# Bott periodicity *Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI. Public domain (CC0).* A projection gives a loop by rotating its range once and leaving its complement fixed. Bott periodicity says that every stable invertible loop comes from a unique difference of projection classes. We prove this for every complex C*-algebra, including nonunital, nonseparable algebras, by a concrete homotopy in the Toeplitz algebra. We use [Lesson 9](KT-OPK-09.md), especially its Toeplitz extension and character \(\chi(T_f+k)=f(1)\). We also import compact stability from [Lesson 5, Theorem 4.1](KT-OPK-05.md#4-compact-operator-stabilization) and [Lesson 6, Corollary 4.2](KT-OPK-06.md#4-realizing-an-invertible-path-at-a-late-stage), split exactness and the index formula from [Lesson 7, Corollary 4.2 and Proposition 5.1](KT-OPK-07.md), and the natural suspension isomorphism from [Lesson 8, Theorem 2.1](KT-OPK-08.md#2-the-idempotent-loop-of-an-invertible). These results precede periodicity. Tensor products below are minimal C*-tensor products. ## 1. A projection makes a based loop Write \(SA=C_0((0,1),A)\), and use \(z(t)=e^{2\pi it}\), with increasing \(t\) the positive circle direction. For a projection \(e\in M_n(D)\) in a unital algebra, put \[ f_e(t)=z(t)e+I_n-e. \tag{1.1} \] It is unitary, its inverse is \(\overline z e+I_n-e\), and its endpoint values are \(I_n\). Thus it represents an element of \(K_1(SD)\). **Proposition 1.1.** Formula (1.1) induces a natural homomorphism \[ \beta_D:K_0(D)\longrightarrow K_1(SD), \qquad [e]\longmapsto[f_e]. \tag{1.2} \] *Proof.* A projection homotopy gives a based loop homotopy by (1.1). Equivalent projections become homotopic after zero stabilization by [Lesson 1, §§3–4](KT-OPK-01.md); adding a zero projection adds an identity loop. Moreover \(f_{e\oplus h}=f_e\oplus f_h\). Hence the map on the projection monoid is well-defined and additive, and extends to its group completion. Applying a *-homomorphism entrywise to the nonconstant part of the loop proves naturality. This also works for a nonunital homomorphism: its extension to the suspension unitization sends the scalar identity to the target scalar identity. \(\square\) For nonunital \(A\), let \(\epsilon:A^+\to\mathbb C\) be the scalar quotient. Split exactness, also after suspension, identifies \[ \begin{aligned} K_0(A)&=\ker\epsilon_*,\\ K_1(SA)&=\ker(S\epsilon)_*. \end{aligned} \tag{1.3} \] Naturality of \(\beta_{A^+}\) therefore restricts it to \(\beta_A\). Explicitly, represent a relative class as \([e]-[P]\), where \(e\) is a projection over \(A^+\) and \(P=\epsilon(e)\) is its constant scalar projection. Then \[ \boxed{\beta_A([e]-[P])=[f_e f_P^{-1}].} \tag{1.4} \] The product has scalar part \(I\) for every \(t\) and endpoints \(I\), so belongs to the invertible matrices over \((SA)^+\). The equality with the restricted map follows from the product/block-sum identity in [Lesson 6, Theorem 2.2](KT-OPK-06.md#2-the-rotation-that-makes-addition-commute) and the injectivity in (1.3). This proves representative independence, rather than imposing it as an additional equivalence relation. Unital consistency follows from the same split diagram. We will prove that every \(\beta_A\) is an isomorphism. ## 2. Exact Toeplitz extensions with coefficients Let \(\mathcal T=C^*(S)\) on \(H^2\), \(p_0=1-SS^*\), \(\mathcal K=K(H^2)\), and \(\sigma:\mathcal T\to C(\mathbb T)\) its symbol map. Put \(\mathcal T_0=\ker\chi\), where \(\chi=\operatorname{ev}_1\sigma\). **Lemma 2.1 (universal isometry).** An isometry \(V\) in a unital C*-algebra determines a unique unital *-homomorphism \(\mathcal T\to C^*(V)\) sending \(S\) to \(V\). A corner isometry \(V^*V=p\), \(VV^*\leq p\), similarly determines a homomorphism with unit image \(p\). *Proof.