(\alpha ))$, for every $\pi \in {\widehat{G}}$, the norm on $A_\pi = L^2_v(A)_\pi$ as a right Hilbert B-submodule of $L^2_v(A)$ is equivalent to the restriction of the $C^*$-norm of A [44, Cor. 2.6], and ${\text {Hom}}_G(V_\pi ,L^2_v(A)) = {\text {Hom}}_G(V_\pi ,A)$.

Appendix B: Hermitian Module Connections from Strong Connections

We present a general method for constructing Hilbert module connections (in the sense of Mesland [80]) from strong connections [56] relative to a spectral triple. This reconciles two prominent notions of connection in the noncommutative geometry literature.

We begin with a minimalistic definition of noncommutative fibration over a spectral triple admitting well-defined integration over the fibres (but without presupposing any noncommutative fibrewise family of Dirac operators).

Definition B.1

Let $({\mathcal {B}},H_0,T)$ be a complete spectral triple for a separable $C^*$-algebra B with adequate approximate identity $\{ \phi _k \}_{k\in {\mathbf {N}}}$. We define a noncommutative fibration over $({\mathcal {B}},H_0,T)$ to be a triple $(A,{\mathbf {E}}_A,{\mathcal {A}})$ consisting of:

• 1. a $C^*$-algebra A together with non-degenerate $*$-monomorphism $B \hookrightarrow A$, such that $\{ \phi _k \}_{k\in {\mathbf {N}}}$ defines an approximate identity of A;

• 2. a faithful conditional expectation ${\mathbf {E}}_A : A \rightarrow B$, such that the resulting completion $L^2(A;{\mathbf {E}}_A)$ of A to a Hilbert B-module admits a countable frame contained in A;

• 3. a dense $*$-subalgebra ${\mathcal {A}}\subset A$, such that ${\mathcal {B}}\subset {\mathcal {A}}$.

We now define a notion of horizontal differential calculus on a noncommutative fibration compatible with the de Rham calculus on the base—this gives us a suitable functional analytic setting for the strong connection condition as identified by Hajac [56].

Definition B.2

Let $({\mathcal {B}},H_0,T)$ be a complete spectral triple for a separable $C^*$-algebra B, and let $(A,{\mathbf {E}}_A,{\mathcal {A}})$ be a noncommutative fibration over $({\mathcal {B}},H_0,T)$. We define a horizontal differential calculus for $(A,{\mathbf {E}}_A,{\mathcal {A}})$ to be a triple $(\Omega ,{\mathbf {E}}_\Omega ,\nabla _0)$ consisting of:

• 1. a $C^*$-algebra $\Omega$ together with a $*$-mononorphism $A \hookrightarrow \Omega$;

• 2. a positive contraction ${\mathbf {E}}_\Omega : \Omega \rightarrow {\mathbf {L}}(H_0)$, such that ${\mathbf {E}}_\Omega |_{A} = {\mathbf {E}}_A$ and

  $$\begin{aligned} \forall b \in B,\, \forall \omega \in \Omega , \quad {\mathbf {E}}_\Omega (b\omega ) = b{\mathbf {E}}_\Omega (\omega ), \quad {\mathbf {E}}_\Omega (\omega b) = {\mathbf {E}}_\Omega (\omega )b; \end{aligned}$$

• 3. a $*$-derivation $\nabla _0 : {\mathcal {A}}\rightarrow \Omega$, such that

  $$\begin{aligned} \forall a \in A, \, \forall b \in {\mathcal {B}}, \quad {\mathbf {E}}_\Omega (a \cdot \nabla _0(b)) = {\mathbf {E}}_A(a) \cdot [T,b]. \end{aligned}$$

  (B.1)

Moreover, we say that $(\Omega ,{\mathbf {E}}_\Omega ,\nabla _0)$ satisfies the strong connection condition whenever

$$\begin{aligned} \forall a \in {\mathcal {A}}, \quad \nabla _0(a) \in \overline{A \cdot \nabla _0({\mathcal {B}})}^{\Omega }. \end{aligned}$$

(B.2)

Finally, we show that the horizontal exterior derivative of a horizontal differential calculus satisfying the strong connection condition canonically induces a Hilbert module connection. Recall that $\mathbin {{\widehat{\otimes }}^{h}}$ denotes the Haagerup tensor product.

