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\begin{document}

\section{Connections and the existence of the Kasparov
product}\label{connections-and-the-existence-of-the-kasparov-product}

\emph{Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).}

A tensor product of represented Hilbert modules gives the right space
for composing two cycles. It does not give an operator automatically.
The second operator usually fails to respect the relation used to form
the interior tensor product. A connection replaces that nonexistent
operator by one that respects elementary tensors up to compact errors.
Kasparov's technical theorem then combines this connection with the
first operator.

We prove existence and uniqueness, including the compactness and
positivity calculations. We also calculate products with homomorphisms,
Morita equivalences, two circle classes and a vector bundle.
Associativity and comparison of the two homotopy relations are proved in
the next lesson; neither is used to construct the product here.

All tensor products and commutators are graded. For homogeneous
operators, \[
[R,T]_{\mathrm{gr}}=RT-(-1)^{|R||T|}TR.
\] In particular the commutator of two odd operators is their
anticommutator. We write \(\mathcal L(E)\) and \(\mathcal K(E)\) for
adjointable and compact module operators, and use the analogous notation
between different modules. The stabilization and compact-operator rules
were proved in \emph{Graded C*-algebras, Clifford algebras and graded
Hilbert modules}. The square-root partition is Theorem 3.1 and Corollary
3.2 of \emph{Kasparov's technical theorem}.

\subsection{1. The connection
conditions}\label{the-connection-conditions}

Let \(E_1\) be a countably generated graded Hilbert \(D\)-module, let
\(E_2\) be a graded Hilbert \(B\)-module, and let
\(\psi:D\to\mathcal L(E_2)\) be a graded homomorphism. Set \[
E=E_1\widehat\otimes_\psi E_2.
\] For homogeneous \(x\in E_1\) define the creation operator \[
T_x y=x\widehat\otimes y,\qquad
T_x^*(z\widehat\otimes y)=\psi(\langle x,z\rangle)y.
\tag{1.1}
\] These formulas give \(\|T_x\|\leq\|x\|\) and degree \(|x|\). In
particular \[
T_x^*T_z=\psi(\langle x,z\rangle),\qquad
T_xT_z^*=\theta_{x,z}\widehat\otimes1.
\tag{1.2}
\]

Let \(F_2\) have degree \(p\), with \([F_2,\psi(d)]_{\mathrm{gr}}\)
compact for every \(d\in D\). An \textbf{\(F_2\)-connection} is an
operator \(G\) of degree \(p\) such that \[
\begin{aligned}
T_xF_2-(-1)^{p|x|}GT_x&\in\mathcal K(E_2,E),\\
F_2T_x^*-(-1)^{p|x|}T_x^*G&\in\mathcal K(E,E_2)
\end{aligned}
\tag{1.3}
\] for every homogeneous \(x\). The two conditions are both imposed; for
self-adjoint \(F_2,G\), either one implies the other by taking adjoints.

\textbf{Lemma 1.1 (the compactness test).} Put \[
I_0=\mathcal K(E_1)\widehat\otimes1\subset\mathcal L(E).
\] An operator \(Z\) is a \(0\)-connection if and only if \[
ZI_0\subset\mathcal K(E),\qquad I_0Z\subset\mathcal K(E).
\tag{1.4}
\] Every connection graded commutes with \(I_0\) modulo
\(\mathcal K(E)\).

\textbf{Proof.} If \(ZT_x\) and \(T_x^*Z\) are compact, multiplication
by \(T_y^*,T_y\) and (1.2) prove (1.4) first on rank-one generators and
then on their norm closure. Conversely take an approximate identity
\(u_\lambda\) of \(\mathcal K(E_1)\). It acts in norm on every vector of
\(E_1\); this follows first on the range of rank-one operators and then
by density. Thus \[
\|(u_\lambda\widehat\otimes1)T_x-T_x\|
\leq\|u_\lambda x-x\|\longrightarrow0.
\] Under (1.4), \(Z(u_\lambda\widehat\otimes1)T_x\) is compact and
converges in norm to \(ZT_x\). Applying the same argument to \(Z^*\)
proves the other condition.

For a connection \(G\), move it across the two creation operators in
\(T_xT_y^*\), using (1.3). The two resulting errors are compact; the
remaining terms have the sign \((-1)^{p(|x|+|y|)}\). This is precisely
the graded commutator with \(\theta_{x,y}\widehat\otimes1\). Norm
approximation proves the assertion for \(I_0\). \(\square\)

The following elementary rules will make the later calculations shorter.

\textbf{Lemma 1.2 (connection calculus).} Adjoints of connections are
connections of the adjoints. Sums and products of homogeneous
connections correspond to sums and products of the second operators,
with the usual degree bookkeeping. Two connections of the same operator
differ by a \(0\)-connection. For self-adjoint connections, continuous
functional calculus has the same property; even and odd functions have
the corresponding parity.

If \(S=R\widehat\otimes1\), for homogeneous \(R\in\mathcal L(E_1)\),
then \[
[S,G]_{\mathrm{gr}}\text{ is a }0\text{-connection}.
\tag{1.5}
\] Finally, connections compose on a triple interior tensor product.

\textbf{Proof.} Taking adjoints in (1.3) gives the first assertion. For
products use, modulo compact operators, \[
GH T_x
\equiv(-1)^{(|G|+|H|)|x|}T_xF_2F'_2.
\] This follows by moving \(H\), then \(G\), past \(T_x\); the errors
remain compact after multiplication by bounded operators. The
adjoint-side calculation is the same. The sum and difference assertions
follow directly from (1.3). Polynomial approximation on a compact
spectral interval proves the assertion about continuous functions, since
the connection conditions are norm closed.

For (1.5), \(ST_x=T_{Rx}\). Moving \(G\) across \(T_x\) and \(T_{Rx}\)
shows \[
SGT_x\equiv(-1)^{p|x|}T_{Rx}F_2,\qquad
GST_x\equiv(-1)^{p(|R|+|x|)}T_{Rx}F_2.
\] Their graded difference vanishes modulo compacts. The second creation
condition follows by the adjoint calculation.

For the last assertion write a creation operator for an elementary
vector \(x\widehat\otimes y\) as \(T_xU_y\), where \(U_y\) is the
creation operator for the second tensor product. Move the final
connection past \(T_x\), then the intermediate connection past \(U_y\).
The total sign is the sum of the two vector degrees. Each error is
compact. Taking adjoints proves the second condition. Finite sums of
elementary vectors are dense and the creation operators depend
continuously in norm on their vectors, completing the proof. \(\square\)

Here and below, an operator on the first factor really does tensor with
\(1\): it is \(D\)-linear and adjointable. An operator on the second
factor generally does not tensor with \(1\).

\textbf{Theorem 1.3 (existence of connections).} An \(F_2\)-connection
exists. It can be chosen self-adjoint when \(F_2\) is self-adjoint, of
the same degree, and with norm at most \(\|F_2\|\). A norm-continuous
path \(F_{2,t}\) admits a norm-continuous path of such connections.

\textbf{Proof.} Regard \(E_1\) as a module over the forced unitization
\(D^+\), and extend \(\psi\) to the unital homomorphism
\(\psi^+(d+\lambda1)=\psi(d)+\lambda1\). Countable generation is
retained. The natural map \[
E_1\widehat\otimes_\psi E_2
\longrightarrow E_1\widehat\otimes_{\psi^+}E_2
\] is unitary: both inner products agree, and the unitization balancing
relations already hold by linearity and the scalar action. Graded
stabilization realizes \(E_1=P\widehat H_{D^+}\) for an even projection
\(P\).

