Crossed products by integers and real numbers: Fourier coordinates and mapping tori

Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI. Public domain (CC0).

A discrete action has an implementing unitary inside a unital crossed product. A flow has a unitary group in the multiplier algebra, whose Fourier functional calculus supplies the corresponding generators. When a flow is periodic, its real crossed product can be unfolded into a family of compact-group crossed products with twisted endpoints. We will construct that family, prove it fills the mapping torus, and use translation on the real line to prepare the Wiener–Hopf extension.

We use the multiplier universal property and exactness in Lesson 1, regular representations and absorption in Lesson 2, and amenability in Lesson 4. Thus full and reduced completions agree for all groups occurring here. The coefficient algebra may be nonunital and nonseparable.

An automorphism and its gauge action

Let \(\alpha\in\operatorname{Aut}(A)\), regarded as the integer action \(n\mapsto\alpha^n\).

Proposition 5.1. If \(A\) is unital, \(A\rtimes_\alpha\mathbb Z\) is the universal unital \(C^*\)-algebra generated by a copy of \(A\) and a unitary \(u\) satisfying

Proof. The canonical multipliers of Lesson 1 lie in the algebra because \(A\) is unital and \(\mathbb Z\) is discrete. Finite sums \(\sum a_nu^n\) are dense. Conversely, a unital representation of \(A\) and a unitary satisfying (1) give a covariant representation by \(n\mapsto u^n\). The universal property of the full crossed product extends the finite-sum formula uniquely. It also applies with a target unital \(C^*\)-algebra, so gives exactly the asserted generator-and-relation property. \(\square\)

For nonunital \(A\), the same formula uses a nondegenerate copy of \(A\) and a unitary in the multiplier algebra; \(u\) itself need not belong to the crossed product. The products \(au^n\) still do.

For \(z\in\mathbb T\), universality gives the gauge automorphism \(\gamma_z(a)=a\), \(\gamma_z(u)=zu\). Its action is norm continuous on finite sums, hence on the completion by density and isometry. With normalized Haar measure define

On polynomials, \(P_n(\sum a_ku^k)=a_nu^n\). The subspace \(Au^n\) is closed because \(a\mapsto au^n\) is isometric. Thus the \(n\)-th spectral subspace is exactly \(Au^n\): if \(\gamma_z(x)=z^nx\), then \(x=P_n(x)\), while every \(au^n\) has that transformation law. In particular the fixed algebra is \(A\).

The average \(P_0\) is completely positive and faithful. Indeed, if \(x\geq0\) and \(P_0(x)=0\), then for every state \(\omega\) the nonnegative continuous function \(z\mapsto\omega(\gamma_z(x))\) has zero integral. Haar measure has full support, so its value at \(z=1\) is zero. States detect positive elements, giving \(x=0\). This proof works without a coefficient unit.

This supplies another proof of full-to-reduced equality for \(\mathbb Z\). The quotient is equivariant for the two gauge actions and injective on \(A\). If \(q(x)=0\), then \(q(P_0(x^*x))=0\), so \(P_0(x^*x)=0\), and faithfulness gives \(x=0\).

The Fourier coefficients need not give a norm-convergent ordinary series. Their Fejér sums do:

Here \(K_N(e^{it})=(N+1)^{-1}|\sum_{j=0}^N e^{ijt}|^2\) is nonnegative with integral one. Away from an arc about \(1\), its integral tends to zero, by the bound \((N+1)^{-1}/\sin^2(t/2)\). On that arc, norm continuity makes \(\|\gamma_z(x)-x\|\) small. Splitting the integral proves (3).

For the rotation action \(\alpha(z)=e^{-2\pi i\theta}z\) on \(C(\mathbb T)\), (1) becomes

This fixes the convention for the rotation algebra. Changing the covariance convention to \(u^*au=\alpha(a)\) replaces the automorphism by its inverse.

Fourier functional calculus for a flow

We fix the positive-sign transform

The real Plancherel and inversion theorems are supplied within the programme by The Plancherel theorem, Theorem 1.1, Proposition 2.1 and Corollary 4.1 and The Pontryagin duality theorem, Theorem 2.1, Proposition 3.1 and Theorem 4.1. Both lessons have been written. Their negative Fourier sign becomes ours by replacing \(\xi\) by \(-\xi\). Lebesgue normalization is proved in The Fourier inversion theorem and the dual Haar measure, Theorem 2.1 and Propositions 3.1–3.3. Emerson, Theorem 1.9.9 and Corollary 1.9.15, give the corresponding literature references.