Theorem B.3

Let $({\mathcal {B}},H_0,T)$ be a complete spectral triple for a separable $C^*$-algebra B, let $(A,{\mathbf {E}}_A,{\mathcal {A}})$ be a noncommutative fibration over $({\mathcal {B}},H_0,T)$, and let $(\Omega ,{\mathbf {E}}_\Omega ,\nabla _0)$ be a horizontal differential calculus for $(A,{\mathbf {E}}_A,{\mathcal {A}})$ that satisfies the strong connection condition. Then $\nabla _0$ canonically induces a Hermitian T-connection $\nabla : {\mathcal {A}}\rightarrow L^2(A;{\mathbf {E}}_A) \mathbin {\widehat{\otimes }^h_B} \Omega ^1_T$ on $L^2(A;{\mathbf {E}}_A)$ by

$$\begin{aligned} \forall a \in {\mathcal {A}}, \quad \nabla (a) :=\sum _{i\in {\mathbf {N}}} \xi _i \mathbin {{\widehat{\otimes }}}{\mathbf {E}}_\Omega \left( \xi _i^*\nabla _0(a)\right) , \end{aligned}$$

(B.3)

where $\{ \xi _i \}_{i \in {\mathbf {N}}}$ is any frame for $L^2(A;{\mathbf {E}}_A)$ contained in A.

Proof

Given a frame $\{ \xi _i \}_{i \in {\mathbf {N}}} \subseteq A$ for $L^2(A;{\mathbf {E}}_A)$, which exists by assumption, we show that (B.3) defines a B-module connection $\nabla$. Let $a \in {\mathcal {A}}$, and write $\nabla (a)=\sum _k a_{k}\nabla _0(b_{k})$ for $a_k \in A$ and $b_k \in {\mathcal {B}}$, so that, by continuity of ${\mathbf {E}}_\Omega$ and closure of $\Omega ^1_T$ in ${\mathbf {L}}(H_0)$,

$$\begin{aligned} \forall i \in {\mathbf {N}}, \quad {\mathbf {E}}_\Omega (\xi _i^*\nabla _0(a)) = {\mathbf {E}}_\Omega \left(\sum _k \xi _i^*a_k \nabla _0(b_k)\right) = \sum _k {\mathbf {E}}_A(\xi _i^*a_k)[T,b_k] \in \Omega ^1_T; \end{aligned}$$

without loss of generality, we may assume that $\Vert \nabla _0(a) \Vert \le 1$.

Choose K large enough that $\left\| \sum _{k> K}a_{k}\nabla _0(b_{k})\right\| ^{2}< \varepsilon /6,$ so that for any n, N,

$$\begin{aligned}&\left\Vert \sum _{n\le |i |\le N} \xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}\nabla _0(a))\right\Vert ^{2}_{h}\\&\le 2\left\Vert \sum _{n\le |i |\le N} \sum _{k\le K}\xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}a_{k}\nabla _0(b_{k}))\right\Vert ^{2}_{h}+2\left\Vert \sum _{n\le |i |\le N} \sum _{k> K}\xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}a_{k}\nabla _0(b_k))\right\Vert ^{2}_{h}\\&\le 2\left\Vert \sum _{n\le |i |\le N} \sum _{k\le K}\xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}a_{k}\nabla _0(b_k))\right\Vert ^{2}_{h}+2\left\Vert \sum _{k> K}a_{k}\nabla _0(b_k)\right\Vert ^{2}_{h}\\&\le 2\left\Vert \sum _{n\le |i |\le N} \sum _{k\le K}\xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}a_{k}\nabla _0(b_k))\right\Vert ^{2}_{h}+\frac{\varepsilon }{3}. \end{aligned}$$

Now choose m and n large enough, so that

$$\begin{aligned}&\left\Vert \sum _{k \le K} (\phi _m a_k - a_k)\nabla _0(b_k)\right\Vert ^2_\Omega<\frac{\varepsilon }{12},\\&\quad \left\Vert \sum _{|i |\ge n} \phi _m \xi _i \mathbf {E}_A(\xi ^*_i \phi _m) \right\Vert _{L^2(A,\mathbf {E}_A)} <\frac{\varepsilon }{12\left\Vert \sum _{k\le K}a_{k}\nabla _0(b_k)\right\Vert ^{2}}. \end{aligned}$$