* Faithfully represent the target algebra on a Hilbert space. For an isometry \(V\), let \(D=\ker V^*\). The spaces \(V^jD\) are pairwise orthogonal, and \[ I-V^NV^{*N}=\sum_{j=0}^{N-1}V^j(I-VV^*)V^{*j}. \tag{2.1} \] The increasing ranges on the right have closed union \(H_s=\bigoplus_{j\geq0}V^jD\); their orthogonal complement is \(H_u=\bigcap_{N\geq0}V^NH\). Both reduce \(V\). It acts as copies of \(S\) on \(H_s\), and as a unitary on \(H_u\): on the latter \(VV^*=I\), and \(V^*\) preserves every \(V^NH\). For any *-polynomial \(r\), the norm on \(H_s\) is at most \(\|r(S)\|\); on \(H_u\), continuous functional calculus gives the bound \(\|r(z)\|_\infty\leq\|r(S)\|\), using the symbol quotient of Lesson 9. Thus evaluation at \(V\) is well-defined and contractive on the dense *-polynomial algebra, and extends to the required homomorphism. Uniqueness follows from the generator. Apply this argument in the corner \(pEp\) for the last assertion; the zero corner gives the zero map. \(\square\) **Lemma 2.2 (coefficient exactness).** For every C*-algebra \(B\), the sequences \[ \begin{aligned} 0&\longrightarrow B\otimes\mathcal K \longrightarrow B\otimes\mathcal T\\ &\xrightarrow{\mathrm{id}\otimes\sigma} C(\mathbb T,B) \longrightarrow0,\\ 0&\longrightarrow B\otimes\mathcal K \longrightarrow B\otimes\mathcal T_0\\ &\xrightarrow{\mathrm{id}\otimes\sigma} SB \longrightarrow0 \end{aligned} \tag{2.2} \] are exact. Here \(SB\) is identified with the functions on the circle vanishing at \(1\). *Proof.* Faithfully represent \(B\) on \(H_B\). For \(F\in C(\mathbb T,B)\), multiply on \(L^2(\mathbb T)\otimes H_B\) and compress to \(H^2\otimes H_B\). This gives a contractive linear map \(s_B\). For a finite \(B\)-valued trigonometric polynomial, its compression is a finite sum \(\sum T_{z^j}\otimes b_j\), with tensor factors flipped if necessary, and its symbol is \(F\). Such polynomials are uniformly dense: a finite partition of unity approximates a continuous Banach-valued function by finite scalar functions times values in \(B\), and scalar trigonometric approximation approximates the scalar functions. Consequently \(s_B\) takes values in \(B\otimes\mathcal T\) and is a right inverse to the symbol map. For an algebraic tensor \(x=\sum b_j\otimes a_j\), each \(a_j-T_{\sigma(a_j)}\) is compact by Lesson 9. Hence \(x-s_B\sigma_B(x)\in B\odot\mathcal K\). If \(\sigma_B(x)=0\) for a general completed tensor, approximate \(x\) by algebraic tensors \(x_m\). Boundedness of \(I-s_B\sigma_B\) shows \(x_m-s_B\sigma_B(x_m)\to x\), proving the exact kernel assertion. Minimal tensor products preserve inclusions, so the indicated kernel is the stated completed tensor algebra. The character extension \(0\to\mathcal T_0\to\mathcal T\xrightarrow\chi\mathbb C\to0\) has *-section \(j(\lambda)=\lambda1\). A *-split extension remains exact after tensoring with any \(B\): for its quotient \(q\) and section \(s\), \(\mathrm{id}_B\otimes sq\) is a bounded *-homomorphism. Applying \(I-\mathrm{id}_B\otimes sq\) to algebraic approximations proves the kernel equality just as above. It follows that the tensors with zero character are precisely \(B\otimes\mathcal T_0\). For a symbol \(F\) with \(F(1)=0\), its compression already has zero character. Restrict the first sequence to obtain the second. No exactness hypothesis on \(B\) has been used. \(\square\) The bounded linear compression section need not be multiplicative. The proof uses its boundedness to identify the kernel; it does not turn the Toeplitz extension into a *-split extension. ## 3. The homotopy that removes the reduced Toeplitz algebra **Theorem 3.1.** For every C*-algebra \(B\), the maps \[ j_B:B\to B\otimes\mathcal T,\quad b\mapsto b\otimes1, \qquad \chi_B=\mathrm{id}_B\otimes\chi \] induce mutually inverse isomorphisms on \(K_0\) and \(K_1\). In particular, \[ K_i(B\otimes\mathcal T_0)=0\quad(i=0,1). \tag{3.1} \] *Proof.