On \(\widehat H_{D^+}\widehat\otimes_{\psi^+}E_2
\cong\widehat\ell^2\widehat\otimes E_2\), define \[
C(e_j\widehat\otimes y)
=(-1)^{p|e_j|}e_j\widehat\otimes F_2y.
\tag{1.6}
\] For a homogeneous vector \(x=\sum_j e_jx_j\) supported on finitely
many coordinates, \(|x_j|=|x|-|e_j|\). The \(j\)-th entry of its
creation error is consequently \[
\begin{aligned}
&(CT_x-(-1)^{p|x|}T_xF_2)_j\\
&\quad=(-1)^{p|e_j|}
 \bigl(F_2\psi^+(x_j)-(-1)^{p|x_j|}\psi^+(x_j)F_2\bigr).
\end{aligned}
\] Every entry is compact by the input commutator condition; the added
scalar coordinates have zero commutator. A finite column of compact
operators is compact, by approximating each entry by finite-rank
operators. Applying the same calculation to \(F_2^*\) and taking
adjoints gives the other creation error. The bound \(\|T_x\|\leq\|x\|\)
extends both errors by norm approximation to every vector. This proves
the full connection condition for \(C\).

Tensor \(P\) with \(1\), and compress: \[
G=(P\widehat\otimes1)C(P\widehat\otimes1).
\tag{1.7}
\] For \(x\in PE_1^{\mathrm{standard}}\), one has
\((P\widehat\otimes1)T_x=T_x\). Thus the first creation error for \(G\)
is the projection times the corresponding error for \(C\); the second is
obtained on the other side. This proves the connection property.
Compression retains the degree, norm bound and self-adjointness. Apply
the same fixed compression to \(C_t\) for a path \(F_{2,t}\); (1.6)
shows norm continuity. \(\square\)

The construction is called a \textbf{Grassmann connection}. It uses the
unitization of the intermediate algebra, so it does not require that
\(\psi(D)E_2\) be all of \(E_2\).

\subsection{2. The product axioms and
normalization}\label{the-product-axioms-and-normalization}

Take cycles \[
\alpha=(E_1,\phi_1,F_1)\in\mathcal E(A,D),\qquad
\beta=(E_2,\psi,F_2)\in\mathcal E(D,B).
\] On \(E=E_1\widehat\otimes_\psi E_2\) put \[
\phi=\phi_1\widehat\otimes1,\qquad S=F_1\widehat\otimes1.
\] A \textbf{product operator} is an odd \(F\in\mathcal L(E)\)
satisfying:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \((E,\phi,F)\) is a Kasparov cycle.
\item
  \(F\) is an \(F_2\)-connection.
\item
  For every \(a\in A\), \[
  \phi(a)[S,F]_{\mathrm{gr}}\phi(a)^*\geq0
  \quad\text{in }\mathcal L(E)/\mathcal K(E).
  \tag{2.1}
  \]
\end{enumerate}

We can normalize \(F_1,F_2\) to odd self-adjoint contractions by Theorem
3.2 of \emph{Kasparov modules and the groups \(KK(A,B)\)}. These
normalizations preserve the product axioms in the following precise
sense.

\textbf{Lemma 2.1 (locally compact changes of the inputs).} Replacing
either input operator by a locally compact perturbation does not change
the connection and localized positivity requirements, modulo compacts. A
product-operator perturbation retains the product axioms if it is both a
\(0\)-connection and two-sided locally compact for \(\phi(A)\).
Self-adjoint normalization and clipping of a product operator have these
properties.

\textbf{Proof.} Suppose \(R\) and \(R^*\) are locally compact for
\(\psi(D)\). Then \[
(T_xR)^*(T_xR)=R^*\psi(\langle x,x\rangle)R
\] is compact. An adjointable operator \(T\) with \(T^*T\) compact is
compact: use \(T(T^*T+\varepsilon)^{-1}T^*T\) and the scalar bound on
its error to approximate \(T\) by compact operators. Thus \(T_xR\) is
compact; the adjoint version proves \(RT_x^*\) compact. This proves the
assertion for the second input.

For a first-input perturbation \(R_1\), write \(R=R_1\widehat\otimes1\).
Each \(R\phi(a)\) and \(\phi(a)R\) lies in \(I_0\). Use Lemma 1.1 to
move the connection \(F\) across these operators, and use its cycle
commutator to move it across \(\phi(a)\). The two terms in
\(\phi(a)[R,F]_{\mathrm{gr}}\phi(a)^*\) then cancel modulo compacts.
Thus (2.1) is unchanged.

For the stated product-operator perturbation \(Z\), both terms of
\(\phi(a)[S,Z]_{\mathrm{gr}}\phi(a)^*\) are compact: one contains
\(Z\phi(a)^*\), the other contains \(\phi(a)Z\). The connection
assertion is its assumed \(0\)-connection condition. In self-adjoint
normalization this condition holds automatically, since \(F-F^*\) is a
\(0\)-connection when \(F_2\) is self-adjoint. A clipping function then
changes \(F\) by a function vanishing at \(\pm1\). Since \(F_2^2-1\) is
locally compact for \(\psi(D)\), that function of \(F_2\) has compact
creation products by the first part. Lemma 1.2 makes the clipping change
a \(0\)-connection. The cycle perturbation rule supplies its local
compactness. \(\square\)

\subsection{3. Existence from the technical
theorem}\label{existence-from-the-technical-theorem}

\textbf{Theorem 3.1 (existence of the product).} If \(A\) is separable,
\(D\) is \(\sigma\)-unital and both input modules are countably
generated, there is a product operator. For self-adjoint input operators
it can be chosen self-adjoint. No \(\sigma\)-unitality hypothesis on
\(B\) is needed.

\textbf{Proof.} Normalize the inputs as above and choose a self-adjoint
odd \(F_2\)-connection \(G\). The module \(E\) is countably generated:
products \(x_i\widehat\otimes y_j\) of two countable generating families
have dense \(B\)-span, using the balancing relation to move the
\(D\)-coefficients into \(\psi(D)y_j\). Put \(J=\mathcal K(E)\), a
\(\sigma\)-unital algebra, and \[
A_1=C^*(I_0,J)=I_0+J.
\] Its two summands are \(\sigma\)-unital, so a sum of their strictly
positive elements is strictly positive in \(A_1\). Set \[
\begin{aligned}
A_2&=C^*(J\cup\mathcal Z),\\
\mathcal Z&=\{G^2-1,[S,G]_{\mathrm{gr}}\}\\
&\quad\cup\{[G,\phi(a)]_{\mathrm{gr}}:a\in A\}.
\end{aligned}
\tag{3.1}
\] All generators can be selected from a countable dense subset of
\(A\). Adjoining them to the \(\sigma\)-unital ideal \(J\) gives a
\(\sigma\)-unital algebra: if \(h_J\) is strictly positive in \(J\), add
a convergent positive sum of suitably weighted \(z_j^*z_j+z_jz_j^*\).
The resulting \(h\) has \(f_n(h)z_j\to z_j\) and \(z_jf_n(h)\to z_j\),
since \(h\) dominates a positive multiple of both \(z_j^*z_j\) and
\(z_jz_j^*\); it also acts as an approximate identity on \(J\). Thus
\(f_n(h)\) approximates every element of the generated algebra.

Each generator in (3.1), other than \(J\), is a \(0\)-connection. For
\(G^2-1\), Lemma 1.2 gives a connection of \(F_2^2-1\), whose creation
errors are compact by Lemma 2.1. The commutators are \(0\)-connections
by (1.5). They and their adjoints therefore annihilate \(I_0\) modulo
\(J\). This proves \[
A_1A_2\subset J.
\tag{3.2}
\] The separable graded space spanned by \(S,G,\phi(A)\) derives
\(A_1\). For \(S,\phi(A)\) this is the adjointable ideal property on the
first module; for \(G\) it is Lemma 1.1.

