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\begin{document}

\section{Graded C*-algebras, Clifford algebras and graded Hilbert
modules}\label{graded-c-algebras-clifford-algebras-and-graded-hilbert-modules}

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

Grading records which operators change parity. Two odd operators acquire
a minus sign when exchanged, and this sign determines the tensor product
used in KK-theory. Clifford algebras supply finite-dimensional models of
this rule. Graded stabilization then places every countably generated
coefficient module inside one standard graded module.

Degrees take values in \(\mathbb Z/2\); all exponents of \((-1)\) are
computed modulo two. Inner products are linear in their second variable.
The ordinary spatial tensor norm and the interior tensor construction
are proved below in Lemmas 3.0a and 6.0a. For the foundational
C*-algebra facts, the earlier programme lesson
\href{https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html}{continuous
functional calculus, homomorphisms, positive cones and quotients}
supplies Theorems 4.2 and 4.4 (contractivity and faithful isometry),
Theorem 5.1 (functional calculus), and Proposition 8.5 (positive order).
The ordinary Hilbert-module and operator foundations are proved in
\href{supporting/hilbert-c-star-modules-and-morita-equivalence/hilbert-module-foundations.html\#mf-001}{Ordinary
Hilbert C*-module foundations}, MF.1--MF.7: coefficient-order
Cauchy--Schwarz and completion in MF.2--MF.3, bounded adjoints and the
operator C*-algebra in MF.4--MF.5, finite Gram positivity in MF.6, and
rank-one bounds in MF.7. The spatial proof uses Hahn--Banach, whose
earlier programme proofs are Theorems 2.1--2.2 and Corollary 6.4 of
\href{https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html}{Hahn--Banach,
Baire, and the basic theorems on Banach spaces}. These inputs establish
the elementary operator tools. The operator-range argument in Lemma 2.1
of \emph{Ext groups, absorption and Brown--Douglas--Fillmore theory}
will be used with its grading checked explicitly. Clifford generators
here are self-adjoint and square to \(+1\).

\subsection{1. Gradings and signs}\label{gradings-and-signs}

A graded C*-algebra is a pair \((A,\alpha)\), where \(\alpha\) is a
*-automorphism and \(\alpha^2=1\). The identity automorphism is allowed.
Its homogeneous subspaces are \[
\begin{aligned}
A^r&=\{a:\alpha(a)=(-1)^ra\},\\
a^r&=\tfrac12(a+(-1)^r\alpha(a)).
\end{aligned}
\tag{1.1}
\] Thus \(A=A^0\oplus A^1\) as a Banach space, \((A^r)^*=A^r\), and
\(A^rA^s\subset A^{r+s}\). A nonzero element of \(A^r\) has degree
\(|a|=r\). Statements about homogeneous zero elements mean the
corresponding multilinear identity in the declared degrees.

The grading is \textbf{inner} if a self-adjoint unitary \(g\in M(A)\)
satisfies \(\alpha(a)=gag\). Blackadar calls this an evenly graded
algebra. The multiplier \(g\) is even: its induced grading fixes it. The
trivial grading is inner, implemented by \(1_{M(A)}\). An inner grading
need not be trivial.

For homogeneous elements put \[
\left[a,b\right]_{\mathrm{gr}}
=ab-(-1)^{|a||b|}ba.
\tag{1.2}
\] This is a commutator if either entry is even, and an anticommutator
if both are odd. In particular \([a,a]_{\mathrm{gr}}=2a^2\) for odd
\(a\), which is positive when \(a=a^*\).

\subsubsection{Proposition 1.1. The graded commutator
identities}\label{proposition-1.1.-the-graded-commutator-identities}

For homogeneous \(x,y,z\), of respective degrees \(p,q,r\), \[
\begin{gathered}
\left[x,y\right]_{\mathrm{gr}}=-(-1)^{pq}[y,x]_{\mathrm{gr}},\\
[x,yz]_{\mathrm{gr}}=[x,y]_{\mathrm{gr}}z\\
+(-1)^{pq}y[x,z]_{\mathrm{gr}},\\
(-1)^{pr}[[x,y]_{\mathrm{gr}},z]_{\mathrm{gr}}\\
+(-1)^{pq}[[y,z]_{\mathrm{gr}},x]_{\mathrm{gr}}\\
+(-1)^{qr}[[z,x]_{\mathrm{gr}},y]_{\mathrm{gr}}=0.
\end{gathered}
\tag{1.3}
\]

\textbf{Proof.} The first formula follows by exchanging the two terms.
In the second formula the two occurrences of \(yxz\) cancel, leaving
\(xyz-(-1)^{p(q+r)}yzx\). For the last formula expand its three
summands: \[
\begin{gathered}
(-1)^{pr}xyz-(-1)^{pr+pq}yxz\\
-(-1)^{qr}zxy+(-1)^{qr+pq}zyx,\\
(-1)^{pq}yzx-(-1)^{pq+qr}zyx\\
-(-1)^{pr}xyz+(-1)^{pr+qr}xzy,\\
(-1)^{qr}zxy-(-1)^{qr+pr}xzy\\
-(-1)^{pq}yzx+(-1)^{pq+pr}yxz.
\end{gathered}
\] Every word appears twice with opposite coefficients. \(\square\)

For \(M_2(\mathbb C)\), conjugation by \(\operatorname{diag}(1,-1)\)
makes diagonal matrices even and off-diagonal matrices odd. For
\(\mathbb C\oplus\mathbb C\), the swap \((a,b)\mapsto(b,a)\) makes
\((a,a)\) even and \((a,-a)\) odd. This second grading is not inner:
every inner automorphism of a commutative algebra is trivial.

\subsection{2. Graded modules and their
operators}\label{graded-modules-and-their-operators}

\subsubsection{Lemma 2.0. Compact module operators and their
multipliers}\label{lemma-2.0.-compact-module-operators-and-their-multipliers}

For every Hilbert \(B\)-module \(E\), the rank-one span has an essential
closed ideal \(K=\mathcal K(E)\) in \(\mathcal L(E)\), with dense action
on \(E\). One has \[
\mathcal L(E)=M(K).
\] The module is countably generated exactly when \(K\) is
\(\sigma\)-unital. On bounded families of adjointable operators, strict
convergence means convergence on every vector for both the operator and
its adjoint. Every bounded strict Cauchy net has an adjointable limit.
Finally, a nondegenerate representation \(\rho:K\to\mathcal L(Y)\)
extends uniquely to a unital representation of \(M(K)\), continuously
for bounded strict convergence.

\textbf{Proof of the compact ideal.} The adjoint and composition
formulas are \[
\begin{gathered}
\theta_{x,y}^*=\theta_{y,x},\\
T\theta_{x,y}=\theta_{Tx,y},\qquad
\theta_{x,y}T=\theta_{x,T^*y}.
\end{gathered}
\] They follow by substituting the inner-product definition. Thus the
rank-one span and its closure are two-sided *-ideals. The map
\(R_x:B\to E\), \(b\mapsto xb\), has adjoint
\(R_x^*z=\langle x,z\rangle\), so \(\theta_{x,x}=R_xR_x^*\) is positive.
If \(a=\langle x,x\rangle\), evaluation at \(x\) gives \[
\|\theta_{x,x}x\|^2=\|a^3\|=\|x\|^6.
\] Together with the Cauchy--Schwarz upper bound this proves
\(\|\theta_{x,x}\|=\|x\|^2\). If \(TK=0\), then
\(T\theta_{x,Tx}=\theta_{Tx,Tx}=0\), so \(T=0\). An ideal disjoint from
\(K\) annihilates it and is consequently zero. This proves essentiality.

The vector \(xa(a+\epsilon)^{-1}\) belongs to \(KE\), because it is
\(\theta_{x,x}(x(a+\epsilon)^{-1})\). Inverses here act through the
scalar unitization of the coefficient. Its squared distance from \(x\)
is at most \[
\sup_{s\geq0}\frac{s\epsilon^2}{(s+\epsilon)^2}=\epsilon/4.
\] Thus \(KE\) has dense span. A contractive approximate identity
\(e_\lambda\) of \(K\) converges to the identity on that span, then on
all vectors by uniform boundedness.

\textbf{Proof of the multiplier identification.} Lesson 01 constructs
the double-centralizer algebra \(M(K)\). Each \(T\in\mathcal L(E)\)
gives its two actions on the ideal \(K\), and essentiality makes this
homomorphism injective. Conversely, let \(m=(L,R)\) be a bounded double
centralizer. Define on the dense span of \(KE\) \[
T_m\Bigl(\sum_jk_jx_j\Bigr)=\sum_jL(k_j)x_j.
\] This is well-defined and bounded: its right side is the vector limit
of \(L(e_\lambda)\sum_jk_jx_j\), since
\(L(e_\lambda)k_j=L(e_\lambda k_j)\). Its norm is at most
\(\|m\|\|\sum_jk_jx_j\|\). The same construction for \(m^*\), using \[
L(k)^*l=k^*L^{\#}(l),
\] gives \(\langle T_m(kx),ly\rangle=\langle kx,T_{m^*}(ly)\rangle\).
Density proves adjointability and \(T_m^*=T_{m^*}\). Finally
\(T_mk=L(k)\), and on vectors \(lx\) one has \(kT_m(lx)=kL(l)x=R(k)lx\).
Thus both multiplier actions are the prescribed ones, proving
surjectivity and the identification.

