T-duality simplifies bulk–boundary correspondence: the noncommutative case

Abstract

We state and prove a general result establishing that T-duality, or the Connes–Thom isomorphism, simplifies the bulk–boundary correspondence, given by a boundary map in K-theory, in the sense of converting it to a simple geometric restriction map. This settles in the affirmative several earlier conjectures of the authors and provides a clear geometric picture of the correspondence. In particular, our result holds in arbitrary spatial dimension, in both the real and complex cases, and also in the presence of disorder, magnetic fields, and H-flux. These special cases are relevant both to string theory and to the study of the quantum Hall effect and topological insulators with defects in condensed matter physics.

Appendices

Appendix A: Paschke’s map and the Connes–Thom isomorphism

This appendix explains why Paschke’s map [37] is, up to a sign convention, the Connes–Thom isomorphism [7] composed with an isomorphism from Green’s imprimitivity theorem. This result belongs to a family of ideas which can be found in [7,8,9, 37] and may be known to experts, but we provide a detailed argument for the reader’s reference. The Paschke map is explicitly defined for each ${\mathbb {Z}}$-algebra $({\mathcal {A}},\alpha )$, whereas the Connes–Thom map is more abstractly defined. As we will see, there is an analogous abstract characterization of the Paschke map, which determines its formula uniquely.

1.1 Generalities on Connes–Thom isomorphism

Recall from [7] that the Connes–Thom isomorphism is a natural transformation between the functors $K_\bullet (\cdot )$ and $K_{\bullet +1}((\cdot )\rtimes {\mathbb {R}})$, $\bullet \in {\mathbb {Z}}_2$, i.e. it assigns to every ${\mathbb {R}}$-$C^*$-algebra $({\mathscr {A}},\alpha )$ an isomorphism $\phi ^\bullet _\alpha : K_\bullet ({\mathscr {A}})\rightarrow K_{\bullet +1}({\mathscr {A}}\rtimes _\alpha {\mathbb {R}})$, such that for any morphism $\rho :({\mathscr {A}},\alpha )\rightarrow (\mathscr {B},\beta )$, the following diagram commutes:

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Here, $\rho \rtimes {\mathbb {R}}$ is the morphism of crossed products induced by the equivariant map $\rho$. Furthermore, Connes shows that $\phi _\alpha ^\bullet$ is an isomorphism, which is determined uniquely by this naturality property together with a normalization condition (for $({\mathcal {A}},\alpha )=({{\mathbb {C}}},{\mathrm {id}})$) and compatibility with suspensions.

1.2 The analogous axioms for Paschke’s map

If $\rho :({\mathcal {A}},\alpha )\rightarrow ({\mathcal {B}},\beta )$ is a morphism of ${\mathbb {Z}}$-$C^*$-algebras (not necessarily unital), the induction functor $\mathrm {Ind}:=\mathrm {Ind}_{\mathbb {Z}}^{\mathbb {R}}(\cdot )$ gives a morphism of induced algebras $\tilde{\rho }:\mathrm {Ind}({\mathcal {A}},\alpha )\rightarrow \mathrm {Ind}({\mathcal {B}},\beta )$ which is equivariant for the respective translation actions $\tau ^\alpha , \tau ^\beta$ of ${\mathbb {R}}$. Elements of $\mathrm {Ind}({\mathcal {A}},\alpha )$ are viewed either as bounded continuous functions $f:{\mathbb {R}}\rightarrow {\mathcal {A}}$ satisfying an equivariance condition⁶, $f(x+1)=\alpha (f(x)), x\in {\mathbb {R}}$, or alternatively, continuous $f:[0,1]\rightarrow {\mathcal {A}}$ such that $f(1)=\alpha (f(0))$ (the mapping torus). The translation action $\tau ^\alpha$ of ${\mathbb {R}}$ on $\mathrm {Ind}({\mathcal {A}},\alpha )$ is $(\tau ^\alpha _t f)(s)=f(s+t), s,t\in {\mathbb {R}}$.

Let $S\alpha$ be the suspended ${\mathbb {Z}}$-action on $S{\mathcal {A}}$ (acting by $\alpha$ on ${\mathcal {A}}$ and trivially on the suspension variable). Note that $S(\mathrm {Ind}({\mathcal {A}},\alpha ))\cong \mathrm {Ind}(S{\mathcal {A}},S\alpha )$ and $S({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})\cong S{\mathcal {A}}\rtimes _{S\alpha }{\mathbb {Z}}$, and write $S^\bullet _{(\cdot )}$ for the natural suspension isomorphisms $K_\bullet (\cdot )\rightarrow K_{\bullet -1}(S(\cdot ))$.