The graded technical theorem supplies even positive contractions \(M,N\)
with \[
M+N=1,\quad MA_1\subset J,\quad NA_2\subset J,
\tag{3.3}
\] and compact commutators with \(S,G,\phi(A)\). Its square-root
corollary gives the same annihilation and commutation properties for the
needed roots. First define \[
F_0=M^{1/2}S+N^{1/2}G.
\tag{3.4}
\] The operator \(M^{1/2}\) is a \(0\)-connection by Lemma 1.1. Hence
its product with \(S\) is a \(0\)-connection. Since \(N-1=-M\) is a
\(0\)-connection, functional calculus gives \(N^{1/2}-1\) a
\(0\)-connection as well. Consequently the second term in (3.4) is an
\(F_2\)-connection. Their sum is the required connection.

All the following equalities are modulo \(J\). Move the cutoffs past
\(S,G,\phi(a)\) to obtain \[
\begin{aligned}
(F_0^2-1)\phi(a)
&=M(S^2-1)\phi(a)\\
&\quad+N(G^2-1)\phi(a)\\
&\quad+M^{1/2}N^{1/2}[S,G]_{\mathrm{gr}}\phi(a)\\
&=0.
\end{aligned}
\tag{3.5}
\] Here \((S^2-1)\phi(a)\in I_0\), while the other two defects belong to
\(A_2\). Also \[
\begin{aligned}
\left[F_0,\phi(a)\right]_{\mathrm{gr}}
&\equiv M^{1/2}[S,\phi(a)]_{\mathrm{gr}}\\
&\quad+N^{1/2}[G,\phi(a)]_{\mathrm{gr}}=0.
\end{aligned}
\tag{3.6}
\] The first commutator is in \(I_0\), and the second is in \(A_2\). The
adjoint defect is compact because the two cutoffs commute with the
self-adjoint \(S,G\) modulo \(J\).

Finally, \[
\begin{aligned}
\phi(a)[S,F_0]_{\mathrm{gr}}\phi(a)^*
&\equiv2\phi(a)M^{1/2}S^2\phi(a)^*\\
&\equiv2\phi(a)M^{1/2}\phi(a)^*\geq0.
\end{aligned}
\tag{3.7}
\] The omitted commutator with \(G\) is killed by \(N^{1/2}\). The
second equality uses \((S^2-1)\phi(a)^*\in I_0\) and the cutoff
commutators. This proves all three product conditions.

For an exactly self-adjoint operator use \[
F=M^{1/4}SM^{1/4}+N^{1/4}GN^{1/4}.
\tag{3.8}
\] It differs from \(F_0\) by a globally compact operator, so all the
verified conditions persist. Undoing the input normalizations with Lemma
2.1 proves the stated generality. \(\square\)

Notice the localization in (3.5). For a nonunital source one only knows
\((F_1^2-1)\phi_1(a)\) is compact. One cannot replace this by a global
assertion \(F_1^2-1\in\mathcal K(E_1)\).

\subsection{4. Uniqueness and descent to homotopy
classes}\label{uniqueness-and-descent-to-homotopy-classes}

\textbf{Theorem 4.1 (uniqueness up to operator homotopy).} Under the
hypotheses of Theorem 3.1, any two product operators on the given tensor
module represent the same operator-homotopy class.

\textbf{Proof.} Normalize \(F_1\) and the two product operators \(F,F'\)
to self-adjoint contractions. Lemma 2.1 preserves their product
conditions. This time apply the technical theorem with the same \(A_1\)
and \[
A_2=C^*(J,\ [S,F]_{\mathrm{gr}},\
[S,F']_{\mathrm{gr}},\ F-F').
\tag{4.1}
\] These generators are \(0\)-connections by Lemma 1.2. Include
\(S,F,F',\phi(A)\) in the separable derivating space. As in Section 3
all hypotheses are satisfied. Let \(M,N\) be the resulting partition and
put \[
H=M^{1/2}S+N^{1/2}F.
\tag{4.2}
\] Use its self-adjoint sandwich version if desired. The existence
calculation shows that \(H\) is a product operator: the defects of \(F\)
are already locally compact, while \(M\) kills the first-module defects
and \(N\) kills \([S,F]_{\mathrm{gr}}\).

We compare \(H\) to both original operators. Fix \(a\), put
\(b=\phi(a)\), and take \(T=F\) or \(T=F'\). Modulo \(J\) the comparison
is \[
\begin{aligned}
b[T,H]_{\mathrm{gr}}b^*
&=bM^{1/2}[T,S]_{\mathrm{gr}}b^*\\
&\quad+2bN^{1/2}b^*.
\end{aligned}
\tag{4.3}
\] For \(T=F\), use the cycle equation for \(F\). For \(T=F'\), use
\(N^{1/2}(F-F')=0\) modulo \(J\) to replace the term
\(N^{1/2}[F',F]_{\mathrm{gr}}\) by \(2N^{1/2}F^2\), then uses the same
localized cycle equation.

These sums are positive. Indeed \(M^{1/2}\) commutes modulo \(J\) with
all operators in the sandwich, so its product with the positive
localized anticommutator is positive. Proposition 3.3 of \emph{Kasparov
modules and the groups \(KK(A,B)\)} supplies an operator homotopy from
\(F\) to \(H\), and one from \(F'\) to \(H\). Reverse and concatenate
the latter to prove the assertion. \(\square\)

The proof needs positivity, rather than smallness of the ordinary
commutator. Both \(S\) and \(F\) are odd, so their relevant bracket is
\(SF+FS\).

\textbf{Theorem 4.2 (the bilinear product).} The construction gives a
well-defined bilinear map \[
KK(A,D)\times KK(D,B)\longrightarrow KK(A,B),\qquad
(x,y)\longmapsto x\widehat\otimes_D y.
\tag{4.4}
\]

\textbf{Proof.} A homotopy of the second cycle is a cycle over
\(C([0,1],B)\). Apply Theorem 3.1 with that coefficient algebra. The
tensor module evaluates to the two endpoint tensor modules; creation
conditions, compact defects and quotient positivity descend under
evaluation. Its endpoints are products of the endpoint input cycles and
hence have the required classes by Theorem 4.1.

For a homotopy of the first cycle, its intermediate algebra is
\(C([0,1],D)\), still \(\sigma\)-unital. Tensor the second cycle
externally with \(C([0,1])\), using the pointwise representation and
constant operator. Compact defects are continuous compact-valued fields.
Apply the same theorem and evaluate. The canonical map identifying the
evaluated interior tensor product with the tensor product of the
evaluated modules is unitary by its inner-product formula and density of
elementary tensors. This proves invariance in both variables.

Direct sums of product operators satisfy every axiom for a direct sum in
either input, by the two block creation conditions and block positivity.
Theorem 4.1 then proves additivity. Degenerate cycles have zero homotopy
class by Proposition 2.3 of the cycle lesson, so the preceding homotopy
invariance also covers their addition. Thus (4.4) is well-defined on the
groups. \(\square\)

\subsection{5. Essential representations and
homomorphisms}\label{essential-representations-and-homomorphisms}

A representation is \textbf{essential} if \(\overline{\psi(D)E_2}=E_2\).
This is useful when identifying \(D\widehat\otimes_\psi E_2\) with
\(E_2\). An arbitrary representation need not be essential, and its
essential submodule need not have an adjointable orthogonal projection.