\textbf{Proof of the countability assertion.} If \(x_j\) generate \(E\),
rescale so \(\|x_j\|\leq1\), and put \[
\begin{gathered}
h=\sum_{j\geq1}2^{-j}\theta_{x_j,x_j},\\
e_n=h(h+n^{-1})^{-1},\qquad r_n=1-e_n.
\end{gathered}
\] The series converges in \(K\). Positivity and the rank-one norm
formula give \[
\begin{aligned}
\|r_nx_j\|^2
&=\|r_n\theta_{x_j,x_j}r_n\|\\
&\leq2^j\|r_nhr_n\|\leq2^j/(4n).
\end{aligned}
\] Hence \(e_n\) converges on all generating vectors and therefore on
all of \(E\). The rank-one composition formulas make \(e_n\) a two-sided
approximate identity of \(K\). Conversely, approximate a sequential
approximate identity of \(K\) by finite rank-one sums \(f_n\), within
\(1/n\). Then \(f_nx\to x\); the countable set of first vectors in these
sums generates \(E\). For a graded module take the homogeneous
components of the generators and rank-one sums. The resulting \(h,e_n\)
can be chosen even. The zero module satisfies the same assertions.

\textbf{Proof of the strict assertions.} If \(T_i,T_i^*\) converge on
every vector and are uniformly bounded, the rank-one formulas imply
convergence of \(T_ik\) and \(kT_i\) for every \(k\in K\). Approximation
by rank-one sums uses the common bound. Conversely, strict convergence
gives vector convergence on \(KE\), then on \(E\) by density and the
common bound; apply it also to the adjoint net. A bounded strict Cauchy
net therefore has vector limits \(T\) and \(S\) for its operator and
adjoint. Passing to limits in the inner-product identity gives
\(T^*=S\), so its limit is adjointable and is the strict limit just
described. This also proves strict convergence of the bounded positive
sums used later, whenever their products on the dense compact-action
span are Cauchy.

\textbf{Proof of the representation extension.} Nondegeneracy means that
the span of \(\rho(K)Y\) is dense. On that span set \[
\widetilde\rho(m)\Bigl(\sum_j\rho(k_j)y_j\Bigr)
=\sum_j\rho(mk_j)y_j.
\] This vector is the limit of \(\rho(me_\lambda)\sum_j\rho(k_j)y_j\).
These operators have norm at most \(\|m\|\), so the formula is
independent of the expression and extends boundedly. The construction
for \(m^*\) supplies the adjoint, by the same double-centralizer
inner-product calculation. Products agree on the dense span, and the
unit acts as the identity; thus this is a unital *-homomorphism. Its
products with \(\rho(k)\) determine it on that span, proving uniqueness.
Bounded strict convergence of the multipliers gives vector and
adjoint-vector convergence on this span and then on every vector, hence
strict convergence on \(\mathcal K(Y)\). No faithfulness of \(\rho\),
fullness, separability or \(\sigma\)-unitality of the coefficients was
needed. \(\square\)

The actual free comparisons are Blackadar's author-posted \emph{K-Theory
for Operator Algebras}, Sections 13.1--13.2, and his revised
\emph{Operator Algebras} notes, Sections II.7.2--II.7.3. References
inside those works are not imported as proofs: all assertions needed
here have been established above using Lesson 01 and the earlier
C*-algebra and adjointable-operator foundations.

A grading on a Hilbert \(B\)-module \(E\), over \((B,\beta)\), is a
complex-linear isometric involution \(\gamma_E\) such that \[
\begin{aligned}
\gamma_E(xb)&=\gamma_E(x)\beta(b),\\
\langle\gamma_E x,\gamma_E y\rangle
 &=\beta(\langle x,y\rangle).
\end{aligned}
\tag{2.1}
\] Its eigenspaces \(E^0,E^1\) satisfy \[
\begin{gathered}
E^pB^q\subset E^{p+q},\\
\langle E^p,E^q\rangle\subset B^{p+q}.
\end{gathered}
\tag{2.2}
\] Conversely these properties for a closed direct-sum decomposition
give (2.1).

The involution \(\gamma_E\) is generally semilinear over \(B\), rather
than \(B\)-linear. Its eigenspaces need not be Hilbert \(B\)-submodules.
For example, take \(E=B=\mathbb C\oplus\mathbb C\) with swap grading;
multiplying the even vector \((1,1)\) by the odd coefficient \((1,-1)\)
produces an odd vector. When \(B\) is trivially graded, \(\gamma_E\) is
a self-adjoint unitary in \(\mathcal L(E)\), and its two eigenspaces are
orthogonal Hilbert submodules.

\subsubsection{Proposition 2.1. Induced operator
gradings}\label{proposition-2.1.-induced-operator-gradings}

The map \[
\alpha_E(T)=\gamma_E T\gamma_E
\tag{2.3}
\] grades \(\mathcal L(E)\) and preserves \(\mathcal K(E)\). For
homogeneous vectors, \[
\begin{gathered}
|\theta_{x,y}|=|x|+|y|,\\
\theta_{x,y}(z)=x\langle y,z\rangle.
\end{gathered}
\tag{2.4}
\] A homogeneous adjointable operator \(T\) of degree \(r\) maps \(E^p\)
into \(E^{p+r}\).

\textbf{Proof.} Twice using semilinearity gives
\(\gamma_E T\gamma_E(xb)=(\gamma_E T\gamma_E x)b\), so (2.3) is
\(B\)-linear. Apply \(\beta\) to the adjoint identity for \(T\), with
arguments \(\gamma_E x,\gamma_E y\), to obtain
\(\langle\gamma_E T\gamma_E x,y\rangle
=\langle x,\gamma_E T^*\gamma_E y\rangle\). Thus (2.3) preserves
adjoints; composition and the involution identity are immediate.
Moreover \[
\gamma_E\theta_{x,y}\gamma_E
=\theta_{\gamma_E x,\gamma_E y}.
\] For homogeneous \(x,y\), their two signs multiply, proving (2.4).
Finite-rank spans and their norm closure are preserved. Finally
\(\gamma_ET=(-1)^rT\gamma_E\) is precisely the stated degree-shift
condition. \(\square\)

A graded representation is a *-homomorphism
\(\pi:(A,\alpha)\to(\mathcal L(E),\alpha_E)\) satisfying
\(\pi(\alpha(a))=\alpha_E(\pi(a))\). Over trivially graded \(B\), even
elements act diagonally on \(E^0\oplus E^1\), and odd elements act
off-diagonally. This is the convention used for an even Kasparov module.

Write \(E^{\mathrm{op}}\) for the same underlying module with grading
\(-\gamma_E\). This reverses vector degrees but leaves the induced
grading on \(\mathcal L(E)\) unchanged. The standard module and its
balanced graded version are \[
\begin{gathered}
H_B=\ell^2(\mathbb N)\otimes B,\\
\gamma_{H_B}((b_j))=(\beta(b_j)),\\
\widehat H_B=H_B\oplus H_B^{\mathrm{op}}.
\end{gathered}
\tag{2.5}
\] Coordinate descriptions do not require a unit of \(B\); a symbol for
a coordinate denotes a copy of \(B\), not a vector containing a
nonexistent unit.

\subsection{3. The graded tensor
product}\label{the-graded-tensor-product}

On the vector space \(A\odot B\), define, for homogeneous entries, \[
\begin{gathered}
(a\widehat\otimes b)(a'\widehat\otimes b')\\
=(-1)^{|b||a'|}aa'\widehat\otimes bb',\\
(a\widehat\otimes b)^*\\
=(-1)^{|a||b|}a^*\widehat\otimes b^*.
\end{gathered}
\tag{3.1}
\] The degree is \(|a|+|b|\). These formulas are extended linearly. In a
triple product the total sign is \(|b||a'|+|b||a''|+|b'||a''|\),
independent of the placement of parentheses. For the involution,
reversing two factors introduces exactly the signs in the second formula
together with the reversed multiplication sign. Thus (3.1) is an
associative *-algebra.

\subsubsection{Lemma 3.0a. The ordinary spatial
norm}\label{lemma-3.0a.-the-ordinary-spatial-norm}

For arbitrary C*-algebras \(A,B\), the formula \[
\|z\|_{\min}=\sup_{\pi,\rho}\|(\pi\odot\rho)(z)\|,
\qquad z\in A\odot B,
\] where the supremum runs over Hilbert-space representations of the two
factors, is a C*-norm. Any pair of faithful representations gives this
same norm. Tensoring homomorphisms is contractive; tensoring injective
homomorphisms is isometric. The ordinary flip and rebracketing are
isometric. No countability, units, or nuclearity are assumed.

\textbf{Proof.} First construct the Hilbert-space tensor product.
Rewrite a finite tensor sum using an orthonormal basis \(e_j\) of the
finite-dimensional span of its first entries as
\(\sum_j e_j\otimes\eta_j\). The product inner product gives squared
norm \(\sum_j\|\eta_j\|^2\), and hence is positive definite on the
algebraic tensor product. Completing gives \(H\otimes K\). Expansion in
an orthonormal basis of the second entries proves
\(\|T\otimes1\|\leq\|T\|\); the corresponding expansion in the first
entries proves \(\|1\otimes S\|\leq\|S\|\). Thus \(T\otimes S\) is
bounded, has adjoint \(T^*\otimes S^*\), and has norm \(\|T\|\|S\|\),
the reverse inequality following from unit product vectors approaching
the two operator norms. The algebraic flip and rebracketing preserve
these inner products and have dense range, so extend to unitaries.