* We first construct the homotopy before introducing \(B\). Inside \(\mathcal T\otimes\mathcal T\), define \[ \widehat{\mathcal T} =C^*(\mathcal K\otimes\mathcal T,\mathcal T\otimes1), \qquad v=S\otimes1. \tag{3.2} \] Apply Lemma 2.2 with coefficient algebra \(\mathcal T\), then flip the factors. The restriction of \(\sigma\otimes\mathrm{id}\) to (3.2) has kernel \(\mathcal K\otimes\mathcal T\) and image \(C(\mathbb T)\otimes1\). Denote the resulting quotient by \(\widehat\sigma\), so \(\widehat\sigma(v)=z\). Its quotient is \(C(\mathbb T)\). Form the pullback \[ \mathcal P= \{(x,y)\in\widehat{\mathcal T}\oplus\mathcal T: \widehat\sigma(x)=\sigma(y)\}. \tag{3.3} \] Projection onto \(y\) gives a *-split extension with kernel \(\mathcal K\otimes\mathcal T\) and section \(y\mapsto(y\otimes1,y)\). Set \[ \begin{aligned} f&=p_0\otimes1,& w&=p_0\otimes S,& g&=p_0\otimes p_0,\\ h&=vfv^*,& a&=v(1-f)v^*. \end{aligned} \tag{3.4} \] The projections \(a,h,f\) are orthogonal and sum to \(1\). Also \(w^*w=f\), \(ww^*=f-g\). The partial isometry \(wv^*\) exchanges \(h\) with \(f-g\), and \(fv^*\) exchanges \(h\) with \(f\). Therefore \[ \begin{aligned} Z_0&=a+wv^*+vw^*+g,\\ Z_1&=a+fv^*+vf \end{aligned} \tag{3.5} \] are self-adjoint unitaries. Indeed each exchanged pair contributes its two support projections to the square, all cross-products with the complementary identity projections are zero, and those supports exhaust \(1\). Their quotient symbols are both \(1\). Let \(E_j=(1-Z_j)/2\), and put \[ Z_t=e^{i\pi tE_1}e^{i\pi(1-t)E_0}, \qquad v_t=Z_tv\quad(0\leq t\leq1). \tag{3.6} \] These are norm-continuous unitaries and isometries, respectively; \(Z_t\) has quotient \(1\), and its endpoints are \(Z_0,Z_1\). Lemma 2.1 defines a homotopy \(\phi_t:\mathcal T\to\widehat{\mathcal T}\), \(\phi_t(S)=v_t\). It is pointwise norm-continuous because this is true on *-polynomials and all these homomorphisms are contractive. Multiplication of (3.5) by \(v\) gives \[ \begin{aligned} \phi_0(S)&=v(1-f)+w,\\ \phi_1(S)&=v(1-f)+f. \end{aligned} \tag{3.7} \] In (3.3), define \(\psi_t(S)=(v_t,S)\). Also define the two corner homomorphisms \[ \begin{aligned} \psi(S)&=(v(1-f),S),&\psi(1)&=(1-f,1),\\ \omega(S)&=(w,0),&\omega(1)&=(f,0). \end{aligned} \tag{3.8} \] The displayed generators are isometries in their indicated corners. Their quotient symbols satisfy the pullback condition. The two corners are orthogonal and their units sum to \((1,1)\). Formula (3.7) therefore implies equalities of homomorphisms \[ \psi_0=\psi+\omega, \qquad \psi_1=\psi+\omega j\chi. \tag{3.9} \] For orthogonal-range homomorphisms \(\rho,\eta:D\to E\), one has \((\rho+\eta)_*=\rho_*+\eta_*\) on either K-group. To see this even for nonunital algebras, factor the sum through \(D\xrightarrow{\Delta}D\oplus D\xrightarrow{\rho\oplus\eta}E\). Direct-sum K-theory identifies \(\Delta_*x=(x,x)\); the last induced map is the sum of its two restrictions. These are the direct-sum statements of Lessons 4 and 6. Now tensor all these maps with \(B\). Their homotopies remain pointwise norm-continuous, by density of finite tensors and contractivity. The map \(\omega_B\) factors as \[ B\otimes\mathcal T \xrightarrow{\ c\ }B\otimes\mathcal K\otimes\mathcal T \xrightarrow{\ \iota\ }B\otimes\mathcal P, \qquad c(b\otimes y)=b\otimes p_0\otimes y. \tag{3.10} \] Compact stability makes \(c_*\) an isomorphism. The *-split pullback extension stays exact after tensoring by Lemma 2.2's split-extension argument, and split K-exactness makes \(\iota_*\) injective. Thus \((\omega_B)_*\) is injective. Homotopy invariance and (3.9) give \[ (\psi_B)_*+(\omega_B)_* =(\psi_B)_*+(\omega_B)_*(j_B\chi_B)_*. \] Subtract