For $\bullet \in {\mathbb {Z}}_2$, let $\gamma ^\bullet$ be a natural transformation of the functors $({\mathcal {A}},\alpha )\mapsto K_\bullet (\mathrm {Ind}({\mathcal {A}},\alpha ))$ and $({\mathcal {A}},\alpha )\mapsto K_{\bullet +1}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})$ from ${\mathbb {Z}}$-algebras to abelian groups, satisfying the following three axioms:

Axiom A.1

(Normalization) If $({\mathcal {A}},\alpha )=({{\mathbb {C}}},{\mathrm {id}})$, then $\gamma ^0_{{\mathrm {id}}}:K_0(C({\mathbb {R}}/{\mathbb {Z}}))=K_0(C({\mathbb {T}}))\rightarrow K_1(C^*({\mathbb {Z}}))$ takes $[1_{C({\mathbb {T}})}]$ to the (Bott) generator [b] of $K_1(C^*({\mathbb {Z}}))$ corresponding to $1\in {\mathbb {Z}}$ regarded as an element of $C^*({\mathbb {Z}})$.

Axiom A.2

(Naturality) If $\rho :({\mathcal {A}},\alpha )\rightarrow ({\mathcal {B}},\beta )$ is a morphism of ${\mathbb {Z}}$-algebras, then the following diagram commutes:

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Axiom A.3

(Suspension) $\gamma$ is compatible with S in the sense that

$$\begin{aligned} S^{(\bullet +1)}_{\mathcal {A\rtimes _\alpha {\mathbb {Z}}}}\circ \gamma ^\bullet _\alpha =\gamma ^{(\bullet -1)}_{S\alpha }\circ S^\bullet _{\mathrm {Ind}({\mathcal {A}},\alpha )}. \end{aligned}$$

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Proposition A.4

There is a unique $\gamma$ satisfying Axioms A.1–A.3.

Proof

Outline: We use a modification of Connes’ argument for the uniqueness of $\phi$. Axiom A.3 and Bott periodicity ensure that we only need to look at $\gamma ^0$ (which we will simply denote by $\gamma$ subsequently). We need to show that $\gamma _\alpha [p]\in K_1({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})$ is uniquely determined for any projection $p\equiv \{p_\theta \}_{\theta \in [0,1]}$ in $\mathrm {Ind}({\mathcal {A}},\alpha )$. The basic idea is to construct a ${\mathbb {Z}}$-action $\alpha '$, exterior equivalent to $\alpha$, which fixes $p_0$. Furthermore, the new induced algebra $\mathrm {Ind}({\mathcal {A}},\alpha ')$, with its translation action $\tau ^{\alpha '}$, is ${\mathbb {R}}$-equivariantly isomorphic to the original $\mathrm {Ind}_{\alpha }{\mathcal {A}}$ with a modified action $\tau '$, where $\tau '$ is exterior equivalent to the original $\tau ^\alpha$ and fixes p. Thus, the projection $p'\in \mathrm {Ind}({\mathcal {A}},\alpha ')$ corresponding to $p\in \mathrm {Ind}({\mathcal {A}},\alpha )$ under this isomorphism is itself translation invariant under $\tau ^{\alpha '}$. Then, the Axioms determine what $\gamma _{\alpha '}[p']$ has to be. Finally, we need to argue that the modification $\alpha \mapsto \alpha '$ can be assumed because the above exterior equivalences determine unique maps, consistent with the Axioms, which yield the desired $\gamma _\alpha [p]$ from $\gamma _{\alpha '}[p']$.

Details: As in [7], we may assume without loss that ${\mathcal {A}}$ is unital and $p\in \mathrm {Ind}({\mathcal {A}},\alpha )$. By [7] Proposition 4 (c.f. [9] Lemma 10.16, [44]), we may also assume that on $\mathrm {Ind}({\mathcal {A}},\alpha )$ there is an action ${\mathbb {R}}$-action $\tau '$, exterior equivalent to $\tau ^\alpha$, which fixes p. Let $U\equiv \{U_t\}_{t\in {\mathbb {R}}}$ be the 1-cocycle on $\mathrm {Ind}({\mathcal {A}},\alpha )$ which relates $\tau ^\alpha$ and $\tau '$ via $\tau '_t=\mathrm {Ad}(U_t)\circ \tau ^\alpha _t$. Thus, $\{U_t\}_{t\in {\mathbb {R}}}$ satisfies the cocycle condition and equivariance condition,

$$\begin{aligned} U_{t_1+t_2}=U_{t_1}\tau ^\alpha _{t_2}(U_{t_2}),\quad U_t(1)=\alpha (U_t(0)), \quad t,t_1,t_2\in {\mathbb {R}}. \end{aligned}$$