We recall explicitly why faithful representations exist. In the
unitization, a nonzero positive element \(c\) has a state on
\(C^*(1,c)\) taking value \(\|c\|\) on \(c\), by evaluation at the
maximal spectral value. Hahn--Banach extends this functional with norm
one and value one at the identity. Such an extension is positive: for
self-adjoint \(h\), expanding \(f(e^{ith})\) at \(t=0\) and using
\(|f(e^{ith})|\leq1\) shows that \(f(h)\) is real; for \(0\leq h\leq1\),
the bound \(|1-f(h)|\leq\|1-h\|\leq1\) then gives \(f(h)\geq0\). Scaling
proves positivity in general. For a state \(f\), set
\(\langle[x],[y]\rangle=f(x^*y)\). Nonnegativity of
\(f((x+ty)^*(x+ty))\), minimized over complex \(t\), gives
Cauchy--Schwarz (with a positive \(\varepsilon\) added to the
denominator if \(f(y^*y)=0\)). Null vectors are therefore orthogonal to
everything and the quotient is an inner-product space. Left
multiplication respects the null space and is bounded because \[
f(x^*a^*ax)\leq\|a\|^2f(x^*x),
\] and its adjoint is left multiplication by \(a^*\). This is a
representation with cyclic vector the class of \(1\). The direct sum
over states separates all positive elements, hence is faithful.
Restricting to the nondegenerate part of the original algebra preserves
faithfulness. The functional calculus, order, and C*-homomorphism facts
in this argument have the earlier programme proofs named in the
introduction.

For \(z=\sum_i a_i\otimes b_i\), every product representation has norm
at most \(\sum_i\|a_i\|\|b_i\|\). The supremum is therefore finite, is
submultiplicative, preserves involution, and satisfies \[
\|z^*z\|_{\min}=\sup_{\pi,\rho}\|(\pi\odot\rho)(z)\|^2
=\|z\|_{\min}^2.
\] It is a norm: if the factors are represented faithfully, their
product representation is algebraically injective. Indeed vector
functionals separate operators, and their restrictions span the dual of
any finite-dimensional operator subspace. Applying product vector
functionals to an image of \(z\) that is zero therefore recovers every
algebraic tensor coefficient. This proves \(z=0\).

It remains to prove that one faithful pair realizes the supremum.
Discard the zero representation parts, and extend representations
canonically to unitizations; for an already unital algebra use its
existing identity. A faithful nondegenerate representation of a
nonunital algebra has a faithful extension: if \(\pi(a)+\lambda I=0\)
with \(\lambda\ne0\), faithfulness would make \(-a/\lambda\) an identity
of the original algebra. We can therefore prove the assertion in the
unital case.

For a faithful unital representation \(\pi:A\to\mathcal B(H)\), the
convex hull of its unit vector states is weak-* dense in the state space
of \(A\). Otherwise real Hahn--Banach separation gives a self-adjoint
\(h\in A\) and a state \(f\) with \[
f(h)>\sup_{\|\xi\|=1}\langle\xi,\pi(h)\xi\rangle.
\] The right side is the maximal spectral value of \(h\): faithfulness
preserves spectrum and order, and a smaller upper bound would contradict
the least scalar upper bound for the self-adjoint operator. Positivity
gives \(f(h)\) at most that same spectral value, a contradiction. Here
weak-* separation is evaluation at a self-adjoint element because a
continuous real linear functional involves only finitely many
evaluations.

Fix a faithful pair \(\pi,\rho\) and put \(c=\|(\pi\odot\rho)(z)\|\).
For \(w\in A\odot B\), the operator inequality in this representation
gives \[
(f_\xi\otimes g_\eta)(w^*z^*zw)
\leq c^2(f_\xi\otimes g_\eta)(w^*w)
\] for its unit vector states. This holds for finite convex combinations
of each family, and then for all states \(f,g\) by weak-* approximation:
the two expressions are finite sums of products of evaluations. In the
product of their cyclic GNS representations, vectors of the form
\(w(\xi_f\otimes\xi_g)\) are dense, and their squared norms and squared
image norms are exactly these two expressions. The inequality thus
bounds that product representation of \(z\) by \(c\).

Every nondegenerate representation is a direct sum of cyclic
representations: choose a maximal orthogonal family of cyclic invariant
subspaces; their orthogonal complement is invariant and, if nonzero,
supplies another cyclic subspace. Each normalized cyclic vector defines
a state and the isometric map from its GNS space onto the cyclic
subspace. The product of two such sums is the sum of the product cyclic
representations, first on algebraic vectors and then by completion.
Hence all pairs have norm at most \(c\). The reverse bound is part of
the supremum, proving independence.

Composing representations with homomorphisms proves contractivity of the
tensor map. For inclusions, restrict faithful representations of the
larger factors to the smaller ones and their nondegenerate parts; they
remain faithful, so independence proves isometry. The Hilbert-space flip
and rebracketing unitaries, applied to faithful pairs and triples, give
the asserted algebra isometries. The same proof identifies \(M_n(A)\)
with \(M_n(\mathbb C)\otimes_{\min}A\). Completing the algebraic tensor
product now gives \(A\otimes_{\min}B\). If a factor is zero, all the
assertions reduce to the zero algebra. \(\square\)

We give a concrete C*-norm rather than leaving positivity of the graded
construction implicit. Set \(D=A\otimes_{\min}B\), the ordinary spatial
tensor product, and \(s=\alpha\otimes1\). For \(d\in A\odot B\), write
\(d=d_0+d_1\), according to the degree of its \(B\)-factor. Define \[
\begin{gathered}
\iota(d)=
\begin{pmatrix}
d_0&d_1\\ s(d_1)&s(d_0)
\end{pmatrix}
\\\in M_2(D).
\end{gathered}
\tag{3.2}
\]

\subsubsection{Lemma 3.1. A faithful
C*-completion}\label{lemma-3.1.-a-faithful-c-completion}

Formula (3.2) is an injective *-homomorphism for (3.1). Its image has a
closed completion inside \(M_2(D)\). The induced norm is the minimal
graded tensor norm.

\textbf{Proof.} In multiplier notation put \[
\lambda(a)=\operatorname{diag}(a,\alpha(a)),\qquad
u=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
\] Then \(u\lambda(a)=\lambda(\alpha(a))u\), and
\(\iota(a\widehat\otimes b)=\lambda(a)u^{|b|}\otimes b\). Moving
\(u^{|b|}\) across \(\lambda(a')\) gives the multiplication sign in
(3.1). Taking its adjoint gives the involution sign. Its top row
recovers \(d_0,d_1\), so it is injective.

The parity projections \(d\mapsto d_r\) extend as contractions on \(D\),
since \(1\otimes\beta\) is an isometric involution. Consequently \[
\tfrac12\|d\|_D\leq\|\iota(d)\|\leq2\|d\|_D.
\tag{3.3}
\] For the lower bound, use \(\|d_r\|\leq\|\iota(d)\|\); for the upper
bound, each diagonal or off-diagonal matrix has norm \(\|d_r\|\). Hence
the image extends to a closed linear image of \(D\). It is a
C*-subalgebra by the already checked product and adjoint rules.

To identify the spatial norm, choose faithful nondegenerate graded
Hilbert-space representations \((\pi_A,H_A,\Gamma_A)\) and
\((\pi_B,H_B',\Gamma_B)\). Here the primes distinguish this Hilbert
space from a standard coefficient module. Such representations exist:
from an ordinary faithful nondegenerate \(\pi\), use
\(\pi\oplus\pi\alpha\) with the swapping grading operator. The spatial
graded representation is \[
\Pi(a\widehat\otimes b)
=\pi_A(a)\Gamma_A^{|b|}\otimes\pi_B(b).
\tag{3.4}
\] It has precisely the signs (3.1).

Represent (3.2) faithfully on \((H_A\oplus H_A)\otimes H_B'\).
Conjugation by \(\operatorname{diag}(1,\Gamma_A)\) makes the two copies
of \(\lambda(a)\) equal to \(\pi_A(a)\), and turns \(u\) into
\(\begin{pmatrix}0&\Gamma_A\\\Gamma_A&0\end{pmatrix}\). The scalar
Hadamard unitary then splits this representation into (3.4) and the same
formula with an additional \((-1)^{|b|}\). Conjugating the second
summand by \(1\otimes\Gamma_B\) removes that additional sign. Thus its
norm equals the norm of (3.4). In particular (3.4) is faithful and its
norm is independent of the faithful graded representations chosen. This
is the minimal spatial graded completion. \(\square\)

We denote it \(A\widehat\otimes B\). Its grading is
\(\alpha\widehat\otimes\beta\), implemented in (3.4) by
\(\Gamma_A\otimes\Gamma_B\).