\textbf{Proposition 5.1 (essential replacement).} If \(D\) is
\(\sigma\)-unital, every cycle \((E_2,\psi,F_2)\) is homotopic to a
cycle on \(E_2^0=\overline{\psi(D)E_2}\), with its essential
representation.

\textbf{Proof.} Normalize \(F_2\) to be self-adjoint. Set \[
L=\{f\in C([0,1],D^+):f(1)\in D\}.
\] It is a countably generated Hilbert ideal of \(C([0,1],D^+)\). For
example, if \(h\) is strictly positive in \(D\), the function
\(\ell(t)=(1-t)1+th\) is strictly positive in \(L\): its standard
functional-calculus approximate identity converges on every fiber, and
uniformly on a given continuous section by compactness of the interval
and a finite-neighborhood argument. Thus \(L\) is \(\sigma\)-unital and
generated as a Hilbert ideal by a countable approximate identity.

Tensor \(L\) with the constant interval module
\(\overline E_2=C([0,1],E_2)\), using the pointwise unital extension
\(\psi^+\). The map \[
f\widehat\otimes \xi\longmapsto
\bigl(t\mapsto\psi^+(f(t))\xi(t)\bigr)
\] identifies this tensor module with \[
\mathcal M=\{\xi\in C([0,1],E_2):\xi(1)\in E_2^0\}.
\tag{5.1}
\] It preserves inner products. Its image is dense: first approximate
the endpoint value by finite sums \(\psi(d)y\), subtract their constant
sections, and approximate the remaining section by sections vanishing
near \(1\). Those sections are obtained with scalar cutoffs supported
away from \(1\), which belong to \(L\). The approximation is uniform.
Products of countable generating families show countable generation of
\(\mathcal M\).

Choose a self-adjoint Grassmann \(F_2\)-connection \(G\) on this tensor
module. Let \(D\) act through its constant embedding in \(L\), followed
by the left action on the tensor product. Left multiplication by an
element of \(L\) is compact on the Hilbert ideal \(L\): rank-one
operators are multiplication by \(fg^*\), and their closed span is
\(L\). Lemma 1.1 therefore gives compact representation commutators.

For the square defect one must retain localization to \(D\), since
\(F_2\) need not be a cycle for the unitization \(D^+\). Approximate
\(d\in D\) by finite sums \(d_1d_2^*\) with \(d_i\in D\), regarded as
constant vectors in \(L\). Its represented left multiplication on the
tensor module is then approximated by \(T_{d_1}T_{d_2}^*\). The
connection identities give \[
(G^2-1)T_{d_1}T_{d_2}^*
\equiv T_{d_1}(F_2^2-1)T_{d_2}^*
\pmod{\mathcal K(\mathcal M)}.
\] The right side is compact by the creation compactness test of Lemma
2.1, because the original square defect is locally compact for
\(\psi(D)\). Its interval field is compact in
\(C([0,1],\mathcal K(E_2))\). Taking the finite-sum limit proves the
localized square condition. Self-adjointness supplies the adjoint
condition. Thus \((\mathcal M,\psi,G)\) is an interval cycle.

Its fiber at \(0\) is \(D^+\widehat\otimes_{\psi^+}E_2\cong E_2\). The
creation operator for \(1\in D^+\) is this unitary identification, so
the connection condition gives \(G_0-F_2\) compact. At \(1\) the fiber
is \(D\widehat\otimes_\psi E_2\cong E_2^0\), with essential
representation. Combining the interval cycle with the initial compact
perturbation proves the assertion. \(\square\)

The same construction supplies the explicit multiplication connections
used later. For homogeneous \(d\in D\), let \(A_d:E_2\to E_2^0\) send
\(y\) to \(\psi(d)y\). Its adjoint is multiplication by \(d^*\),
followed by inclusion in \(E_2\); the inner-product identity verifies
this without an orthogonal projection onto \(E_2^0\). At the final
fiber, the creation operator for the constant vector \(d\in L\) is
exactly \(A_d\). Therefore the Grassmann connection gives \[
F_2^0A_d-(-1)^{|d|}A_dF_2
\in\mathcal K(E_2,E_2^0),
\tag{5.1a}
\] and its adjoint condition. At the initial fiber the creation for
\(1\) identifies \(G_0\) with \(F_2\) modulo compacts, as above.
Self-adjoint normalization and clipping preserve (5.1a): a two-sided
locally compact change \(K\) of the final operator has compact \(KA_d\),
by approximating \(d\) by sums of products and writing
\(A_{rs}=\psi^0(r)A_s\). The changes of the initial operator have
compact \(A_dK\) by the analogous adjoint-side argument. Hence the
replacement can be normalized while retaining both
multiplication-connection conditions.

\textbf{Theorem 5.2 (products with homomorphisms).} For graded
homomorphisms \(f:A\to D\) and \(g:D\to B\), under the countability
hypotheses for their homomorphism cycles, \[
[f]\widehat\otimes_D y=f^*(y),\qquad
x\widehat\otimes_D[g]=g_*(x).
\tag{5.2}
\] Consequently \(1_A=[\operatorname{id}_A]\) is a two-sided identity
whenever the indicated products are defined.

\textbf{Proof.} The homomorphism cycle for \(f\) is \((D,f,0)\). Replace
the representation of \(y\) by an essential one using Proposition 5.1.
The unitary \[
D\widehat\otimes_\psi E_2\longrightarrow E_2,\qquad
d\widehat\otimes y\longmapsto\psi(d)y
\tag{5.3}
\] intertwines the representation with \(\psi f\). Its creation errors
for \(F_2\) are the original compact commutators with \(\psi(d)\). It is
a cycle, and its positivity condition is zero because the first operator
is zero. This gives \(f^*(y)\).

For \(g\), write \(E_g=\overline{g(D)B}\). Since the intermediate
algebra \(D\) is \(\sigma\)-unital, a countable approximate identity
makes this module countably generated, and \(g(d)|_{E_g}\) is compact by
the support-module proof in Lesson 06, Section 5. Thus \((E_g,g,0)\) is
the homomorphism cycle even for arbitrary \(B\). If \(B\) is
\(\sigma\)-unital, Lemma 5.2 there identifies it with the canonical
cycle \((B,g,0)\).

The inclusion of \(E_g\) into \(B\) gives a unitary
\(E_1\widehat\otimes_gE_g\to E_1\widehat\otimes_gB\): it preserves inner
products, and its range is dense because
\(x\widehat\otimes b=\lim_j xe_j\widehat\otimes b=\lim_jx\widehat\otimes g(e_j)b\).
The tensor module is countably generated by the vectors
\(x_i\widehat\otimes g(e_j)\), for generating vectors \(x_i\) of
\(E_1\). Under this unitary the proposed tensor cycle is \[
(E_1\widehat\otimes_g B,\ \phi_1\widehat\otimes1,\
F_1\widehat\otimes1).
\] Each creation operator \(T_x:E_g\to E_1\widehat\otimes_g E_g\) is
compact. Indeed \(T_x^*T_x\) is the compact action of
\(g(\langle x,x\rangle)\) on \(E_g\), and the compactness test used in
Lemma 2.1 applies. Hence \(S=F_1\widehat\otimes1\) is a
\(0\)-connection. Compact input defects tensor to compact operators
here: a rank-one operator \(\theta_{x,z}\) becomes \(T_xT_z^*\). Finally
\[
\phi(a)[S,S]_{\mathrm{gr}}\phi(a)^*
=2\phi(a)S^2\phi(a)^*\geq0
\] already as an operator, since \(S\) is self-adjoint. This is exactly
the coefficient pushforward. Choosing \(f\) or \(g\) to be the identity
proves the last statement. \(\square\)

If \(B\) is not \(\sigma\)-unital, the full Hilbert module \(B\) need
not be countably generated, so \((B,g,0)\) is not automatically an
allowed cycle. For separable \(D\), use instead the countably generated
ideal module \[
B_g=\overline{g(D)B}\subset B.
\] A countable approximate identity of \(D\) acts in norm on this
module, so \(g(u_n)\) generate it as a right module. The left action of
\(g(d)\) is compact: the multipliers \(g(u_n)g(d)g(u_n)\) converge to
\(g(d)\) in norm, and each is multiplication by \(xy^*\) for
\(x=g(u_n)g(d)\), \(y=g(u_n)\) in \(B_g\), hence a rank-one operator on
\(B_g\). The tensor product with \(E_1\) is unchanged, since \(E_1D\) is
dense in \(E_1\). The proof of (5.2) then applies with \(B_g\). For
\(\sigma\)-unital \(B\), the simpler full-module cycle suffices.