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Define the modified ${\mathbb {Z}}$-action $\alpha '$ on ${\mathcal {A}}$ via

$$\begin{aligned} \alpha '_n=\mathrm {Ad}(U_n(0))\circ \alpha _n, \quad n\in {\mathbb {Z}}. \end{aligned}$$

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We can verify, from (31), that the assignment $u:n\mapsto U_n(0)=:u_n,\, n\in {\mathbb {Z}}$ defines a unitary 1-cocycle $\{u_n\}_{n\in {\mathbb {Z}}}$ on $({\mathcal {A}},\alpha )$, i.e. $u_{m+n}=u_m\alpha ^m(u_n)$. Thus, ${\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}\cong {\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}$ (a canonical isomorphism $\varphi _u$ is given in Lemma A.5 later).

Since $\tau '_1=\mathrm {Ad}(U_1)\circ \tau ^\alpha _1$ fixes p, it follows that $p_0:=p(0)\in {\mathcal {A}}$ fixed by $\alpha '$. Define the isomorphism $\Psi _U:\mathrm {Ind}({\mathcal {A}},\alpha )\rightarrow \mathrm {Ind}({\mathcal {A}},\alpha ')$ by

$$\begin{aligned} \Psi _Uf(s)=\mathrm {Ad}(U_s(0))[f(s)],\quad s\in {\mathbb {R}}, f\in \mathrm {Ind}({\mathcal {A}},\alpha ). \end{aligned}$$

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We can verify that $\Psi _Uf$ does satisfy $\alpha '$-equivariance and intertwines $\tau '$ on $\mathrm {Ind}({\mathcal {A}},\alpha )$ with the new translation action $\tau ^{\alpha '}$ on $\mathrm {Ind}({\mathcal {A}},\alpha ')$, i.e. $(\tau ^{\alpha '}_t(\Psi _U f))= (\Psi _U(\tau '_t f))$.

Observe that $p'(\theta )=p'_0=p_0$ since $p'=\Psi _U(p)$ is fixed by $\tau ^{\alpha '}$. The homomorphism

$$\begin{aligned} \omega :({{\mathbb {C}}},{\mathrm {id}})\rightarrow ({\mathcal {A}},\alpha '),\;\;\;\;\;\omega (\lambda )=\lambda p'_0 \end{aligned}$$

is ${\mathbb {Z}}$-equivariant, and we write $\tilde{\omega }$ for the corresponding inflated map $\tilde{\omega }:C({\mathbb {T}})\cong \mathrm {Ind}({{\mathbb {C}}},{\mathrm {id}})\rightarrow \mathrm {Ind}({\mathcal {A}},\alpha ')$. Note that

$$\begin{aligned} (\tilde{\omega } (1_{C({\mathbb {T}})}))(\theta )\equiv \omega (1_{C({\mathbb {T}})}(\theta ))=p'_0=p'(\theta ),\quad \theta \in [0,1], \end{aligned}$$

so $p'=\tilde{\omega }(1_{C({\mathbb {T}})})$. Then naturality and normalization mean that $\gamma _{\alpha '}[p']$ must be determined by the equation

$$\begin{aligned} \gamma _{\alpha '}[p']=\gamma _{\alpha '}\circ \tilde{\omega }_*[1_{C({\mathbb {T}})}]=(\omega \rtimes {\mathbb {Z}})_*\circ \gamma _{{\mathrm {id}}}[1_{C({\mathbb {T}})}]=(\omega \rtimes {\mathbb {Z}})_*[b]. \end{aligned}$$

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Corollary A.7 gives the details of how $\gamma [p]$ is obtained from $\gamma _{\alpha '}[p']$. $\square$