Proposition 5.2. The canonical Fourier identifications are

Proof. Integer Fourier series identify \(\ell^2(\mathbb Z)\) unitarily with \(L^2(\mathbb T)\) by \(\delta_n\mapsto z^n\). The bilateral shift becomes multiplication by \(z\). Laurent polynomials are uniformly dense in \(C(\mathbb T)\), by Stone–Weierstrass. Amenability identifies the regular and full norms.

For \(\mathbb R\), Plancherel conjugates \(\lambda(g)\) to multiplication by \(\mathcal F_+g\), first on smooth compactly supported functions and then on \(L^1\) by the bound \(\|\lambda(g)\|\leq\|g\|_1\). For \(g\in C_c(\mathbb R)\), its transform is continuous and vanishes at infinity: approximate it in \(L^1\) by smooth compactly supported functions and apply integration by parts to those functions. Multiplication has norm \(\|\mathcal F_+g\|_\infty\), because the transform is continuous and Lebesgue measure has full support.

The closure of these transforms is a self-adjoint subalgebra of \(C_0(\mathbb R)\), since convolution becomes multiplication and involution becomes complex conjugation. It is nonzero at any specified frequency: take \(g(t)=e^{-2\pi it\xi}h(t)\) with \(h\geq0\) a nonzero compactly supported bump. It separates two frequencies \(\xi\neq\eta\): choose \(t_0\) where \(e^{2\pi it_0\xi}\neq e^{2\pi it_0\eta}\), and take a nonnegative integral-one bump concentrated near \(t_0\). Its two transform values approach those distinct numbers as its support shrinks. The locally compact Stone–Weierstrass theorem makes the closure all of \(C_0(\mathbb R)\). Again amenability gives the full norm. \(\square\)

For a covariant pair \((\pi,U)\) of an \(\mathbb R\)-action, write

using Stone's theorem. The group-algebra multiplier map of Lesson 1 sends \(g\) to \(\mathcal F_+g(D)\) in this representation. The image of the crossed product under \(\pi\rtimes U\) is the closed span of \(\pi(a)\mathcal F_+g(D)\), \(a\in A\), \(g\in C_c(\mathbb R)\); equivalently one may use all functions of \(D\) from \(C_0(\mathbb R)\). This follows from the product-generation statement in Lesson 1 and (5). It identifies this span with the crossed product itself only when \(\pi\rtimes U\) is faithful. For example, \(A=\mathbb C\) and \(U_t=1\) give the image \(\mathbb C\), whereas the crossed product is \(C_0(\mathbb R)\). Neither \(D\) nor \(U_t\) is asserted to be a bounded element of the crossed product.

A dense smooth convolution algebra

For unital \(A\), the Euclidean version of the smooth convolution and density theorem is Frequency calculus for an action of Euclidean space, Proposition 1.1. We extend it here to nonunital coefficients and give explicit one-dimensional weighted estimates and the differentiated involution formula. These details will be used with this lesson's generator normalization.

Let \(\alpha:\mathbb R\to\operatorname{Aut}(A)\) be a flow. Set

For \(a_\phi=\int\phi(t)\alpha_t(a)\,dt\), \(\phi\in C_c^\infty(\mathbb R)\), differentiation after a change of variable gives \(\delta^k(a_\phi)=(-1)^k\int\phi^{(k)}(t)\alpha_t(a)\,dt\). Integral-one bumps shrinking to zero show that \(A^\infty\) is dense. Differentiating products and adjoints gives a *-derivation \(\delta\).

It is closed: for a smooth \(a\), \(\alpha_t(a)-a=\int_0^t\alpha_s(\delta a)\,ds\); if \(a_j\to a\) and \(\delta a_j\to b\), this identity passes to the limit and differentiation at zero gives \(\delta a=b\). It follows successively that the seminorms \(\sum_{j=0}^k\|\delta^ja\|\) make \(A^\infty\) complete. The action commutes with \(\delta\) and preserves each of these seminorms.