\subsection{6. A useful formula which need not be a
connection}\label{a-useful-formula-which-need-not-be-a-connection}

Suppose \(F_1\) is a self-adjoint contraction and \(G\) is an odd
\(F_2\)-connection, without requiring \(G\) to be self-adjoint. Put \[
R=(1-S^2)^{1/2},\qquad H=S+RG.
\tag{6.1}
\] This formula is useful when its representation commutators are
compact. It does not generally satisfy the connection condition itself.

\textbf{Theorem 6.1 (the special formula).} Under the product
hypotheses, if \([H,\phi(a)]_{\mathrm{gr}}\) is compact for all
\(a\in A\), then \((E,\phi,H)\) is a cycle operator homotopic to a
product operator.

\textbf{Proof.} We first prove the cycle assertion without assuming that
\(S\) graded commutes with \(G\) globally modulo compacts. Let \[
C=C^*(1,J,I_0,S,G,\phi(A)),\qquad Q=C/J,
\] Let \(I\) be the closed two-sided ideal of \(Q\) generated by the
image of \(I_0\). This image need not itself be an ideal. A
\(0\)-connection \(D\in C\) annihilates \(I\), as follows. Suppress
quotient symbols for this calculation. First-factor generators
\(S,\phi(a)\), and generators from \(I_0\), multiply \(I_0\) into itself
on either side. The generators \(G,G^*\) graded commute with \(I_0\) in
\(Q\), by Lemma 1.1. Generators from \(J\) have zero image. The identity
has the first property.

For a homogeneous word \(c=a_1\cdots a_n\) in these generators and
\(i\in I_0\), prove \(Dci=0\) by induction on \(n\). The empty word is
Lemma 1.1. If the last generator is a first-factor generator, combine
\(a_ni\in I_0\) and apply the induction hypothesis to the shorter word.
If it is \(G\) or \(G^*\), write \(c=u a_n\) and move \(i\) past that
last generator: \[
\begin{aligned}
Du a_ni
&=(-1)^{|a_n||i|}Du i a_n\\
&=0.
\end{aligned}
\] The remaining zero-image and identity cases are immediate. Thus
\(Dci=0\) for all words, then for their linear span and norm closure.
Multiplication on the right proves that \(D\) kills every generator
\(ci c'\) of \(I\). Apply the same argument to the \(0\)-connection
\(D^*\), and take adjoints, to obtain \(ID=0\). This proves annihilation
without asserting that arbitrary products of derivating operators still
derive \(I_0+J\). Write \[
Z=\operatorname{Ann}(I)
=\{z\in Q:zI=Iz=0\}.
\] This is a closed two-sided ideal, and \(I\cap Z=0\): an element \(z\)
in the intersection has \(zz^*=0\). In these quotient calculations we
suppress the quotient symbols.

The operators \(G^2-1\), \(G-G^*\) and \([S,G]_{\mathrm{gr}}\) belong to
\(Z\), by the connection calculus and the second input's localized
defects. In \(Q/Z\), \(G\) is self-adjoint and anticommutes with \(S\),
so it commutes with \(S^2\) and with \(R\). Thus \([G,R],[G^*,R]\in Z\).
Direct multiplication gives \[
\begin{aligned}
H-H^*&=R(G-G^*)+[R,G^*]\in Z,\\
H^2-1&=R^2(G^2-1)+R[S,G]_{\mathrm{gr}}\\
&\quad+R[G,R]G\in Z.
\end{aligned}
\tag{6.2}
\]

The first input cycle implies \[
R\phi(a),\ \phi(a)R,\ [S,\phi(a)]_{\mathrm{gr}}\in I.
\tag{6.3}
\] For the roots in (6.3), pass first to
\(\mathcal L(E_1)/\mathcal K(E_1)\): \(F_1\) commutes graded with the
source, and \((1-F_1^2)\phi_1(a)=0\); commuting functional calculus
gives the square-root assertion there. Tensor the resulting compact
operators with \(1\).

Now pass to \(Q/I\). There \(R\phi(a)=0\), \(S\) commutes graded with
\(\phi(a)\), and \(S^2\phi(a)=\phi(a)\). By the assumed pseudolocality
of \(H\), \[
RH\phi(a)=(-1)^{|a|}R\phi(a)H=0
\] for homogeneous \(a\). Since \(R\) commutes with \(S\), this says
\(R^2G\phi(a)=0\). Positivity of \(R\) gives \(RG\phi(a)=0\): the square
of its norm is the norm of \(\phi(a)^*G^*R^2G\phi(a)\), which is zero.
Hence \[
H\phi(a)=H^*\phi(a)=S\phi(a)
\quad\text{in }Q/I.
\] For the square, graded commutation of \(S\) with the source gives \[
\begin{aligned}
H^2\phi(a)&=HS\phi(a)\\
&=(-1)^{|a|}H\phi(a)S\\
&=(-1)^{|a|}S\phi(a)S\\
&=S^2\phi(a)=\phi(a).
\end{aligned}
\] The localized defects therefore belong to \(I\), as well as to \(Z\)
by (6.2). Their intersection is zero. This proves all cycle defects
compact in the original algebra.

To compare with a product, use the Section 3 technical-theorem
construction, adding \([G,R]\) and \(G-G^*\) to \(A_2\), and \(R\) to
the derivating space already containing \(S,G,\phi(A)\). These additions
are permitted: the added defects are \(0\)-connections by the quotient
calculation above, and \(R\) is an even first-factor operator, so it
normalizes \(I_0\) and derives \(I_0+J\). The cutoffs commute with \(H\)
modulo \(J\) because, for \(P=M,N\), \[
\begin{aligned}
\left[P,H\right]&=[P,S]+[P,R]G\\
&\quad+R[P,G]\in J.
\end{aligned}
\] Functional calculus gives the same compact commutator for their
required roots. No derivation condition on \(H\) is needed. The second
input may first be normalized without changing which operators are its
connections, by Lemma 2.1. Let \[
F=M^{1/2}S+N^{1/2}G
\] be the resulting product operator. Its self-adjoint version differs
globally compactly, since \(N^{1/2}(G-G^*)\) is compact; equivalently
the construction may use \((G+G^*)/2\). The extra cutoff conditions and
(6.3) give \[
\begin{aligned}
\phi(a)[H,F]_{\mathrm{gr}}\phi(a)^*
&\equiv2\phi(a)M^{1/2}S^2\phi(a)^*\\
&\quad+2\phi(a)N^{1/2}R\phi(a)^*\geq0
\quad\pmod J.
\end{aligned}
\tag{6.4}
\] Here \([H,S]_{\mathrm{gr}}=2S^2+R[S,G]_{\mathrm{gr}}\); the second
term disappears after sandwiching, because \(\phi(a)R\in I\) and the
bracket annihilates \(I\). Also
\([H,G]_{\mathrm{gr}}=[S,G]_{\mathrm{gr}}+2RG^2+[G,R]G\); the
\(N^{1/2}\) cutoff kills its two bracket terms and the square defect.
The remaining two summands in (6.4) are positive because the cutoffs
commute modulo \(J\) with \(S\) and \(R\).