1.2.1 Connes’ $2\times 2$ matrix trick

Lemma A.5

(c.f. [7] Lemma 2) Let $\alpha , \alpha '$ be exterior equivalent ${\mathbb {Z}}$-actions on ${\mathcal {A}}$ related by a unitary 1-cocycle ${\mathbb {Z}}\ni n \mapsto u_n \in U{\mathcal {M}}{\mathcal {A}}$. Then

$$\begin{aligned} \kappa \begin{pmatrix}a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \end{pmatrix}= \begin{pmatrix} \alpha (x_{11}) &{}\quad \alpha (x_{12})u_1^* \\ u_1\alpha (x_{21}) &{}\quad \alpha '(x_{22}) \end{pmatrix} \end{aligned}$$

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is a ${\mathbb {Z}}$-action on $M_2({\mathcal {A}})$ which restricts to $\alpha$ in the top-left corner and to $\alpha '$ in the bottom-right corner. Let $\iota , \iota '$ be the equivariant inclusions of ${\mathcal {A}}$ into their respective corners in $M_2({\mathcal {A}})$. There is a unique isomorphism $\varphi _u:{\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}\rightarrow {\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}$ such that $(\iota '\rtimes {\mathbb {Z}})(\varphi _u(y))=\mathrm {Ad}(\begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0\end{pmatrix})[(\iota \rtimes {\mathbb {Z}})(y)]$ for all $y\in {\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}$.

Similarly, there are inclusions $\tilde{\iota }, \tilde{\iota '}$ of the induced algebras into the respective corners of $\mathrm {Ind}_\kappa M_2({\mathcal {A}})$. Let $\tau ^\alpha , \tau '$ be the exterior equivalent ${\mathbb {R}}$-actions on $\mathrm {Ind}({\mathcal {A}},\alpha )$ as in the proof of Proposition A.4, related by the 1-cocycle $\{U_t\}_{t\in {\mathbb {R}}}$. A straightforward computation shows:

Lemma A.6

Define $X\in \mathrm {Ind}_{\kappa } M_2({\mathcal {A}})$ by

$$\begin{aligned} X(s)=\begin{pmatrix} 0 &{}\quad U_s^*(0)\\ U_s(0) &{}\quad 0 \end{pmatrix}. \end{aligned}$$

Then the isomorphism $\Psi _U$ satisfies $\tilde{\iota '}(\Psi _U(f))=\mathrm {Ad}(X)[\tilde{\iota }f], f\in \mathrm {Ind}({\mathcal {A}},\alpha )$.

Corollary A.7

Suppose $\gamma$ satisfies Axioms A.1–A.3, and $\alpha , \alpha ', U$ are as above. Then $\gamma ^\bullet _{\alpha '}=(\varphi _u)_*\circ \gamma ^\bullet _\alpha \circ (\Psi _U^{-1})_*$.

Proof

Recall that inner automorphisms induce the identity map in K-theory. Thus, Lemmas A.5 and A.6 show that $(\iota \rtimes {\mathbb {Z}})_*=(\iota '\rtimes {\mathbb {Z}})_*\circ (\varphi _u)_*$ and $\tilde{\iota }_*=\tilde{\iota '}_*\circ (\Psi _U)_*$. Using the naturality axiom, and dropping the superscript ${}^\bullet$ for now,

$$\begin{aligned}&(\iota '\rtimes {\mathbb {Z}})_*\circ \gamma _{\alpha '}= \gamma _\kappa \circ \tilde{\iota '}_*= \gamma _\kappa \circ \tilde{\iota }_*\circ (\Psi _U^{-1})_*\\&\quad = (\iota \rtimes {\mathbb {Z}})_*\circ \gamma _\alpha \circ (\Psi _U^{-1})_*= (\iota '\rtimes {\mathbb {Z}})_*\circ (\varphi _u)_*\circ \gamma _\alpha \circ (\Psi _U^{-1})_*, \end{aligned}$$

so injectivity of $(\iota '\rtimes {\mathbb {Z}})_*$ (see [7] Proposition 2–3) gives the required result. $\square$

1.3 Paschke’s isomorphisms

Paschke’s isomorphisms [37]

$$\begin{aligned} \gamma ^{\bullet ,{{{\mathrm{Paschke}}}}}_\alpha : K_\bullet (\mathrm {Ind}({\mathcal {A}},\alpha ))\rightarrow K_{\bullet +1}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}) \end{aligned}$$

are first defined for unital ${\mathcal {A}}$ and $\bullet =0$ on [p], by exhibiting the existence of a path of unitaries $\theta \mapsto w_\theta \in {\mathcal {A}}$ such that

$$\begin{aligned} p(\theta )=\mathrm {Ad}(w_\theta )(p_0),\quad \theta \in [0,1]. \end{aligned}$$