One can instead complete in the largest norm obtained from pairs of
representations whose homogeneous images supercommute. This is
\(A\widehat\otimes_{\max}B\). The norm exists: (3.4) is a faithful
candidate, and any such pair bounds the norm of a finite sum by the sum
of the products of the factor norms. We specify the minimal norm unless
a maximal subscript is present.

\subsubsection{Theorem 3.2. Associativity and the graded
flip}\label{theorem-3.2.-associativity-and-the-graded-flip}

Rebracketing and the signed flip give graded isomorphisms \[
\begin{aligned}
(A\widehat\otimes B)\widehat\otimes C
 &\cong A\widehat\otimes(B\widehat\otimes C),\\
\tau(a\widehat\otimes b)
 &=(-1)^{|a||b|}b\widehat\otimes a.
\end{aligned}
\tag{3.5}
\] The same statements hold for maximal completions.

\textbf{Proof.} Use faithful graded representations of all three
factors. For homogeneous \(a,b,c\), both bracketings act, after the
ordinary Hilbert-space rebracketing unitary, by \[
\pi_A(a)\Gamma_A^{|b|+|c|}
\otimes\pi_B(b)\Gamma_B^{|c|}
\otimes\pi_C(c).
\tag{3.6}
\] Lemma 3.1 makes both representations faithful, proving equality of
the two norms. The algebraic rebracketing therefore extends to the
asserted isomorphism.

For the flip, define a unitary on homogeneous vectors by \[
U(\xi\otimes\eta)=(-1)^{|\xi||\eta|}\eta\otimes\xi.
\] On vectors of degrees \(r,s\), conjugating (3.4) gives the operator
associated to \((-1)^{pq}b\widehat\otimes a\), where \(p=|a|,q=|b|\):
the total sign is \(pq+ps\). This is the sign from its scalar prefactor
and the graded action in the new order. Thus the flip preserves the
faithful spatial norm. Formula (3.1) verifies its product and adjoint
identities, and applying it twice gives the identity.

For maximal completions, a representation of either bracketing is a
triple of representations with pairwise supercommuting homogeneous
images. Indeed the first two extend uniquely to their maximal product
and then supercommute with the third, and conversely restriction
recovers that triple. The two universal norms consequently coincide.
Exchanging the first two representations with the sign in (3.5) gives
the maximal flip. \(\square\)

The sign in the involution matters. The tensor of two self-adjoint odd
elements is skew-adjoint; multiplying it by \(i\) gives a self-adjoint
element.

\subsection{4. Removing an inner grading from the
product}\label{removing-an-inner-grading-from-the-product}

\subsubsection{Theorem 4.1. The inner-grading
isomorphism}\label{theorem-4.1.-the-inner-grading-isomorphism}

If \(\alpha=\operatorname{Ad}g\), with \(g\in M(A)\) a self-adjoint
unitary, then \[
\begin{gathered}
\Phi:A\widehat\otimes B\longrightarrow A\otimes_{\min}B,
\\
\Phi(a\widehat\otimes b)=ag^{|b|}\otimes b
\end{gathered}
\tag{4.1}
\] is a *-isomorphism. On the right the grading is
\(\alpha\otimes\beta\). If \(\beta=\operatorname{Ad}h\) is also inner,
it is implemented by \(g\otimes h\). Formula (4.1) also identifies the
maximal graded and ordinary maximal tensor products.

\textbf{Proof.} Since \(ga'=(-1)^{|a'|}a'g\), \[
(ag^q)(a'g^{q'})
=(-1)^{q|a'|}aa'g^{q+q'}.
\] This verifies multiplication. The identity
\((ag^q)^*=(-1)^{|a|q}a^*g^q\) verifies the involution. The same linear
formula gives the inverse, because \(g^2=1\) and \(g\) is even.

For the norm, choose faithful nondegenerate graded representations as in
Lemma 3.1, and extend \(\pi_A\) to multipliers. Put \(g_A=\pi_A(g)\) and
\(c=\Gamma_Ag_A\). Both \(g_A\) and \(\Gamma_A\) implement \(\alpha\),
so \(c\) commutes with \(\pi_A(A)\). They commute with each other, since
\(g\) is even; hence \(c\) is a self-adjoint unitary. Let
\(P_\pm=(1\pm c)/2\). The unitary \[
W=P_+\otimes1+P_-\otimes\Gamma_B
\] conjugates (3.4) into \(\pi_A(a)g_A^{|b|}\otimes\pi_B(b)\). On the
negative eigenspace of \(c\), the sign from \(\Gamma_A=cg_A\) is
canceled by conjugation with \(\Gamma_B\). The resulting representation
is the faithful ordinary spatial representation of (4.1). Thus \(\Phi\)
is isometric.

The grading assertion follows by applying \(\alpha\otimes\beta\) to the
formula, using \(\alpha(g)=g\).

For the maximal assertion, take a nondegenerate supercommuting pair
\(\pi,\rho\). The extended multiplier \(\pi(g)\) commutes with
\(\rho(B)\): approximate \(g\) strictly by \(ge_i\), where \(e_i\) is an
even approximate identity of \(A\), and use that these are even.
Replacing \(\rho(b)\) by \(\pi(g)^{|b|}\rho(b)\) produces an ordinary
representation of \(B\) commuting with \(\pi(A)\). Its multiplicativity
and adjoint identities use the fact that \(\pi(g)\) commutes with
\(\rho(B)\). Conversely an ordinary commuting pair yields a
supercommuting pair by the same replacement. Degenerate representations
can be restricted to the essential subspace of \(\pi(A)\); the
complementary subspace contributes zero to the product. The universal
norms are therefore identified. \(\square\)

This is an isomorphism of the underlying C*-algebras with the specified
grading on the ordinary product; it does not discard parity information.

For example, if \(A\) is inner graded and \(M_2\) has its diagonal
grading, the product first becomes \(M_2(A)\) with grading operator
\(\operatorname{diag}(g,-g)\). Write \(p_\pm=(1\pm g)/2\). The
multiplier matrix \[
\begin{gathered}
V=\begin{pmatrix}p_+&p_-\\p_-&p_+\end{pmatrix},
\\
V\operatorname{diag}(g,-g)V^*=\operatorname{diag}(1,-1),
\end{gathered}
\tag{4.2}
\] is a self-adjoint unitary. The equality follows separately on the two
eigenspaces of \(g\): it is the identity on the first and exchanges the
two coordinates on the second. Thus the product is isomorphic to
\(M_2(A)\) with the standard diagonal grading.

\subsection{5. Complex Clifford
algebras}\label{complex-clifford-algebras}

For a real Euclidean space \(V\), the complex Clifford algebra
\(\mathrm{Cl}(V)\) is the universal unital C*-algebra with a real-linear
map \(c:V\to\mathrm{Cl}(V)\) such that \[
\begin{gathered}
c(v)^*=c(v),\\
c(v)c(w)\\+c(w)c(v)=2(v,w)1.
\end{gathered}
\tag{5.1}
\] The grading sends \(c(v)\) to \(-c(v)\). For an orthonormal basis of
\(\mathbb R^n\), write \(C_n=\mathrm{Cl}(\mathbb R^n)\) and
\(e_j=c(v_j)\). Its generators are odd self-adjoint anticommuting
unitaries. Put \(C_0=\mathbb C\) with trivial grading.

The universal construction exists because the relations have
Hilbert-space representations, constructed below, and any word reduces
to a scalar multiple of \[
e_{i_1}\cdots e_{i_r},\qquad
i_1<\cdots<i_r.
\tag{5.2}
\] There are \(2^n\) such words. Their universal representation norms
are bounded by one, so the universal norm of any linear combination is
bounded by the sum of the absolute coefficients.

Use the Pauli matrices \[
\begin{aligned}
\sigma_1&=\begin{pmatrix}0&1\\1&0\end{pmatrix},\\
\sigma_2&=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\\
\sigma_3&=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\end{aligned}
\tag{5.3}
\] They are self-adjoint unitaries, anticommute pairwise, and satisfy
\(\sigma_1\sigma_2=i\sigma_3\).

\subsubsection{Theorem 5.1. The Clifford algebra
computation}\label{theorem-5.1.-the-clifford-algebra-computation}

As graded C*-algebras, \[
\begin{gathered}
C_1\\\cong\mathbb C\oplus\mathbb C\\
\text{with swap grading},\\
C_{2k}\\\cong M_{2^k}(\mathbb C)\\
\text{with grading }\operatorname{Ad}\Gamma_k,\\
C_{2k+1}\\\cong M_{2^k}(\mathbb C)\oplus M_{2^k}(\mathbb C)\\
\text{with swap grading},
\end{gathered}
\tag{5.4}
\] where \(\Gamma_k=\sigma_3^{\otimes k}\). For \(k\geq1\), its positive
and negative eigenspaces both have dimension \(2^{k-1}\).

\textbf{Proof.} The normal forms in (5.2) bound the dimension by
\(2^n\). We construct a quotient of exactly that dimension, which
simultaneously proves that all normal forms are independent.

In dimension one, evaluate the generator at \(+1\) and \(-1\). The
images of \(1,e_1\) form a basis of \(\mathbb C\oplus\mathbb C\), so the
dimension bound makes this map an isomorphism. Negating the generator
swaps the two evaluations.