The positive comparison criterion proves the operator homotopy to \(F\).
This proves the formula without an unverified interpolation path.
\(\square\)

\subsection{7. Morita and external
products}\label{morita-and-external-products}

\textbf{Proposition 7.1 (imprimitivity bimodules).} Let \(X\) be an
imprimitivity \(A\)-\(D\) bimodule, with \(A,D\) \(\sigma\)-unital. Its
cycle \((X,\lambda,0)\) has an inverse represented by the conjugate
\(D\)-\(A\) bimodule \(X^*\). Products with this cycle are the usual
Morita tensor constructions.

\textbf{Proof.} Compact left action makes the zero operator a valid
cycle; \(\sigma\)-unitality of \(\mathcal K(X)=A\) gives countable
generation. For a cycle on \(E_2\), every \(F_2\)-connection on
\(X\widehat\otimes_D E_2\) is a cycle: its defects are \(0\)-connections
and the source action lies in \(\mathcal K(X)\widehat\otimes1\).
Positivity is zero because the first operator is zero. Thus it computes
the indicated Morita construction.

For two zero-operator bimodule cycles use the zero product operator. The
inner-product maps \[
\begin{aligned}
X^*\widehat\otimes_A X&\longrightarrow D,
&\bar x\widehat\otimes y&\longmapsto\langle x,y\rangle_D,\\
X\widehat\otimes_D X^*&\longrightarrow A,
&x\widehat\otimes\bar y&\longmapsto{}_A\langle x,y\rangle
\end{aligned}
\tag{7.1}
\] are well-defined bimodule isometries. For example the squared
inner-product calculation for the first uses \[
\langle y_1,{}_A\langle x_1,x_2\rangle y_2\rangle_D
=\langle y_1,x_1\rangle_D\langle x_2,y_2\rangle_D.
\] Fullness makes their ranges dense and hence surjective. The
corresponding product cycles are \((D,\operatorname{id},0)\) and
\((A,\operatorname{id},0)\), the identities of Theorem 5.2. This proves
the assertion. \(\square\)

For separable source algebras and \(\sigma\)-unital coefficient
algebras, dilation by an auxiliary algebra \(C\) is \[
\tau_C(E,\phi,F)
=(E\widehat\otimes C,\ \phi\widehat\otimes1,\
F\widehat\otimes1).
\] The auxiliary algebra must be \(\sigma\)-unital to guarantee
countable generation of this displayed module. Compact defects are
checked after multiplying by \(a\widehat\otimes c\): they are original
compact operators tensored with multiplication by \(c\in\mathcal K(C)\).
The same construction over an interval shows homotopy invariance.

Thus for \(x\in KK(A_1,B_1)\) and \(y\in KK(A_2,B_2)\) we define the
\textbf{external product} \[
x\boxtimes y
=\tau_{A_2}(x)\widehat\otimes_{B_1\widehat\otimes A_2}
\tau_{B_1}(y)
\in KK(A_1\widehat\otimes A_2,B_1\widehat\otimes B_2).
\tag{7.2}
\] The second dilation places \(B_1\) on the left, with the graded flip
used to order its factors. Countable generation and the intermediate
\(\sigma\)-unitality hold under the stated hypotheses. Theorem 4.2 makes
(7.2) bilinear and well-defined. Its associativity and graded symmetry
will follow in the next lesson from the internal product.

Coefficient Morita reduction is compatible with this construction
directly, without invoking associativity. If \(Y\) is a compact-left
correspondence and \(F\) is a product operator, \(F\widehat\otimes1\)
satisfies the same product axioms after tensoring the coefficient
modules with \(Y\). Compact creation errors remain compact by the
compact-left tensor rule, the canonical tensor-associating unitary
identifies the creation operators, and the induced homomorphism on
adjointable operators preserves quotient positivity. This also justifies
using the \(C_2\)-\(\mathbb C\) matrix correspondence to express the
product of two odd classes as an even cycle.

\subsection{8. The two circle classes}\label{the-two-circle-classes}

Let \(e_n(z)=z^n\) be the Fourier basis of \(L^2(\mathbb T)\), and let
the odd circle cycle have operator \[
F_{\mathbb T}e_n=f(n)e_n,\qquad
f(n)=\frac{n}{\sqrt{1+n^2}}.
\tag{8.1}
\] This is the bounded transform of \(-i\,d/d\theta\). Multiplication by
\(z\) is the bilateral shift, and its commutator with (8.1) is compact
because \(f(n+1)-f(n)\to0\) at both ends. Laurent-polynomial
approximation gives compact commutators for all continuous functions.

\textbf{Proposition 8.1 (the torus product).} With the order of the two
circle factors fixed, their external product is the even cycle on \[
H=\ell^2(\mathbb Z^2)\otimes\mathbb C^2,\qquad
\Gamma=\sigma_3,
\] with multiplication representation of \(C(\mathbb T^2)\) and \[
F e_{n,m}
=\frac{n\sigma_1-m\sigma_2}{\sqrt{1+n^2+m^2}}e_{n,m}.
\tag{8.2}
\] Here \[
\begin{gathered}
\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\\
\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\end{gathered}
\]

\textbf{Proof.} In the Clifford model for the product, take \[
E_1=\ell^2(\mathbb Z)\widehat\otimes C(\mathbb T)
\widehat\otimes C_1,\qquad
F_1=f(n)\varepsilon.
\] The first source circle acts by shifting \(n\); the second acts on
the coefficient \(C(\mathbb T)\). After the \(C_2\) matrix
correspondence, the second cycle is on
\(\ell^2(\mathbb Z)\otimes\mathbb C^2\), with \[
\psi(g\widehat\otimes\varepsilon^r)=M_g\sigma_1^r,
\qquad F_2=-f(m)\sigma_2,\quad r=0,1.
\tag{8.3}
\] Here the odd product uses right Clifford dilation of the second
cycle. Its coefficient generators are ordered as the second cycle's
original generator, followed by the auxiliary generator supplied by the
first cycle. We represent these by \(-\sigma_2,\sigma_1\), respectively.
Their chirality is \((-i)(-\sigma_2)\sigma_1=\sigma_3\), agreeing with
the chosen grading. This identification is an even unitary conjugate of
the usual ordered pair \(\sigma_1,\sigma_2\). It fixes the minus sign in
(8.2) and (8.3), and hence the orientation of the product.

These are valid graded cycles by the circle compactness calculation and
\(\sigma_1\sigma_2=-\sigma_2\sigma_1\). Their tensor module is \(H\),
and \(S=F_1\widehat\otimes1=f(n)\sigma_1\).

The operator (8.2) is odd and self-adjoint. Its square defect is \[
F^2-1=-\frac{1}{1+n^2+m^2}1_2,
\] a compact diagonal operator: outside any finite lattice set its
entries tend uniformly to zero. The two coordinate shifts have compact
commutators with \(F\). Indeed the gradient of
\((x,y)\mapsto(x\sigma_1-y\sigma_2)/\sqrt{1+x^2+y^2}\) has norm bounded
by \(C/\sqrt{1+x^2+y^2}\); finite differences in either coordinate tend
to zero at infinity. Laurent-polynomial approximation again proves all
source commutators compact.