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In particular, $p_1=\mathrm {Ad}(w_1)(p_0)$. Then $\gamma ^{0,\mathrm {Paschke}}_\alpha [p]$ is defined to be

$$\begin{aligned} \gamma ^{0,{{{\mathrm{Paschke}}}}}_\alpha [p]=[L^*w_1p_0+1-p_0], \end{aligned}$$

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where $L\in {\mathcal {M}}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})$ is the unitary implementing $\alpha$ (i.e. $\alpha (a)=LaL^*, a\in {\mathcal {A}}$), with this map shown to be well-defined in K-theory.⁷

The $\bullet =1$ case is defined by compatibility with suspensions. Since $\gamma ^{0,\mathrm {Paschke}}_\alpha$ may be expressed using representative projections and unitaries (rather than their K-theory classes), it is easy to see that the naturality Axiom A.2 is satisfied. It is also clear that normalization Axiom A.1 is satisfied up to a minus sign.

On the other hand, there is a commutative diagram due to the Connes–Thom natural isomorphisms,

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Composing $\phi _{\tau ^\alpha }^\bullet$ with the natural K-theory isomorphisms

$$\begin{aligned} M_\alpha ^{\bullet +1,{\mathrm{Green}}}:K_{\bullet +1}(\mathrm {Ind}({\mathcal {A}},\alpha )\rtimes _{\tau ^\alpha }{\mathbb {R}})\cong K_{\bullet +1}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}) \end{aligned}$$

given by the Morita equivalence $\mathrm {Ind}({\mathcal {A}},\alpha )\rtimes _{\tau ^\alpha }{\mathbb {R}}\sim _{\mathrm{M.E.}}{\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}$ (this is a special case of Green’s imprimitivity theorem [16]), we obtain another $\gamma$ satisfying Axioms A.1–A.3: The normalization axiom follows from the fact that $\phi _{\tau ^{{\mathrm{id}}}}$ is an isomorphism, so there is only one possibility (apart from a sign choice) for $M_{\mathrm{id}}^{0,{\mathrm{Green}}}\circ \phi _{\tau ^{\mathrm{id}}}:{\mathbb {Z}}[1_{C({\mathbb {T}})}]\rightarrow {\mathbb {Z}}[b]$. Compatibility with suspensions is inherited from the Connes–Thom map, while naturality follows from naturality of implementing the Morita equivalence (see [10] Chapter 4). By Proposition A.4, we have

Theorem A.8

The map $\gamma ^{\bullet ,\,{{{\mathrm{Paschke}}}}}_\alpha :K_\bullet (\mathrm {Ind}({\mathcal {A}},\alpha ))\rightarrow K_{\bullet +1}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})$ is, up to a sign, the composition $M_\alpha ^{\bullet +1,\,{\mathrm{Green}}}\circ \phi ^\bullet _{\tau ^\alpha }$.

1.3.1 Paschke’s unitaries and Connes’ cocycle

Given a $p\in \mathrm {Ind}({\mathcal {A}},\alpha )$, we may start from Connes’ cocycle $\{U_t\}_{t\in {\mathbb {R}}}$ in (31) above and obtain a path of unitaries $w_\theta :=U^*_\theta (0), \theta \in [0,1]$ in ${\mathcal {A}}$. Then $w_0=1$ and

$$\begin{aligned} p_\theta =p(\theta )\,{=}\,\tau ^\alpha _\theta (p(0))\,{=}\,(\mathrm {Ad}(U^*_\theta )\circ \tau '(p))(0)=\mathrm {Ad}(U^*_\theta (0))(p(0))=\mathrm {Ad}(w_\theta )(p_0), \end{aligned}$$

thus $\{w_\theta \}_{\theta \in [0,1]}$ is precisely a Paschke path whose existence was proved in [37].

Conversely, given a Paschke path $\{w_\theta \}_{\theta \in [0,1]}$ such that $w_0=1$ and $\mathrm {Ad}(w_\theta )(p_0)=p_\theta$, we can reconstruct Connes’ cocycle for the action $\tau ^\alpha$ on $\mathrm {Ind}({\mathcal {A}},\alpha )$ as follows. Regard $p\in \mathrm {Ind}({\mathcal {A}},\alpha )$ as a projection-valued function ${\mathbb {R}}\rightarrow {\mathcal {A}}$ through the equivariance condition $p(s)\equiv p_s=\alpha (p_{s-1}), s\in {\mathbb {R}}$. Similarly, extend Paschke’s $w:[1,0]\mapsto {\mathcal {A}}$ to a continuous unitary function $w:{\mathbb {R}}\ni s\mapsto w_s\in {\mathcal {A}}$ by the recursion relation $w_s=\alpha (w_{s-1})\cdot w_1$. It is easy to check that $p_s=\mathrm {Ad}(w_s)(p_0)$ for all $s\in {\mathbb {R}}$ and that $n\mapsto w_n$ defines a 1-cocycle for $({\mathcal {A}},\alpha )$. The projection $p_0$ is then fixed by $\alpha '_n=\mathrm {Ad}(w^*_n)\circ \alpha , n\in {\mathbb {Z}}$, since