For \(n=2k\), work on \(S_k=(\mathbb C^2)^{\otimes k}\), and set \[
\begin{aligned}
c_{2j-1}&=\sigma_3^{\otimes(j-1)}\otimes\sigma_1\otimes1^{\otimes(k-j)},\\
c_{2j}&=\sigma_3^{\otimes(j-1)}\otimes\sigma_2\otimes1^{\otimes(k-j)}.
\end{aligned}
\tag{5.5}
\] Each is a self-adjoint unitary. Two in the same slot anticommute. For
two in distinct slots, the earlier non-diagonal Pauli factor meets
exactly one \(\sigma_3\) in the later operator; that is the only
negative interchange. Thus all Clifford relations hold. Adjacent
products recover the diagonal operator in each slot: \[
\begin{gathered}
-ic_{2j-1}c_{2j}\\=1^{\otimes(j-1)}\otimes\sigma_3\otimes1^{\otimes(k-j)}.
\end{gathered}
\tag{5.6}
\] Multiplying by these diagonal operators in earlier slots isolates
\(\sigma_1\) and \(\sigma_2\) in slot \(j\). Together with
\(1,\sigma_3\), they span the full \(M_2\) there. Their slot products
consequently span every tensor of matrix units, so the representation is
onto \(\operatorname{End}(S_k)\). The latter has dimension \(2^{2k}\);
the normal-form bound proves injectivity. Its grading is conjugation by
\(\Gamma_k=\sigma_3^{\otimes k}\), since this negates every displayed
generator. Among the \(2^k\) tensor-basis vectors, flipping the first
factor pairs opposite grading eigenvalues, proving the equal
multiplicities for \(k\geq1\).

For \(n=2k+1\), use the elements \[
\begin{gathered}
\omega_k=(-i)^ke_1\cdots e_{2k},\\
 w=\omega_ke_{2k+1}.
\end{gathered}
\tag{5.7}
\] Reversing \(2k\) generators introduces \((-1)^k\), exactly the factor
needed for \(\omega_k^*=\omega_k\) and \(\omega_k^2=1\). It anticommutes
with each of the first \(2k\) generators and commutes with the last, so
\(w\) is a central odd self-adjoint unitary. Extend (5.5) in two ways by
sending the last generator to \(+\Gamma_k\) and \(-\Gamma_k\). In both,
(5.6) gives \(\omega_k=\Gamma_k\); hence \(w\) takes the values \(+1\)
and \(-1\). The central projections \((1\pm w)/2\) isolate the two
matrix summands. Each summand contains the full matrix algebra already
generated by (5.5), making the combined representation onto
\(M_{2^k}\oplus M_{2^k}\). Its dimension \(2^{2k+1}\) attains the
normal-form bound, so it is faithful.

Negating every generator acts in these coordinates as \[
(X,Y)\longmapsto(\Gamma_kY\Gamma_k,\Gamma_kX\Gamma_k).
\tag{5.8}
\] Checking the first \(2k\) generators uses their anticommutation with
\(\Gamma_k\); checking the last uses the exchange of its two opposite
values. Conjugating the second coordinate by \(\Gamma_k\) changes this
grading to pure swap. The argument includes \(k=0\), with an empty
tensor product and \(\Gamma_0=1\). \(\square\)

\subsubsection{Corollary 5.2. Clifford tensor
products}\label{corollary-5.2.-clifford-tensor-products}

The generators from the two factors give \[
\begin{gathered}
C_p\widehat\otimes C_q\cong C_{p+q},
\\
C_1\widehat\otimes C_1\cong(M_2,\operatorname{Ad}\sigma_3).
\end{gathered}
\tag{5.9}
\] These identities hold for both minimal and maximal graded products.
Consequently \(C_{n+2}\cong C_n\widehat\otimes C_2\).

\textbf{Proof.} The two copies of the generators in the product are odd
self-adjoint unitaries. The multiplication rule (3.1) makes generators
from different factors anticommute, so they define a surjective graded
homomorphism from \(C_{p+q}\). The minimal product has vector-space
dimension \(2^p2^q=2^{p+q}\) by Lemma 3.1; the surjection is therefore
injective. For the maximal product, the algebraic product already has
that dimension, and its maximal norm is faithful because it dominates
the minimal norm. Its completion is the same finite-dimensional
*-algebra. Finite-dimensional C*-algebras have their unique C*-norm,
since the norm squared of an element is the spectral radius of its
square modulus. Finally the \(p=q=1\) representation sends the two
generators to \(\sigma_1,\sigma_2\), giving the displayed grading.
\(\square\)

The self-adjoint \textbf{chirality element} in even dimension is
\(\omega_k\) in (5.7). Reversing an orientation reverses its sign. More
generally an orthogonal change of orthonormal basis multiplies the top
word by the determinant: distinct orthogonal vectors have zero pairwise
inner products, so their Clifford product is their exterior top product
under the linear normal-form identification (5.2). In (5.5), chirality
is exactly \(\Gamma_k\), fixing our orientation and sign convention.

For a Euclidean vector bundle \(V\to X\), each orthogonal change of
frame acts on (5.1) by a graded *-automorphism. These actions satisfy
the transition cocycle, so the fibers \(\mathrm{Cl}(V_x)\) assemble into
a locally trivial graded algebra bundle \(\mathrm{Cl}(V)\). The normal
forms (5.2) show that its complex rank is \(2^{\operatorname{rank}V}\).
For locally compact Hausdorff \(X\), continuous sections vanishing at
infinity form the graded C*-algebra \(C_0(X,\mathrm{Cl}(V))\), with
supremum norm and pointwise operations. A Hermitian graded vector bundle
carrying an odd Clifford action is a graded Clifford-module bundle. A
global irreducible spinor bundle is additional data; its existence does
not follow merely from choosing local matrix descriptions.

\subsection{6. Interior tensor products and graded
stabilization}\label{interior-tensor-products-and-graded-stabilization}

Let \(E\) be a graded Hilbert \(A\)-module, \(F\) a graded Hilbert
\(B\)-module, and \(\phi:A\to\mathcal L(F)\) a graded *-homomorphism.
Its \textbf{interior} tensor product is the ordinary Hilbert-module
tensor product with \[
\begin{gathered}
\langle x\otimes y,x'\otimes y'\rangle\\
=\langle y,\phi(\langle x,x'\rangle)y'\rangle,\\
\gamma(x\otimes y)=\gamma_E x\otimes\gamma_F y.
\end{gathered}
\tag{6.1}
\] There is no extra sign in the inner-product formula. The left action
belongs in \(\mathcal L(F)\); an action into \(B\) alone would not
describe a general correspondence.

\subsubsection{Lemma 6.0a. Constructing the ordinary interior
product}\label{lemma-6.0a.-constructing-the-ordinary-interior-product}

For Hilbert modules \(E_A,F_B\) and any homomorphism
\(\phi:A\to\mathcal L(F)\), the first formula in (6.1) defines a
positive semidefinite form on the balanced algebraic tensor product. Its
null-space quotient and completion are a Hilbert \(B\)-module. Every
\(T\in\mathcal L(E)\) gives an adjointable \(T\otimes1\), with norm at
most \(\|T\|\) and adjoint \(T^*\otimes1\). This assignment preserves
products and adjoints. For three compatible correspondences, the map
\((x\otimes y)\otimes z\mapsto x\otimes(y\otimes z)\) is a unitary.
Degenerate left actions and nonunital coefficient algebras are allowed.

\textbf{Proof.} Balancing respects the form because
\(\langle xa,x'\rangle=a^*\langle x,x'\rangle\),
\(\langle x,x'a\rangle=\langle x,x'\rangle a\), and \(\phi\) preserves
products and adjoints. It is conjugate symmetric and right \(B\)-linear
in the second variable.

The nonunital matrix inclusion, row adjoint and finite Gram positivity
used next are proved in
\href{supporting/hilbert-c-star-modules-and-morita-equivalence/hilbert-module-foundations.html\#mf-006}{Ordinary
Hilbert C*-module foundations, MF.6}. For \(x_1,\ldots,x_n\in E\),
define \(V:A^n\to E\) by \(V(a_i)=\sum_i x_ia_i\). It is bounded by the
triangle inequality and has adjoint \(V^*x=(\langle x_i,x\rangle)_i\).
Its Gram matrix is \[
G=V^*V=[\langle x_i,x_j\rangle]\in M_n(A)_+.
\] Apply the homomorphism \(\phi_n\) to obtain a positive operator on
\(F^n\). For \(u=\sum_i x_i\otimes y_i\) this gives \[
\langle u,u\rangle
=\langle (y_i),\phi_n(G)(y_i)\rangle\geq0.
\] The same argument applied simultaneously to two finite sums shows
that their two-by-two Gram matrix is positive in \(M_2(B)\). This also
proves that the form descends through every balanced relation.

For completeness, a positive block matrix
\(\begin{pmatrix}a&b\\b^*&d\end{pmatrix}\) in \(M_2(B)\) satisfies \[
b^*(a+\varepsilon1)^{-1}b\leq d,
\qquad b^*b\leq(\|a\|+\varepsilon)d.
\] The first inequality follows by multiplying the matrix, with
\(\varepsilon\) added to its first corner, on both sides by the column
\((-(a+\varepsilon)^{-1}b,1)\) and its adjoint. The second uses the
scalar lower bound for \((a+\varepsilon)^{-1}\). Letting
\(\varepsilon\downarrow0\) proves
\(\|\langle u,v\rangle\|\leq\|\langle u,u\rangle\|^{1/2}\|\langle v,v\rangle\|^{1/2}\).
Thus the form norm is a seminorm with the triangle inequality; every
null vector is orthogonal to every vector. The null vectors form a
submodule, since \(\langle ub,ub\rangle=b^*\langle u,u\rangle b\). The
quotient is a pre-Hilbert module. Its completion retains the inner
product and bounded right action by these inequalities.