For the connection condition first take a vector
\(x=e_n\widehat\otimes g\widehat\otimes\varepsilon^r\). Its creation
operator is \(J_nM_g\sigma_1^r\), where \(J_n\) embeds the second
Fourier space at the fixed first coordinate \(n\). At this fixed
coordinate, \[
\left\|
\frac{n\sigma_1-m\sigma_2}{\sqrt{1+n^2+m^2}}
+f(m)\sigma_2
\right\|\longrightarrow0
\quad(|m|\longrightarrow\infty).
\tag{8.4}
\] Thus the difference times \(J_n\) is compact. Move \(\sigma_2\)
across \(\sigma_1^r\), giving the sign \((-1)^r\), and move \(f(m)\)
across \(M_g\), giving the already proved compact circle commutator.
These are exactly the two terms of the first creation error.
Self-adjointness gives the adjoint condition. Finite sums of these
vectors are dense in \(E_1\), so the norm-closed connection condition
holds for every vector.

Finally the positivity is particularly simple: \[
[S,F]_{\mathrm{gr}}
=\frac{2n^2}{\sqrt{1+n^2}\sqrt{1+n^2+m^2}}1_2\geq0.
\tag{8.5}
\] Sandwiching this positive operator by any \(\phi(a)\) stays positive.
All three product axioms hold, and uniqueness proves the assertion.
\(\square\)

The untwisted scalar index of (8.2) is zero: only the zero Fourier mode
lies in the kernel, with one vector in each parity. This does not make
its \(C(\mathbb T^2)\)-homology class zero. A bundle test supplies more
information than the scalar test.

The simpler expression \(f(n)\sigma_1-
\sqrt{1-f(n)^2}\,f(m)\sigma_2\) illustrates the hypothesis in Theorem
6.1. Its commutator with the first coordinate shift does not tend to
zero when \(n\) is fixed and \(|m|\to\infty\). Hence it cannot simply
replace (8.2). A formula for the operator still needs the
representation-commutator test.

\subsection{9. Bundles and twisted Dirac
operators}\label{bundles-and-twisted-dirac-operators}

Let \(M\) be a closed even-dimensional spin\(^{c}\) manifold with graded
spinor bundle \(S_M\), and let \(D_M\) be its self-adjoint Dirac
operator with domain \(H^1(M,S_M)\). Let \(V\) be a smooth Hermitian
vector bundle with a Hermitian connection. The section module
\(\Gamma(V)\) is a finitely generated projective \(C(M)\)-module,
concentrated in even degree. Pointwise multiplication gives the cycle \[
[[V]]=[(\Gamma(V),\lambda,0)]\in KK(C(M),C(M)).
\] Its scalar pullback is the K-theory class
\([V]\in KK(\mathbb C,C(M))\).

We will use the compact-resolvent and self-adjointness assertions for
closed-manifold Dirac operators proved in Corollary 4.2 of
\emph{Fredholm modules and analytic K-homology}. The analytic index of
the bounded transform is the Sobolev Dirac index, by Theorem 5.2 of
\emph{The index pairing between K-theory and K-homology}.

\textbf{Lemma 9.1 (bounded intertwining and transforms).} Suppose
\(D,D'\) are self-adjoint operators with compact resolvents on two
Hilbert spaces. Let \(T\) be bounded, map the domain of \(D\) into that
of \(D'\), and satisfy \[
D'T-TD=C
\] there, for a bounded operator \(C\). Then, for
\(b(t)=t/\sqrt{1+t^2}\), \[
b(D')T-Tb(D)\text{ is compact}.
\tag{9.1}
\]

\textbf{Proof.} For \(\alpha=\sqrt{1+s^2}\), the domain assumption gives
the resolvent identity \[
(D'\pm i\alpha)^{-1}T-T(D\pm i\alpha)^{-1}
=-(D'\pm i\alpha)^{-1}C(D\pm i\alpha)^{-1}.
\tag{9.2}
\] Every right side is compact and has norm at most \(\|C\|/(1+s^2)\).
Use \[
\begin{gathered}
b(D)=\frac2\pi\int_0^\infty D(D^2+1+s^2)^{-1}\,ds,\\
D(D^2+\alpha^2)^{-1}
=\frac12\bigl((D-i\alpha)^{-1}+(D+i\alpha)^{-1}\bigr).
\end{gathered}
\tag{9.3}
\] The individual truncated integrals converge strongly, as follows from
the scalar integral and bounded spectral calculus. Their intertwined
difference, by (9.2), converges in norm to a compact operator, since
\(\int_0^\infty(1+s^2)^{-1}ds<\infty\). Its strong limit is the
difference in (9.1). \(\square\)

\textbf{Theorem 9.2 (twisting is the Kasparov product).} If \(D_V\) is
the Dirac operator on \(S_M\otimes V\) for the tensor connection, then
\[
[[V]]\widehat\otimes_{C(M)}[D_M]=[D_V].
\tag{9.4}
\] Also \[
[V]\widehat\otimes_{C(M)}[D_M]
=\operatorname{index}D_V^+\in KK(\mathbb C,\mathbb C)=\mathbb Z.
\tag{9.5}
\]

\textbf{Proof.} The map \[
\Gamma(V)\widehat\otimes_{C(M)}L^2(M,S_M)
\longrightarrow L^2(M,S_M\otimes V)
\] given by pointwise tensoring is an isometry by integration of the
fiber inner products. Its range is dense: use finitely many local
frames, a partition of unity and density of smooth spinor sections.
Hence it is unitary.

For a smooth section \(v\) of \(V\), creation \(T_v\) maps \(H^1\) to
\(H^1\), and the tensor connection formula gives \[
D_VT_v-T_vD_M
=\sum_j c(e_j)\otimes\nabla^V_{e_j}v.
\tag{9.6}
\] The right side is a smooth bundle homomorphism on a compact manifold,
hence bounded on \(L^2\). Lemma 9.1 proves that \(b(D_V)\) is a
\(b(D_M)\)-connection for every smooth \(v\). Smooth sections are dense
in \(\Gamma(V)\) in the uniform norm; creation is norm-continuous in
that norm. Thus the connection property holds for all sections, and
self-adjointness gives its second condition.

Ellipticity supplies the remaining cycle conditions for \(b(D_V)\).
Positivity is zero because the first operator is zero. Theorem 4.1
proves (9.4).

For (9.5) use the same tensor product and operator, with scalar first
representation. It is a product by the same connection proof. The scalar
Fredholm calculation of the cycle lesson makes its class the even-kernel
dimension minus the odd-kernel dimension. The bounded-transform domain
calculation in the index-pairing lesson identifies this with
\(\operatorname{index}D_V^+\). This proves the integer formula without
an associativity argument or a topological index theorem. \(\square\)

A Grassmann connection gives a second concrete representative. Choose a
smooth isometric embedding \(V\subset M\times\mathbb C^N\), with smooth
projection \(p\). The operator \[
p\,b(D_M)^{\oplus N}p
\quad\text{on }pL^2(M,S_M)^{\oplus N}
\tag{9.7}
\] is a connection and a cycle. Its representation is compact on the
first projective module, and its positivity condition is again zero. It
has the same class as the twisted Dirac transform by the product
uniqueness theorem. Thus the choice of embedding and the choice of
connection on \(V\) do not change the class.

\subsection{10. Exercises}\label{exercises}

\textbf{10.1. Exact tensoring.} Suppose a homogeneous \(F_2\) graded
commutes exactly with \(\psi(D)\). Prove that \(1\widehat\otimes F_2\)
is well-defined on the interior tensor product and is an
\(F_2\)-connection.

\textbf{10.2. A projective Grassmann connection.} Let \(E_1=pD^N\), for
an even projection \(p\) in the graded matrix algebra over \(D\). Write
the Grassmann connection explicitly and prove both creation conditions.
If the first representation is compact, prove that it is already a
product operator with first operator zero.