$$\begin{aligned} \alpha '(p_0)=\mathrm {Ad}(w^*_1)(\alpha (p_0))=\mathrm {Ad}(w^*_1)\circ \mathrm {Ad}(w_1)(p_0)=p_0. \end{aligned}$$

The assignment $t\mapsto U_t\in \mathrm {Ind}({\mathcal {A}},\alpha )$ defined by

$$\begin{aligned} U_t(s)=w_sw_{s+t}^*, \quad s,t\in {\mathbb {R}} \end{aligned}$$

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is the desired Connes 1-cocycle for $(\mathrm {Ind}({\mathcal {A}},\alpha ),\tau ^\alpha )$; we can verify that each $U_t$ is $\alpha$-equivariant and satisfies the cocycle condition. If we define the ${\mathbb {R}}$-action $\tau '$ on $\mathrm {Ind}({\mathcal {A}},\alpha )$ by $\tau '_t=\mathrm {Ad}(U_t)\circ (\tau ^\alpha _t)$, we find that the projection p is now fixed by $\tau '$.

1.4 Paschke’s formula from axioms

Recall that $\gamma ^0_\alpha [p]$ can be computed by passing to $\alpha '$, using $\gamma ^0_\alpha [p]=(\varphi _u^{-1})_*\circ \gamma ^0_{\alpha '}\circ (\Psi _U)_*[p]$. In (34), we saw that $\gamma ^0_{\alpha '}\circ (\Psi _U)_*[p]=\gamma ^0_{\alpha '}[p']=(\omega \rtimes {\mathbb {Z}})_*[b].$ Note that even if ${\mathcal {A}}$ is unital, the homomorphism $\omega$ (and thus $\omega \rtimes {\mathbb {Z}}$) is nonunital in general, so the induced map $(\omega \rtimes {\mathbb {Z}})_*$ in $K_1$ is slightly tricky to compute (c.f. Proposition 8.1.6 of [46]). Namely, we have to append a formal unit $\mathbf{1}$ to $C({\mathbb {T}})$ and ${\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}$, replace [b] by $[\mathbf{1}-1_{C({\mathbb {T}})}+b]$, then compute $(\omega \rtimes {\mathbb {Z}})_*$ using the unital extension $(\omega \rtimes {\mathbb {Z}})^+:C({\mathbb {T}})^+\rightarrow ({\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}})^+$. This gives

$$\begin{aligned} ((\omega \rtimes {\mathbb {Z}})^+)_*[\mathbf{1}-1_{C({\mathbb {T}})}+b]&=[\mathbf{1}+(\omega \rtimes {\mathbb {Z}})(-1_{C({\mathbb {T}})}+b)]\\&=[\mathbf{1}-L'p'_0+p'_0]\\&=[\mathbf{1}-1_{{\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}}+1_{{\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}}+L'p'_0-p'_0], \end{aligned}$$

and mapping back to unitaries in $\mathrm {Ind}({\mathcal {A}},\alpha ')$, we obtain

$$\begin{aligned} (\omega \rtimes {\mathbb {Z}})_*[b]=[1+L'p'_0-p'_0]=[L'p_0+(1-p_0)]. \end{aligned}$$

Finally, recall that the isomorphism $\varphi _u$ implements the isomorphism ${\mathcal {A}}\rtimes _\alpha {\mathbb {Z}}\cong {\mathcal {A}}\rtimes _{\alpha '}{\mathbb {Z}}$ induced by the exterior equivalence $\alpha '_n=\mathrm {Ad}(w^*_n)\circ \alpha _n$, so it effects $w_1^*L\mapsto L'$. Thus,

$$\begin{aligned} \gamma ^0_\alpha [p]=(\varphi _u^{-1})_*\circ \gamma ^0_{\alpha '}\circ (\Psi _U)_*[p]=(\varphi _u^{-1})_*[L'p_0+(1-p_0)]=[w_1^*Lp_0+(1-p_0)], \end{aligned}$$

which, up to a sign, concides with $\gamma ^{0,{{{\mathrm{Paschke}}}}}_\alpha [p]$.