For \(T\in\mathcal L(E)\), positivity of \(\|T\|^2-T^*T\) gives \[
[\langle Tx_i,Tx_j\rangle]
\leq\|T\|^2[\langle x_i,x_j\rangle],
\] by applying \(V^*(\|T\|^2-T^*T)V\). Apply \(\phi_n\) and evaluate at
\((y_i)\). The resulting inequality proves that \(T\otimes1\) respects
null vectors and has norm at most \(\|T\|\). The elementary
inner-product identity gives its adjoint \(T^*\otimes1\). Products,
linearity, and the identity are checked on elementary tensors and extend
by density.

If \(G_C\) has left action \(\psi:B\to\mathcal L(G)\), the induced left
action of \(A\) on \(F\otimes_\psi G\) is \(\phi(a)\otimes1\), which has
just been constructed. Both bracketings are balanced, and the inner
product between two elementary threefold tensors in either bracketing is
\[
\langle z,\psi(\langle y,\phi(\langle x,x'\rangle)y'\rangle)z'\rangle.
\] Consequently rebracketing is well defined and isometric on the
null-space quotients. Elementary tensors have dense span on both sides,
so its extension is a unitary. No step used nondegeneracy or a unit.
\(\square\)

\subsubsection{Proposition 6.1. The interior
grading}\label{proposition-6.1.-the-interior-grading}

Formula (6.1) defines a grading on \(E\otimes_\phi F\). For homogeneous
tensors its degree is \(|x|+|y|\). The map \(T\mapsto T\otimes1\) is a
graded *-homomorphism from \(\mathcal L(E)\) to
\(\mathcal L(E\otimes_\phi F)\). Rebracketing three compatible
correspondences preserves grading.

\textbf{Proof.} The balanced relation is respected because \[
\begin{gathered}
\gamma_E(xa)\otimes\gamma_Fy
\\=\gamma_Ex\otimes\phi(\alpha(a))\gamma_Fy
\\=\gamma_Ex\otimes\gamma_F(\phi(a)y).
\end{gathered}
\] The inner product of the two graded tensors in (6.1) equals
\(\beta(\langle y,\phi(\langle x,x'\rangle)y'\rangle)\), by (2.1) and
the gradedness of \(\phi\). Thus the grading is isometric, respects null
vectors, and extends to the completion. It is a complex-linear
involution with the required coefficient semilinearity. Lemma 6.0a
supplies the adjointable map \(T\otimes1\); on elementary tensors
conjugation by the grading gives \((\gamma_ET\gamma_E)\otimes1\).
Finally both ways of rebracketing have grading
\(\gamma_E\otimes\gamma_F\otimes\gamma_G\), and the rebracketing unitary
constructed in Lemma 6.0a therefore preserves it. \(\square\)

\subsubsection{Theorem 6.2. Graded
stabilization}\label{theorem-6.2.-graded-stabilization}

Let \(B\) be any graded C*-algebra and \(E\) a countably generated
graded Hilbert \(B\)-module. There is a degree-zero adjointable unitary
\[
E\oplus\widehat H_B\cong\widehat H_B.
\tag{6.2}
\] No unitality or \(\sigma\)-unitality hypothesis on \(B\) is needed.

\textbf{Proof.} Split a countable generating family into its even and
odd components. The resulting homogeneous family still generates: the
span of the original generators times \(B\) lies in the span of the
components times \(B\). Rescale its nonzero members to contractions. For
a homogeneous generator \(\eta_j\) of degree \(p_j\), put \[
X_j=H_B^{[p_j]},\qquad
R_j((b_m))=\eta_jb_1,
\tag{6.3}
\] where the bracket means use the natural grading if \(p_j=0\) and its
opposite if \(p_j=1\). This map is adjointable, with \[
R_j^*x=(\langle\eta_j,x\rangle,0,\ldots),
\] and has degree zero between the indicated modules. For example an
input coordinate of coefficient degree \(r\) has total degree \(r+p_j\),
exactly the degree of \(\eta_jb_1\). The adjoint degree check follows
from \(|\langle\eta_j,x\rangle|=p_j+|x|\). The ranges have dense joint
span by countable generation.

Repeat each map at arbitrarily late indices, retaining its source
parity. Insert zero maps of both source parities as necessary so that
there are infinitely many summands of each parity. Then \[
X=\bigoplus_jX_j\cong\widehat H_B
\tag{6.4}
\] by separate countable coordinate bijections in the two parities.
Define the same operator as in the operator-range stabilization proof:
\[
\begin{gathered}
T:X\to E\oplus X,\\
T((x_j))=\\
\left(\sum_j2^{-j}R_jx_j,(4^{-j}x_j)_j\right).
\end{gathered}
\tag{6.5}
\] It is adjointable and degree zero. Its adjoint is
\(T^*(y,(z_j))=(2^{-j}R_j^*y+4^{-j}z_j)_j\). Every finite-support vector
lies in its range by taking \(y=0\); hence that range is dense.

The range of \(T\) is dense too. Approximate \(y\in E\) by a finite sum
\(\sum R_iu_i\), and realize each term in a sufficiently late copy
\(j_i\) of the same graded source, with input \(2^{j_i}u_i\). The second
coordinate has norm at most \(\sum_i2^{-j_i}\|u_i\|\), as small as
desired. To realize a general second coordinate, first approximate it by
a finite-support vector, produce it with \(x_j=4^jz_j\), and correct its
first coordinate at later indices. Thus both density arguments respect
the typed source summands.

The preceding lesson's operator-range lemma proves that these two dense
ranges give a unitary by the rule \[
U(|T|x)=Tx.
\tag{6.6}
\] For clarity, its required polar step uses
\(\overline{\operatorname{ran}(T^*T)}
=\overline{\operatorname{ran}T^*}=X\), so \(|T|\) has dense range; (6.6)
preserves inner products and has dense, closed range \(E\oplus X\). This
is a valid polar decomposition in the present circumstances. Since \(T\)
has degree zero, \(T^*T\) and its positive square root are even. The
rule (6.6) therefore intertwines the gradings on a dense subspace, and
hence everywhere. It is a degree-zero unitary \(X\to E\oplus X\). Use
(6.4) and invert this unitary to obtain (6.2). \(\square\)

In particular \(E\) is degree-preservingly isomorphic to
\(P\widehat H_B\) for an even projection
\(P\in\mathcal L(\widehat H_B)\): transport the projection onto the
first summand of (6.2). Its compact operators identify with
\(P\mathcal K(\widehat H_B)P\). Indeed transporting \(\theta_{x,y}\)
gives the corresponding rank-one operator on the range, and cutting
arbitrary rank-one operators by \(P\) produces \(\theta_{Px,Py}\).

\subsubsection{Proposition 6.3. Exterior tensor
products}\label{proposition-6.3.-exterior-tensor-products}

For graded Hilbert modules \(E\) over \(B\) and \(F\) over \(C\), their
exterior product is a graded Hilbert \(B\widehat\otimes C\)-module with
homogeneous formulas \[
\begin{gathered}
(x\widehat\otimes y)(b\widehat\otimes c)\\
=(-1)^{|y||b|}xb\widehat\otimes yc,\\
\langle x\widehat\otimes y,x'\widehat\otimes y'\rangle\\
=(-1)^{|y|(|x|+|x'|)}
   \langle x,x'\rangle\widehat\otimes\langle y,y'\rangle.
\end{gathered}
\tag{6.7}
\] Its grading is the sum of the vector degrees. Homogeneous adjointable
operators act by \[
(S\widehat\otimes T)(x\widehat\otimes y)
=(-1)^{|T||x|}Sx\widehat\otimes Ty,
\tag{6.8}
\] and \((S\widehat\otimes T)^*=(-1)^{|S||T|}S^*\widehat\otimes T^*\).
Moreover \[
\mathcal K(E\widehat\otimes F)
\cong\mathcal K(E)\widehat\otimes\mathcal K(F).
\tag{6.9}
\]

\textbf{Proof.} We realize the product as a corner, which proves
positivity as well as the sign formulas. Let
\(L_E=\mathcal K(E\oplus B)\), with the grading induced by
\(\gamma_E\oplus\beta\). Let \(p_E,q_E\) be the even multiplier
projections onto \(E,B\). For \(x\in E\), the map \(R_x:B\to E\),
\(b\mapsto xb\), is compact and satisfies \(R_x^*z=\langle x,z\rangle\).
To check compactness when \(B\) is nonunital, let \(e_i\) be a
contractive approximate identity: \(\theta_{x,e_i}\) is the map
\(b\mapsto xe_ib\), and it converges in norm to \(R_x\) since
\(xe_i\to x\). The latter follows by applying the approximate identity
to \(\langle x,x\rangle\). Thus the corners of \(L_E\) are \(B\), \(E\),
its adjoint corner, and \(\mathcal K(E)\). Use the analogous notation
for \(F\).