Then, for any $N\ge n$ we can estimate

$$\begin{aligned}&\left\Vert \sum _{n\le |i |\le N} \sum _{k\le K}\xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}a_{k}\nabla _0(b_k))\right\Vert ^{2}_{h}\\&\le \left\Vert {\mathbf {E}}_A\left(\sum _{i} \xi _{i}\xi _{i}^{*}\right)\right\Vert \left\Vert \sum _{n\le |i |\le N} \sum _{k,\ell \le K}\nabla _0(b_k)^{*}{\mathbf {E}}_A\left( a_{k}^{*}\xi _{i}\right){\mathbf {E}}_A\left(\xi _{i}^{*}a_{\ell }\right)\nabla _0(b_\ell )\right\Vert \\&\le 2\left\Vert \sum _{n\le |i |\le N} \sum _{k,\ell \le K}\nabla _0(b_k)^{*}{\mathbf {E}}_A(a_{k}^{*}\phi _{m}\xi _{i}){\mathbf {E}}_A\left(\xi _{i}^{*}\phi _{m}a_{\ell }\right)\nabla _0(b_\ell )\right\Vert \\&\quad \quad +2\left\Vert \sum _{k,\ell \le K} \nabla _0(b_k)^{*}(\phi _{m}a_{k}-a_{k})^{*}(\phi _{m}a_{\ell }-a_{\ell })\nabla _0(b_{\ell })\right\Vert \\&\le 2\left\Vert \sum _{n\le |i |\le N} \sum _{k,\ell \le K} \nabla _0(b_k)^{*}{\mathbf {E}}_A(a_{k}^{*}\phi _{m}\xi _{i}){\mathbf {E}}_A\left(\xi _{i}^{*}\phi _{m}a_{\ell }\right)\nabla _0(b_\ell )\right\Vert + \frac{\varepsilon }{6} \end{aligned}$$

Now observe that by [68, Lemma 4.2], we can estimate

$$\begin{aligned} \left( \sum _{|i |\ge n}{\mathbf {E}}_A(a_{k}^{*}\phi _{m}\xi _{i}){\mathbf {E}}_A\left(\xi _{i}^{*}\phi _{m}a_{\ell }\right) \right) _{k,\ell \le K}&=\left( {\mathbf {E}}_A\left( a_{k}^{*}\left( \sum _{|i|\ge n}\phi _{m}\xi _{i}{\mathbf {E}}_A\left( \xi _{i}^{*}\phi _{m}\right) \right) a_{\ell }\right) \right) _{k,\ell \le K}\\&\le \left\Vert \sum _{|i |\ge n}\phi _{m}\xi _{i}{\mathbf {E}}_A\left(\xi _{i}^{*}\phi _{m}\right)\right\Vert \left( {\mathbf {E}}_A(a_{k}^{*}a_{\ell })\right) _{k,\ell \le K}\\&\le \frac{\varepsilon }{12\left\Vert \sum _{k\le K}a_{k}\nabla _0(b_k)\right\Vert ^{2}} \left( {\mathbf {E}}_A(a_{k}^{*}a_{\ell })\right) _{k,\ell \le K}, \end{aligned}$$

as matrices. Therefore

$$\begin{aligned} \left\Vert \sum _{n\le |i |\le N}\sum _{k,\ell \le K}\nabla _0(b_k)^{*} {\mathbf {E}}(a_{k}^{*}\phi _{m}\xi _{i}){\mathbf {E}}\left(\xi _{i}^{*}\phi _{m}a_{\ell }\right)[T,b_{\ell }]\right\Vert \le \frac{\varepsilon }{12}, \end{aligned}$$

and we continue to estimate

This proves that the series is convergent in the Haagerup norm. Independence of the choice of frame $\{ \xi _{i} \} \subset A$ now follows, for if $\{ \eta _j \} \subset A$ is another countable frame we write

$$\begin{aligned} \sum _{i} \xi _{i}\otimes {\mathbf {E}}_\Omega (\xi _{i}^{*}\nabla _0(a))&=\sum _{i,j}\eta _{j}\otimes {\mathbf {E}}_A(\eta _{j}^{*}\xi _{i}){\mathbf {E}}_\Omega (\xi _{i}^{*}\nabla _0(a))=\sum _{i,j}\eta _{j}\otimes {\mathbf {E}}_\Omega \left({\mathbf {E}}_A(\eta _{j}^{*}\xi _{i})\xi _{i}^{*}\nabla _0(a)\right)\\&=\sum _{j}\eta _{j}\otimes {\mathbf {E}}_\Omega \left(\sum _{i}\left( \xi _{i}{\mathbf {E}}_A(\xi _{i}^*\eta _{j})\right) ^*\nabla _0(a)\right)=\sum _{j}\eta _{j}\otimes {\mathbf {E}}_\Omega (\eta _{j}^{*}\nabla _0(a)), \end{aligned}$$

where convergence of the relevant sums follows from continuity of ${\mathbf {E}}_A$ and ${\mathbf {E}}_\Omega$. $\square$