1.5 The real $C^*$-algebra case

A general reference for the K-theory of real $C^*$-algebras is [49]. The Connes–Thom isomorphisms and Pimsner–Voiculescu exact sequence also hold for real $C^*$-algebras ${\mathcal {A}}$ and real crossed products, see [48, 49]. Similarly, Paschke’s map $\gamma ^{0,{{{{{\mathrm{Paschke}}}}}}}_\alpha$ still makes sense on real ${\mathbb {Z}}$-algebras $({\mathcal {A}},\alpha )$. We can then repeat the arguments in this appendix to obtain the real version of Theorem A.8. The only significant change is in the normalization axiom, and in taking $\bullet \in {\mathbb {Z}}_8$ rather than in ${\mathbb {Z}}_2$.

Recall that ${\mathbb {C}}\rtimes _{\mathrm{id}}{\mathbb {Z}}=C_{\mathbb {R}}^*({\mathbb {Z}})\cong C({\mathbb {T}};\varsigma )$ where $\varsigma$ is the involution $\theta \mapsto -k\theta$ on ${\mathbb {T}}=\widehat{{\mathbb {Z}}}$ inherited from complex conjugation of characters (parametrized by $\theta \in [0,1]$), and $C({\mathbb {T}};\varsigma )$ is the real $C^*$-algebra of continuous functions $f\in {\mathbb {T}}\rightarrow {{\mathbb {C}}}$ such that $\overline{f(\theta )}=f(-\theta )$. These functions are precisely the ones with real Fourier coefficients. It is known (pp. 40 of [49]) that $K_1(C({\mathbb {T}};\varsigma ))\cong {\mathbb {Z}}[z]\oplus {\mathbb {Z}}_2[-1_{C({\mathbb {T}},\varsigma )}]$, where z denotes the unitary function $\theta \mapsto e^{2\pi i\theta }$ with winding number 1. Furthermore, [z] is the Bott element implementing (1,1) periodicity in KR-theory (c.f. Theorem 1.5.4 of [49], Theorem 10.3 of [27]), so we also write it as $[b_r]$. We can also identify it with the unitary L implementing the (trivial) automorphism in ${\mathbb {R}}\rtimes _{\mathrm{id}}{\mathbb {Z}}$.

On the other hand, the induced algebra for $({\mathbb {R}},{\mathrm{id}})$ is just the continuous real-valued functions on the circle $C({\mathbb {T}};{\mathbb {R}})$. It is also known that

$$\begin{aligned} KO_0(C({\mathbb {T}};{\mathbb {R}}))={\mathbb {Z}}[1_{C({\mathbb {T}};{\mathbb {R}})}]\oplus {\mathbb {Z}}_2\left[ [P_{{\mathrm{Mob}}}]-[1_{C({\mathbb {T}};{\mathbb {R}})}]\right] , \end{aligned}$$

where $P_{\mathrm{Mob}}$ is the Möbius projection in $M_2(C({\mathbb {T}};{\mathbb {R}}))$ given by

$$\begin{aligned} P_{\mathrm{Mob}}(\theta )=\mathrm {Ad}(R(\theta )) \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}=\mathrm {Ad}\begin{pmatrix}\cos \,\pi \theta &{} -\sin \,\pi \theta \\ \sin \,\pi \theta &{} \cos \,\pi \theta \end{pmatrix}\begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

It is straightforward to compute the real Paschke map on $KO_0(C({\mathbb {T}};{\mathbb {R}}))$. For the constant projection 1, the Paschke path of unitaries is trivial, so we have $\gamma ^{0,{{{\mathrm{Paschke}}}}}_{\mathrm{id}}[1]=[L+1-1]=[L]=[b_r]$. For the Möbius projection, the Paschke path of unitaries is $w_\theta =R(\theta )$, so $w_1$ is minus the identity. Then

$$\begin{aligned}&\gamma ^{0,{\mathrm{Paschke}}}_{\mathrm{id}} [P_{\mathrm{Mob}}]=[-LP_{\mathrm{Mob}}(0)+1_2-P_{\mathrm{Mob}}(0)]\\&\quad =\left[ \begin{pmatrix} -z &{}\quad 0 \\ 0 &{}\quad 1_{C({\mathbb {T}},\varsigma )}\\ \end{pmatrix}\right] =[-z]=[-1_{C({\mathbb {T}},\varsigma )}]+[z], \end{aligned}$$

so $\gamma ^{0,{{{\mathrm{Paschke}}}}}_{\mathrm{id}}([P_{\mathrm{Mob}}]-[1_{C({\mathbb {T}};{\mathbb {R}})}])=[-1_{C({\mathbb {T}},\varsigma )}]$, i.e. the torsion generators are correctly mapped to each other.