In \(L=L_E\widehat\otimes L_F\), put \[
P=p_E\widehat\otimes p_F,\qquad
Q=q_E\widehat\otimes q_F.
\] These even multiplier projections are well defined by the spatial
representation (3.4) and nondegenerate multiplier extensions. The
coefficient corner \(QLQ\) is \(B\widehat\otimes C\), and \(PLQ\) is a
Hilbert module over it with inner product \(z^*z'\). Corner inclusions
preserve the minimal tensor norm: restrict faithful graded spatial
representations to the ranges of the even corner projections, and apply
Lemma 3.1.

Identify \(x\widehat\otimes y\) with \(R_x\widehat\otimes R_y\) in
\(PLQ\). These tensors have dense span, since compressing the dense
elementary tensors of \(L\) by \(P,Q\) compresses each factor to its
indicated corner. Formula (3.1) gives the right action in (6.7). For the
inner product, the adjoint introduces \((-1)^{|x||y|}\), and moving
\(R_y^*\) past \(R_{x'}\) introduces \((-1)^{|y||x'|}\). Their product
is the displayed inner-product sign. Positivity, conjugate symmetry,
boundedness of the action and completeness follow from this Hilbert
corner. Its induced grading has the stated degree.

Extend \(S\in\mathcal L(E)\) and \(T\in\mathcal L(F)\) as block
operators with zero on the coefficient summands. They define multiplier
corners: multiplying a compact rank-one operator by such a block
operator, on either side, is again compact, with the adjoint giving the
second multiplier action. Their graded tensor multiplier acts on
\(PLQ\); multiplication in the corner gives (6.8) and the stated
adjoint. These multipliers are bounded, since the faithful spatial
representations extend nondegenerately to multipliers and a homogeneous
elementary tensor has norm at most \(\|S\|\|T\|\).

Finally \(\mathcal K(PLQ)\) is the norm-closed span of left
multiplications by \(zz'^*\). For elementary corner tensors these
products are \[
(-1)^{|x'|(|y|+|y'|)}
\theta_{x,x'}\widehat\otimes\theta_{y,y'}.
\] Their span is dense in the corner
\(PLP=\mathcal K(E)\widehat\otimes\mathcal K(F)\). Left multiplication
of this corner on \(PLQ\) is faithful: an element annihilating \(PLQ\)
annihilates the dense span of the products \(zz'^*\), hence annihilates
\(PLP\); taking its product with its own adjoint shows it is zero. A
faithful C*-representation is isometric. This proves (6.9). \(\square\)

\subsection{7. Clifford modules and the Bott
symbol}\label{clifford-modules-and-the-bott-symbol}

Let \(\widehat M_n\) denote the Grothendieck group of finite-dimensional
unital graded \(C_n\)-representations, with isomorphisms preserving
grading. Restriction along \(C_n\subset C_{n+1}\) gives a homomorphism
\[
\begin{gathered}
r_n:\widehat M_{n+1}\to\widehat M_n,\\
Q_n=\widehat M_n/r_n(\widehat M_{n+1}).
\end{gathered}
\tag{7.1}
\] This is an algebraic quotient of representation groups, before any
K-theory theorem is applied.

\subsubsection{Proposition 7.1. The representation
quotient}\label{proposition-7.1.-the-representation-quotient}

For \(k\geq0\), \[
\begin{aligned}
\widehat M_{2k}&\cong\mathbb Z^2,&
\widehat M_{2k+1}&\cong\mathbb Z,\\
Q_{2k}&\cong\mathbb Z,& Q_{2k+1}&=0.
\end{aligned}
\tag{7.2}
\] The generator of \(Q_{2k}\) can be represented by \(S_k\) with
grading \(\Gamma_k\); reversing its grading changes its class to its
negative.

\textbf{Proof.} Let \(d=2^k\). If \(M_d\) acts on a finite-dimensional
space, its matrix units identify that space with
\(\mathbb C^d\otimes W\): take \(W=E_{11}S\) and send \(e_i\otimes\xi\)
to \(E_{i1}\xi\). These images are mutually orthogonal and exhaust the
space because \(\sum_iE_{ii}=1\). In these coordinates the action is
\(X\otimes1\), and its commutant is \(1\otimes\operatorname{End}(W)\),
as follows by commuting with all \(E_{ij}\).

For the grading \(\operatorname{Ad}\Gamma_k\), a module grading
\(\gamma\) therefore has the form \(\Gamma_k\otimes J\): the product
\((\Gamma_k\otimes1)\gamma\) commutes with the matrix algebra. The
element \(\Gamma_k\) is even, so it commutes with \(\gamma\); hence
\(J\) is a self-adjoint unitary. Its two eigenspace dimensions classify
graded modules and add under direct sum. The representation monoid is
\(\mathbb N^2\), and its group completion is \(\mathbb Z^2\). This also
holds at \(k=0\), where the two integers are the even and odd
dimensions.

For a swap-graded \(M_d\oplus M_d\), the module grading is a unitary
from the first central summand to the second, intertwining their matrix
actions, and its inverse in the other direction. The two multiplicity
spaces consequently have equal dimension. Choosing this unitary as their
identification puts the grading in swap form. Thus the monoid is
\(\mathbb N\), with group completion \(\mathbb Z\).

To compute restriction to \(C_{2k}\), use the odd representation in
(5.8) on \(S_k\oplus S_k\), with grading
\((\xi,\eta)\mapsto(\Gamma_k\eta,\Gamma_k\xi)\). The diagonal and
anti-diagonal subspaces are invariant for the first \(2k\) generators,
with gradings \(\Gamma_k\) and \(-\Gamma_k\). Hence \(r_{2k}(1)=(1,1)\).
Each of the two graded irreducibles for \(C_{2k+2}\) has dimension
\(2^{k+1}\); restricted to \(C_{2k+1}\), that is exactly the dimension
of its unique graded irreducible, so \(r_{2k+1}(1,0)=r_{2k+1}(0,1)=1\).
The two quotients are therefore
\(\mathbb Z^2/\mathbb Z(1,1)\cong\mathbb Z\) and
\(\mathbb Z/\mathbb Z=0\). Reversing the even grading exchanges
\((1,0)\) and \((0,1)\), whose quotient classes are opposite.
\(\square\)

A graded module \(c:C_n\to\operatorname{End}(S)\) supplies the symbol \[
\sigma_c(v)=c(v)|_{S^+}:S^+\to S^-.
\tag{7.3}
\] For \(v\ne0\), its inverse is \(\|v\|^{-2}c(v)|_{S^-}\), by (5.1).
Thus the two trivial bundles and (7.3) define a compactly supported
K-theory class in \[
K_0(C_0(\mathbb R^n)).
\] Explicitly, restrict the two trivial bundles and this map to the
closed unit ball \(D^n\); the boundary map on \(S^{n-1}\) is invertible.
The difference-bundle theorem identifies that relative triple with
\(K^0(D^n,S^{n-1})=K_0(C_0(\mathbb R^n))\).
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-difference}{Relative
difference bundles, radial pairs and the planar normalization, Theorem
RK.3} proves both inverse maps for every compact Hausdorff closed pair.
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-radial}{Theorem
RK.4} proves the radial disc/sphere identification in both degrees,
including rank-zero components. For \(n=0\) the class is
\(\dim S^+-\dim S^-\).

\subsubsection{Lemma 7.2. Extending a module kills its
symbol}\label{lemma-7.2.-extending-a-module-kills-its-symbol}

If a graded \(C_n\)-module extends to a graded \(C_{n+1}\)-module, the
class of (7.3) is zero. Every finite-dimensional graded
\(C_{2k+1}\)-module has such an extension.

\textbf{Proof.} Let \(h=c(e_{n+1})\) be the extra odd self-adjoint
unitary. For \(\|v\|\leq1\), replace (7.3) by the positive-to-negative
block of \[
c(v)+\sqrt{1-\|v\|^2}\,h;
\tag{7.4}
\] outside the unit ball leave it equal to \(c(v)\). Since \(h\)
anticommutes with \(c(v)\), the square of (7.4) is \(1\). The
replacement is continuous at the unit sphere, agrees with the old symbol
outside a compact set, and is invertible everywhere. It represents zero
in relative K-theory, by the global-extension relation proved in
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-symbol-rules}{Corollary
RK.4a}. More explicitly, linearly interpolating the symbols is still
fixed and invertible outside the ball, so it is an allowed relative
homotopy to this everywhere invertible triple. The same argument for
\(n=0\) uses \(h:S^+\to S^-\).

For odd \(n=2k+1\), use the central odd unitary \(w\) from (5.7), and
let \(\gamma\) be the module grading. Put \[
h=i\gamma c(w).
\tag{7.5}
\] Since \(c(w)\) is odd, it anticommutes with \(\gamma\); hence
\(h=h^*\) and \(h^2=1\). Since \(w\) is central and \(\gamma\)
anticommutes with every \(c(e_j)\), \(h\) anticommutes with all
\(c(e_j)\). It also anticommutes with \(\gamma\), so it is odd. These
are precisely the relations for the extra generator. \(\square\)

Consequently (7.3) induces a well-defined homomorphism \[
\mathrm{ABS}_n:Q_n\longrightarrow K_0(C_0(\mathbb R^n)).
\tag{7.6}
\] For \(n=2\) the positive-to-negative block of \(x\sigma_1+y\sigma_2\)
is multiplication by \(x+iy\). Denote its class by \(\beta_2\). In
dimension \(2k\), (5.5) gives the corresponding spinor symbol
\(\beta_{2k}\).

