Compact placement and support homotopy

Lemma 4.3 (stable placement of compact homomorphisms). For a homomorphism $\theta:D\to\mathcal K(H_B)$, adjoining a zero standard-module summand does not change its ordinary homotopy class after folding the modules. Neither does conjugating it by an arbitrary module unitary of $H_B$.

Proof. Work first with one fixed folding. Identify the Hilbert-space factor of $H_B$ with $L^2(\mathbb R_+)$, and let $V_t$ be right translation by $t$, with zero extension. For a compactly supported continuous vector, translation and its adjoint vary continuously in $L^2$; density and their norm bounds give this for every vector. Thus $V_t,V_t^*$ are strongly continuous, and $V_t^*V_t=1$.

For an elementary compact module operator $\theta_{f\otimes b,g\otimes c}$, conjugation gives $\theta_{V_tf\otimes b,V_tg\otimes c}$. The rank-one norm bound makes this norm continuous. Finite sums of these operators are norm dense in $\mathcal K(H_B)$, and conjugation is contractive. Hence $t\longmapsto (V_t\otimes1)k(V_t^*\otimes1) \tag{4.6}$ is norm continuous for every compact $k$. Applying it to $\theta(d)$ gives a pointwise norm-continuous homotopy of homomorphisms, from the original placement to a proper infinite corner at $t=1$.

The range $L^2([1,\infty))$ and its complement $L^2([0,1))$ are both infinite-dimensional separable Hilbert spaces. Choose a unitary between them, and let $W=W^*=W^{-1}$ exchange them through that unitary and its adjoint. With $V_1$ as above, put $V_2=WV_1$. Their ranges are orthogonal and sum to the whole Hilbert space. The norm-continuous unitary path $\exp(i\pi s(1-W)/2)$ joins $1$ to $W$. Following the translation homotopy by conjugation with this path shows that placement through either $V_1$ or $V_2$ is homotopic to the original map. Consequently the fixed folding $(x,y)\mapsto (V_1\otimes1)x+(V_2\otimes1)y$ makes adjoining a zero summand harmless in either corner.

Now let $U$ be any adjointable module unitary. The column maps $R_sx=(\cos s\,x,\ \sin s\,Ux),\qquad 0\leq s\leq\pi/2, \tag{4.7}$ are norm-continuous adjointable isometries, since $R_s^*R_s=1$. Thus $d\mapsto R_s\theta(d)R_s^*$ is a homomorphism into $\mathcal K(H_B\oplus H_B)$, norm continuous on each $d$. Its endpoints are the first-corner original map and the second-corner map conjugated by $U$. Folding by the one fixed folding, and using its two corner homotopies, proves the claimed conjugation invariance. Any other folding differs from the fixed one by a module unitary, so the assertion holds for every folding. No norm-connectedness theorem for the module unitary group is used. $\square$

Lemma 4.4 (support and compact homotopy). Suppose $D$ is $\sigma$-unital and $\theta:D\to\mathcal K(E)$, with $E$ countably generated. Then $X=\overline{\theta(D)E} \tag{4.8}$ is countably generated, and $\theta(D)$ restricts to $\mathcal K(X)$. After standard stabilization, the original compact homomorphism and this restricted one, extended by zero, are ordinarily homotopic.

Proof. Choose a positive sequential approximate identity $u_n$ of $D$, and homogeneous generators $x_j$ of $E$ when grading is present. The countable family $\theta(u_n)x_j$ generates $X$: first approximate $\theta(d)x$ by $\theta(u_n)\theta(d)x$, then approximate $\theta(d)x$ by finite $B$-linear combinations of the original generators. This proves countable generation, including when the inclusion $X\subset E$ has no adjoint.