the common summand and cancel this injective map. It follows that \((j_B\chi_B)_*=\mathrm{id}\). The reverse composite \(\chi_Bj_B\) is literally \(\mathrm{id}_B\). Finally the split character extension identifies \(K_i(B\otimes\mathcal T)\) with \(K_i(B\otimes\mathcal T_0)\oplus K_i(B)\). Since its section already induces an isomorphism, the first summand is zero. \(\square\) The injectivity in (3.10) uses both compact stability and the split ideal inclusion. Split exactness alone would not make the corner map an isomorphism. In (3.7), the final summand is \(f\): replacing it by \(g\) gives initial projection \(1-f+g\), so fails the isometry equation. These checks also fix the pullback (3.3) over the actual quotient circle algebra. ## 4. The boundary proves periodicity and fixes its sign **Theorem 4.1 (Bott periodicity).** The map \(\beta_A:K_0(A)\to K_1(SA)\) is a natural isomorphism for every complex C*-algebra \(A\). Under compact stability, the index boundary of the second extension in (2.2) satisfies \[ \boxed{\partial_A\beta_A=-\mathrm{id}_{K_0(A)}.} \tag{4.1} \] *Proof.* The exact segment from Lesson 7 for the second extension in (2.2) is \[ \begin{aligned} K_1(B\otimes\mathcal T_0) &\longrightarrow K_1(SB)\\ &\xrightarrow{\partial_B}K_0(B\otimes\mathcal K)\\ &\longrightarrow K_0(B\otimes\mathcal T_0). \end{aligned} \tag{4.2} \] The outside groups vanish by Theorem 3.1; compact stability identifies the middle target with \(K_0(B)\). Thus \(\partial_B\) is an isomorphism for every \(B\). For unital \(B\) and a projection \(e\in M_n(B)\), lift \(f_e\) to \[ V_e=S\otimes e+1\otimes(I_n-e). \tag{4.3} \] Its character is \(I_n\), so \(V_e-I_n\) belongs to \(M_n(B\otimes\mathcal T_0)\), after the tensor flip. Direct multiplication gives \[ V_e^*V_e=I_n, \qquad I_n-V_eV_e^*=p_0\otimes e. \tag{4.4} \] The partial-isometry boundary is initial defect minus final defect, by Lesson 7, Proposition 5.1. Hence \(\partial_B[f_e]=0-[p_0\otimes e]=-[e]\). This already gives \(\partial_\mathbb C[z]=-1\), the shift convention of Lesson 9. Additivity proves (4.1) on \(K_0(B)\). Since \(\partial_B\) is an isomorphism, \(\beta_B=-\partial_B^{-1}\) is one too. For nonunital \(A\), use the two split kernel identifications (1.3). The natural maps \(\beta_{A^+}\) and \(\beta_\mathbb C\) are isomorphisms, so their commuting diagram restricts to an isomorphism of kernels: injectivity is inherited; for surjectivity, an element of the lower kernel has a unique preimage upstairs, whose scalar image must be zero by injectivity of \(\beta_\mathbb C\). Naturality of the coefficient Toeplitz boundaries, followed by the injective map \(K_0(A)\to K_0(A^+)\), proves (4.1) for \(A\). There is no need to assume that a product of two relative partial-isometry lifts is itself a partial isometry. \(\square\) ## 5. The polynomial-loop route: an outline with explicit reductions Here is the finite-dimensional algebra behind the alternative polynomial proof. The reductions work over any unital complex Banach algebra \(D\); for the present course \(D\) is a C*-algebra. Reduction from a nonunital algebra is the split kernel diagram (1.3). First, a based continuous invertible loop \(u:\mathbb T\to GL_n(D)\) is based-homotopic to a Laurent polynomial loop. To verify the analytic step, convolve with the Fejér kernel \[ F_N(\theta)=\frac1{N+1} \left|\sum_{j=0}^N e^{ij\theta}\right|^2. \] It is nonnegative, has normalized integral one, and its integral outside any fixed arc about zero tends to zero, by the geometric-sum bound \(F_N(\theta)\leq((N+1)\sin^2(\theta/2))^{-1}\). Uniform continuity gives uniform convergence of these Banach-valued convolutions to \(u\). If \(M=\sup_z\|u(z)^{-1}\|\), choose a polynomial \(g\) with \(\|g-u\|_\infty