The remaining \textbf{complex Atiyah--Bott--Shapiro theorem} to prove is
that (7.6) is an isomorphism for every \(n\). With our orientation
convention, \([S_k]\) maps to \(\beta_{2k}\). Its asserted conclusion is
that the target is \(\mathbb Z\) in even dimensions, generated by this
symbol, and zero in odd dimensions. The multiplicative formulation
identifies \(\bigoplus_{n\geq0}Q_n\) with \(\mathbb Z[t]\),
\(\deg t=2\), and identifies \(t^k\) with \(\beta_{2k}\); products on
the target use exterior products and the ordered decomposition of
Euclidean space.

The representation calculation and the vanishing lemma have been proved
here. The assertion that these symbols generate all compactly supported
K-theory classes, and the stated compatibility of products, are the
topological content of Bott periodicity. Their full proof remains an
obligation in \emph{Bott periodicity in KK: the Bott and Dirac
elements}. That proof must establish that the oriented spinor symbol
corresponds to the Bott element and that its product with the Dirac
element is the identity, without using (7.6) as a premise. The algebraic
calculation in this lesson alone is not a proof of surjectivity in
(7.6).

\subsection{8. Exercises}\label{exercises}

\textbf{8.1. Commutator signs.} Expand the graded Leibniz and Jacobi
identities in (1.3). Also show that for homogeneous \(x,y\), \[
\left[x,y\right]_{\mathrm{gr}}^*
=-(-1)^{|x||y|}[x^*,y^*]_{\mathrm{gr}}.
\] What do these formulas say when both entries are odd and
self-adjoint?

\textbf{8.2. Two odd generators.} Construct explicitly the isomorphism
\(C_1\widehat\otimes C_1\to M_2\), including the involution of the
product of the two generators and its grading.

\textbf{8.3. Dimension three.} Give explicit representations of \(C_2\)
and \(C_3\). Determine the central projections of \(C_3\), its grading
in those coordinates, and a coordinate change making it pure swap.

\textbf{8.4. Spinors and chirality.} For an oriented Euclidean space
\(V\) of dimension \(2k\), construct a graded space \(S\) of dimension
\(2^k\) and prove \(\mathrm{Cl}(V)\cong\operatorname{End}(S)\). Compute
the chirality element with the self-adjoint convention (5.1), and
describe the effect of reversing orientation. Include the case \(k=0\).

\subsection{9. Solutions}\label{solutions}

\textbf{Solution to 8.1.} Write \(p=|x|,q=|y|,r=|z|\). The Leibniz right
side expands as \[
xyz-(-1)^{pq}yxz
+(-1)^{pq}yxz-(-1)^{pq+pr}yzx,
\] which is the left side. The three Jacobi expansions in the proof of
Proposition 1.1 exhibit all six word cancellations, including their
coefficients; no ungraded skew-symmetry was used. Finally \[
\begin{aligned}
\left[x,y\right]_{\mathrm{gr}}^*
 &=y^*x^*-(-1)^{pq}x^*y^*\\
 &=-(-1)^{pq}
   (x^*y^*-(-1)^{pq}y^*x^*).
\end{aligned}
\] For odd self-adjoint \(x,y\), the bracket is the self-adjoint
anticommutator \(xy+yx\); for \(y=x\) it is \(2x^2\geq0\). The Jacobi
and Leibniz signs remain those in (1.3), with \(p=q=1\) where
appropriate.

\textbf{Solution to 8.2.} Put \(a=\varepsilon\widehat\otimes1\) and
\(b=1\widehat\otimes\varepsilon\). They are odd self-adjoint unitaries
and \(ab=-ba\). The four basis vectors \(1,a,b,ab\) map respectively to
\[
1,\quad \sigma_1,\quad\sigma_2,\quad i\sigma_3.
\] They are linearly independent matrices, so the homomorphism is
bijective. The tensor involution gives \((ab)^*=-ab\), agreeing with
\((i\sigma_3)^*=-i\sigma_3\). The even span is that of \(1,ab\), which
maps to diagonal matrices; the odd span is that of \(a,b\), which maps
to off-diagonal matrices. Conjugation by \(\sigma_3\) is therefore the
grading. Finite-dimensionality ensures the same C*-isomorphism for both
completions.

\textbf{Solution to 8.3.} For \(C_2\), use
\(e_1=\sigma_1,e_2=\sigma_2\), with grading
\(\operatorname{Ad}\sigma_3\). They generate \(M_2\), and its
four-dimensional basis exhausts (5.2). For \(C_3\), send \[
\begin{gathered}
e_1\mapsto(\sigma_1,\sigma_1),\\
e_2\mapsto(\sigma_2,\sigma_2),\\
e_3\mapsto(\sigma_3,-\sigma_3).
\end{gathered}
\] The central self-adjoint unitary \(w=-ie_1e_2e_3\) maps to
\((1,-1)\), so its two spectral projections map to \((1,0)\) and
\((0,1)\). Each summand contains a full \(M_2\); this gives an
eight-dimensional faithful algebra, as required. The grading is \[
\alpha(X,Y)=(\sigma_3Y\sigma_3,\sigma_3X\sigma_3).
\] This sends each displayed generator to its negative. Under
\(\Psi(X,Y)=(X,\sigma_3Y\sigma_3)\), one has
\(\Psi\alpha\Psi^{-1}(X,Y)=(Y,X)\).

\textbf{Solution to 8.4.} Choose a positively oriented orthonormal basis
and put \(S=(\mathbb C^2)^{\otimes k}\). Define its generators by (5.5).
The anticommutation calculation there verifies (5.1); equations (5.6)
then produce each isolated Pauli matrix, so the representation is onto
\(\operatorname{End}(S)\). Its dimension equals the normal-form bound
\(2^{2k}\), proving injectivity. Multiplying each adjacent pair in (5.5)
gives \(i\) times the isolated \(\sigma_3\), so \[
(-i)^kc_1\cdots c_{2k}=\sigma_3^{\otimes k}.
\] This self-adjoint unitary anticommutes with every generator and
implements the grading. A negatively oriented orthonormal basis has top
exterior product of the opposite sign, so chirality reverses, exchanging
\(S^+\) and \(S^-\). For \(k=0\), \(S=\mathbb C\), the empty product is
\(1\), \(C_0=\mathbb C=\operatorname{End}(S)\), and the chosen grading
is entirely even.

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

Lemmas 3.0a and 6.0a prove the ordinary tensor norms and interior
construction, including independence, positivity, adjointability, and
rebracketing. The graded tensor norm, Clifford structure,
arbitrary-coefficient graded stabilization, representation quotient, and
symbol-vanishing statements have their proofs here. The dense-range
polar step in (6.6) uses Lemma 2.1 of this course's earlier lesson
\href{KT-KK-02.html}{Ext groups, absorption and Brown--Douglas--Fillmore
theory}. The earlier Hilbert-module and C*-algebra proofs are identified
in the introduction. The relative symbol group in Section 7 is
identified with operator K-theory by the earlier
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-difference}{Relative
difference bundles, radial pairs and the planar normalization, Theorems
RK.3--RK.4 and Corollary RK.4a}.

The full Bott-periodicity and Atiyah--Bott--Shapiro isomorphism remains
an explicit proof obligation in this course's \emph{Bott periodicity in
KK: the Bott and Dirac elements}. Its required generality, oriented
spinor generator, and product compatibility are stated after (7.6).
Neither the symbol map nor the representation quotient calculation uses
that theorem. No spin-c existence or classification theorem is invoked.

\subsection{References}\label{references}

\begin{itemize}
\item
  B. Blackadar, \emph{K-Theory for Operator Algebras}, second edition,
  1998.
  \href{https://www.bruceblackadar.com/Mathematics/book6.pdf}{Actual
  freely readable author text}, Sections 13.5 and 14.1--14.6.1. The
  general interior action here has codomain \(\mathcal L(F)\), and the
  stabilization proof uses coefficient-module maps when there is no
  unit.
\item
  B. Blackadar, \emph{Operator Algebras: Theory of C*-Algebras and von
  Neumann Algebras}, corrected author edition dated February 8, 2017.
  \href{https://www.bruceblackadar.com/Mathematics/Cycr.pdf}{Actual
  freely readable edition}, Section II.9.1. The
  representation-independent spatial norm has a complete proof in Lemma
  3.0a.
\item
  \href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-difference}{Relative
  difference bundles, radial pairs and the planar normalization},
  Theorems RK.3--RK.4 and Corollary RK.4a, for the complete earlier
  programme proof of relative difference bundles, radial compact support
  and symbol vanishing. The free Hatcher and Blackadar sources are
  identified there.
\item
  \emph{Kasparov's KK-theory}, \href{KT-KK-02.html}{Ext groups,
  absorption and Brown--Douglas--Fillmore theory}, Lemma 2.1, for the
  earlier operator-range proof used in graded stabilization.
\end{itemize}

\end{document}