\textbf{10.3. A homomorphism on the right.} Verify the connection, cycle
and positivity conditions for \(F_1\widehat\otimes1\) in the product
with \((B,g,0)\). Explain the countability qualification when \(B\) is
not \(\sigma\)-unital.

\textbf{10.4. Two odd circle classes.} Construct their product as an
even cycle using \(C_1\widehat\otimes C_1\cong M_2\). Prove compactness
of its source commutators, both connection conditions and positivity,
with the signs fixed. Compute its scalar index.

\subsection{11. Solutions}\label{solutions}

\textbf{Solution to 10.1.} On homogeneous elementary tensors set \[
K(x\widehat\otimes y)=(-1)^{|F_2||x|}
x\widehat\otimes F_2y.
\] For homogeneous \(d\), applying the formula to
\(xd\widehat\otimes y\) gives the sign \((-1)^{|F_2|(|x|+|d|)}\).
Applying it to \(x\widehat\otimes\psi(d)y\) gives the same result
because \(F_2\psi(d)=(-1)^{|F_2||d|}\psi(d)F_2\). Thus it respects
balancing.

To see boundedness without a formal manipulation of null vectors, extend
\(\psi\) to \(D^+\) and use the standard-module operator (1.6). Exact
commutation implies that operator commutes with the even stabilized
projection \(P\), by first checking finite matrix coefficients and then
strong convergence of the matrix truncations on vectors. Its restriction
is the formula above, has norm at most \(\|F_2\|\), and has adjoint
given by the same formula with \(F_2^*\). Finally \[
KT_x=(-1)^{|F_2||x|}T_xF_2
\] exactly; the adjoint-side identity is also exact. Thus both creation
errors are zero.

\textbf{Solution to 10.2.} Let \(C\) be the signed diagonal
amplification (1.6) on the finite free module tensor \(E_2\), and let
\(\Pi=p\widehat\otimes1\). Set \(G=\Pi C\Pi\) on
\(\Pi(E_2^{\oplus N})\), with the appropriate coordinate parities. For a
homogeneous \(x=(x_j)\) in \(pD^N\), \(T_x\) is the column of the
represented coordinates, and \(\Pi T_x=T_x\). Hence \[
GT_x-(-1)^{|F_2||x|}T_xF_2
=\Pi\bigl(CT_x-(-1)^{|F_2||x|}T_xF_2\bigr).
\] Its entries are compact commutators of \(F_2\) with \(\psi(x_j)\).
This is a finite compact column. Taking adjoints proves the second
condition when \(F_2\) is self-adjoint; in general apply the same finite
row computation using the second connection condition for \(C\).

For a compact first representation, every \(\phi(a)\) lies in the image
of \(\mathcal K(pD^N)\). The defect \(G^2-1\) is a \(0\)-connection, so
its product with that image is compact. Lemma 1.1 gives the compact
representation commutators; the adjoint defect is handled by adjoints or
by choosing self-adjoint \(G\). The first operator is zero, making the
positivity sandwich zero. Thus this connection is a product without an
additional cutoff partition.

\textbf{Solution to 10.3.} On \(E=E_1\widehat\otimes_g B\), the creation
operator from \(B\) has
\(T_x^*T_x=g(\langle x,x\rangle)\in\mathcal K(B)\). The \(T^*T\)
compactness test makes \(T_x\) compact. Therefore \(ST_x\) and
\(T_x^*S\) are compact for \(S=F_1\widehat\otimes1\), proving the
\(0\)-connection condition. Tensoring a compact rank-one operator on
\(E_1\) gives \(T_xT_z^*\), so all three input defects descend to
compact defects. With normalized self-adjoint \(F_1\), the positivity
expression is \(2\phi(a)S^2\phi(a)^*\), positive already before
quotienting.

The full module \(B\) is countably generated exactly when it is
\(\sigma\)-unital. If this fails and \(D\) is separable, replace it by
\(B_g=\overline{g(D)B}\), as in Section 5. On this module
\(g(\langle x,x\rangle)\) still acts compactly, and the approximate
identity of \(D\) makes it countably generated. The tensor product and
resulting pushforward class agree with the usual coefficient tensor
construction.

\textbf{Solution to 10.4.} Use the Pauli matrices in Section 8 as the
two odd Clifford generators and \(\sigma_3\) as the grading. The tensor
module is \(\ell^2(\mathbb Z^2)\otimes\mathbb C^2\), with operator
(8.2). Its square is \((n^2+m^2)/(1+n^2+m^2)\), so the defect tends to
zero outside finite lattice sets. Each source coordinate shift has a
commutator whose entries are finite differences of the coefficient
vector in (8.2), bounded by \(C/\sqrt{1+n^2+m^2}\) at infinity. They are
compact; uniform Laurent-polynomial approximation handles all source
functions.

For the dense creation vectors
\(e_n\widehat\otimes g\widehat\otimes\varepsilon^r\), the operators are
\(J_nM_g\sigma_1^r\). Equation (8.4) supplies the compact
fixed-coordinate error, the original circle commutator supplies the
compact error from \(M_g\), and
\(\sigma_2\sigma_1^r=(-1)^r\sigma_1^r\sigma_2\) supplies exactly the
creation sign. Taking adjoints gives the second condition. Norm
approximation handles arbitrary creation vectors. The positivity bracket
is (8.5), positive at every lattice point. The zero mode has one even
and one odd vector, while no other mode has a kernel. Hence the scalar
index is \(1-1=0\). This completes every product axiom and the scalar
calculation.

\subsection{What this lesson does not
prove}\label{what-this-lesson-does-not-prove}

The product's associativity, its naturality for general changes of the
intermediate algebra, graded symmetry of external products and the
comparison \(KK_h=KK_{\mathrm{oh}}\) are theorems of \emph{Homotopy,
associativity, the index pairing and KK-equivalence}. The present
construction and examples do not assume them.

The elementary module calculus uses the programme's Hilbert-module
prerequisites: \emph{Tensor products of Hilbert modules and
C*-correspondences} in \emph{Hilbert C*-modules and Morita equivalence}
gives positivity and completion of the interior tensor product;
\href{KT-KK-05.html\#lemma-2-0-compact-module-operators-and-their-multipliers}{Lesson
05, Lemma 2.0} proves \(\mathcal L(E)=M(\mathcal K(E))\) and the
equivalence of countable generation with \(\sigma\)-unitality of
\(\mathcal K(E)\). Graded stabilization and the compact-left tensor
rules also have complete proofs in the preceding graded lesson of this
course. The closed-manifold analytic input is precisely Corollary 4.2 of
the Fredholm-module lesson and Theorem 5.2 of the index-pairing lesson,
both already proved there.

No characteristic-class index formula, Bott equivalence, Thom
isomorphism or universal coefficient theorem is used here.

\subsection{References and source
credit}\label{references-and-source-credit}

\begin{itemize}
\tightlist
\item
  B. Blackadar,
  \href{https://www.bruceblackadar.com/Mathematics/book6.pdf}{\emph{K-Theory
  for Operator Algebras}, author-posted corrected second edition}. The
  \href{https://www.bruceblackadar.com/mathpubs.html}{author identifies
  this freely downloadable version} as a slightly corrected second
  edition. The source retains its author copyright and the usage terms
  stated on that page. Sections 18.3--18.4, printed pp.~169--175, for
  connections, existence, uniqueness and homomorphism cycles. The
  required proofs and all exercise solutions are given in this lesson
  and its exact preceding programme prerequisites.
\end{itemize}

\end{document}