The normalization axiom in the real $C^*$-algebra case needs to be replaced by

Axiom A.9

(Normalization) If $({\mathcal {A}},\alpha )=({\mathbb {R}},{\mathrm {id}})$, then $\gamma ^0_{{\mathrm {id}}}:KO_0(C({\mathbb {T}};{\mathbb {R}}))\cong KO^0({\mathbb {T}})\rightarrow KO_1(C_{\mathbb {R}}^*({\mathbb {Z}}))\cong KO_1(C({\mathbb {T}};\varsigma ))$ takes $[1_{C({\mathbb {T}};{\mathbb {R}})}]$ to the (Bott) generator $[b_r]$ and $([P_{\mathrm{Mob}}]-[1_{C({\mathbb {T}};{\mathbb {R}})}])$ to $[-1_{C({\mathbb {T}},\varsigma )}]$.

Theorem A.10

There is a unique natural transformation $\gamma$ of the functors $({\mathcal {A}},\alpha )\mapsto KO_\bullet (\mathrm {Ind}({\mathcal {A}},\alpha ))$ and $({\mathcal {A}},\alpha )\mapsto KO_{\bullet +1}({\mathcal {A}}\rtimes _\alpha {\mathbb {Z}})$, $\bullet \in {\mathbb {Z}}_8$ , from the category of real ${\mathbb {Z}}$-algebras to abelian groups, which satisfy Axiom A.9 and Axioms A.2–A.3 with K replaced by KO. Thus, the real version of Corollary A.8 holds: $\gamma ^{\bullet ,\,{{{{{\mathrm{Paschke}}}}}}}_\alpha =M_\alpha ^{\bullet +1,\,{\mathrm{Green}}}\circ \phi ^\bullet _{\tau ^\alpha }$ up to a sign convention.

Appendix B: Multipliers for lattice groups

Let $N={\mathbb {Z}}^d$ and for a multiplier $\sigma$, let $\widetilde{\sigma }(n,n^\prime ) = \sigma (n,n^\prime )/ \sigma (n^\prime ,n)$ be the antisymmetric bicharacter labelling its class in $H^2(N,{\mathbb {T}})$. We reconstruct a canonical representative multiplier in this class starting from $\widetilde{\sigma }$. Choose generators $\{e_j\}_{j\in J}$, where J is a totally ordered set, and, expanding $n = \sum _j n_je_j \in N$, and $n^\prime$ similarly, we define the multiplier

$$\begin{aligned} \sigma _J(n,n^\prime ) = \prod _{j<k}\widetilde{\sigma }(e_j,e_k)^{n_jn_k^\prime }, \end{aligned}$$

which is also a bicharacter (but not antisymmetric). Then we see that

$$\begin{aligned} \widetilde{\sigma }_J(n,n^\prime )= & {} \frac{\sigma _J(n,n^\prime )}{\sigma _J(n^\prime ,n)}\\= & {} \prod _{j<k}\widetilde{\sigma }(e_j,e_k)^{n_jn_k^\prime }\prod _{k<j}\widetilde{\sigma }(e_k,e_j)^{-n_jn_k^\prime }\\= & {} \prod _{j\ne k}\widetilde{\sigma }(e_j,e_k)^{n_jn_k^\prime }\\= & {} \widetilde{\sigma }(n,n^\prime ). \end{aligned}$$

Thus, we see that $\widetilde{\sigma }_J = \widetilde{\sigma }$, and consequently, $\sigma _J$ is cohomologous to $\sigma$, and so also to any $\sigma _{J^\prime }$ for any other totally ordered set $J^\prime$. We note that when $N = {\mathbb Z}^{2}$ and $J = \{1,2\}$ indexes the usual generators with the natural ordering, the antisymmetric bicharacter $\widetilde{\sigma }(n,n^\prime ) = \exp (i(n_1n_2^\prime - n_2n_1^\prime ))$ produces $\sigma _J(n,n^\prime ) = \exp (i(n_1n_2^\prime ))$.