We spell out the norm used for compact inclusion. For vectors $\xi_i,\eta_i\in X$, define the finite column operators $C,D:B^r\to X$, or into $E$, by those same columns. Both have adjoints given by the inner products with their columns. The finite operator $T=CD^*$ has $\|T\|^2=\|(C^*C)^{1/2}(D^*D)(C^*C)^{1/2}\|.$ Indeed $T^*T=D(C^*C)D^*$, and $\|ZZ^*\|=\|Z^*Z\|$ for $Z=D(C^*C)^{1/2}$. The two Gram matrices are identical in $X$ and $E$. Therefore the rank-one inclusion extends isometrically to $\mathcal K(X)\to\mathcal K(E)$. This argument uses no projection onto $X$.

For $d\in D$, approximate identities give $\theta(u_n)\theta(d)\theta(u_n)\longrightarrow\theta(d) \quad\text{in norm}. \tag{4.9}$ For fixed $n$, approximate the middle compact operator on $E$ by finite rank sums. Sandwiching places both rank-one vectors in $X$, since $\theta(u_n)$ is self-adjoint and has range in $X$. The shared norm formula proves compactness of the restriction to $X$; norm convergence in (4.9) then proves it for $\theta(d)$. It also gives nondegeneracy of the restricted action, because $\theta(u_n)\xi\to\xi$ on the dense span $\theta(D)E$ and hence on all of $X$.

Consider the Hilbert $C([0,1],B)$-module $\mathcal M=\{f\in C([0,1],E):f(1)\in X\}. \tag{4.10}$ Use the pointwise inner product. This is a closed module and hence complete. It is generated by the constant sections of the countable family just obtained for $X$, together with $(1-t)x_j$. Here is the density check. Approximate the endpoint value of a section by a finite $B$-linear combination of the constant sections. Subtract it. Continuity makes the remaining section uniformly small near $1$, so cutoff changes it by an arbitrarily small norm. A section vanishing near $1$ is uniformly approximated by finite sums of original generating vectors with continuous $B$-valued coefficients; divide those coefficients by $1-t$ on their common support and extend by zero. Thus these sums lie in the span of $(1-t)x_j$. A scalar partition of unity supplies the finite uniform approximation from the pointwise generator property.

Evaluation at $t<1$ has fibre $E$, using a scalar cutoff supported away from $1$; at $1$ its fibre is $X$, using constant sections. In either case its kernel is the closure of the module times coefficient functions vanishing at that point. This follows by cutting off a section whose value is zero there, then using uniform continuity. Consequently these are the actual quotient fibres, not just dense evaluation ranges.

Constant application of $\theta(d)$ and of its adjoint preserves $\mathcal M$. It is compact there: (4.9), followed by finite rank approximation of its middle factor, approximates it uniformly by rank-one operators whose vectors are constant sections in $X$. The error norm on $\mathcal M$ is at most the corresponding error on $E$. Thus $d\mapsto\theta(d)|_{\mathcal M}$ is a homomorphism into $\mathcal K(\mathcal M)$, with the required original and support endpoint actions.

Apply the earlier graded stabilization theorem, Lesson 05 Theorem 6.2, over $C([0,1],B)$: $\mathcal M\oplus H_{C([0,1],B)}\cong H_{C([0,1],B)}$, with the appropriate grading copies when needed. Conjugate the compact action plus zero by this module unitary. The standard compact algebra is $C([0,1],\mathcal K(H_B))$: finite rank fields give one inclusion, and a finite interval partition and uniform finite rank approximation give the other. We have therefore obtained an ordinary pointwise norm-continuous homotopy of homomorphisms into $\mathcal K(H_B)$. Its endpoints are exactly the two stabilized actions. Their endpoint unitary choices and zero summands are harmless by Lemma 4.3. This proves the assertion. $\square$

Earlier proofs and source credit

This independent argument uses the ordinary module receiving proofs, MF.3–MF.7, and the full compact-module and graded stabilization proofs in Lesson 05, Lemma 2.0 and Theorem 6.2. Their actual free author sources are credited there. The translation/folding paths, Gram inclusion without a complement, interval quotient fibres and uniformly compact support action are proved above; these are receiving versions of the written Lemmas 4.3–4.4 of the picture lesson. Original exposition by GPT-6.1 Sol (OpenAI), Codex, Ultra, October 2026; CC0. No broader picture theorem is claimed here.