Define \(\mathcal S(\mathbb R,A^\infty)\) to consist of Fréchet-smooth functions with finite seminorms

The twisted operations are

These preserve the space. To check convolution, differentiate under the integral, apply the Leibniz rule for \(\delta^j\), and use \((1+|t|)^N\leq(1+|s|)^N(1+|t-s|)^N\). The result is bounded by a finite sum of

with the binomial factors included. These integrable bounds also justify all differentiations. For involution,

applying \(\delta^j\) and using its commutation with \(\alpha_t\) gives the needed seminorm estimates. Completeness follows from completeness of \(A^\infty\), convergence of derivatives on compact sets, and the seminorm bounds. Thus this is a Fréchet *-algebra.

It is dense in \(A\rtimes_\alpha\mathbb R\): finite sums of \(C_c^\infty(\mathbb R)\)-functions times elements of \(A^\infty\) approximate \(C_c(\mathbb R,A)\) in \(L^1\), by compact-image approximation, scalar smoothing, and the density just proved. The completion norm is bounded by the \(L^1\) norm.

There are two useful derivations on this smooth algebra:

The first is a derivation by the coefficient Leibniz rule and commutation with \(\alpha\). The second is a derivation because \(t=s+(t-s)\) in convolution, and it is the infinitesimal generator of the character action \(f(t)\mapsto e^{2\pi i\eta t}f(t)\). Both respect involution. Ordinary differentiation \(\partial_t\) is a seminorm operation in (7); it is not itself the asserted convolution derivation.

In a covariant representation, a smooth \(a\) preserves \(\operatorname{Dom}D\), and

Indeed, use \(U_t\pi(a)v=\pi(\alpha_ta)U_tv\) and differentiate the vector at \(t=0\). The differentiability of the right side shows domain preservation and gives (11). This identifies the coefficient generator with the implementing commutator, including its normalization.

A periodic flow becomes a mapping torus

For an automorphism \(\sigma\) of a \(C^*\)-algebra \(C\), define

Equivalently extend \(F\) continuously to \(\mathbb R\) by \(F(s+1)=\sigma(F(s))\). Its norm is the supremum on a fundamental interval.

Theorem 5.3. Suppose \(\beta:\mathbb R\to\operatorname{Aut}(B)\) is strongly continuous and \(\beta_1=\operatorname{id}\). Let \(C=B\rtimes_\beta\mathbb T\), where \(\mathbb T=\mathbb R/\mathbb Z\). Denote its canonical group multipliers by \(v_r\), with \(v_{r+1}=v_r\), and define the dual automorphism

Then \(B\rtimes_\beta\mathbb R\cong M_\sigma\).

Proof. The character twist \(r\mapsto e^{2\pi isr}v_r\), together with \(i_B\), is a nondegenerate covariant pair in \(M(C)\). For \(f\in C_c(\mathbb R,B)\) its integrated value belongs to \(C\), since it is the compact-group convolution function obtained by periodization:

The sum is locally finite, and relabeling \(n\) makes the expression in parentheses a continuous periodic \(B\)-valued function of \(t\). On \(s\in[0,1]\), it varies continuously in its \(L^1\) norm. Formula (14) obeys \(\Phi(f)(s+1)=\sigma(\Phi(f)(s))\), is contractive for the full norm, and preserves products and adjoints because each character-twisted pair is covariant. Thus it extends to a homomorphism into \(M_\sigma\).

To prove faithfulness and the norm, represent \(B\) faithfully and nondegenerately by \(\rho\) on \(H\). On \(L^2(\mathbb R,H)\), its regular real crossed product has

Apply integer Fourier series separately on each coset \(t+\mathbb Z\):

This notation means the \(L^2\) Fourier transform of the sequence; pointwise convergence is not asserted. Decomposition of Lebesgue measure into the intervals \(n+[0,1)\), followed by Fourier-series Plancherel, makes \(Z\) unitary onto \(L^2([0,1]^2,H)\).

If \(t-r=t'+k\), \(0\leq t'<1\), \(k\in\mathbb Z\), then \((Z\lambda_rZ^*\psi)(s,t)=e^{-2\pi iks}\psi(s,t')\). The coefficient operator is \(\rho(\beta_{-t}(b))\), since \(\beta_n=\operatorname{id}\). Conjugate once more by the unitary

Since \(t-t'-k=r\), the conjugated group operator is

For each \(s\), this is precisely the compact-group regular representation of (14). That representation of \(C\) is faithful by Lesson 4. Consequently,

The first equality uses faithfulness of the real regular representation, again by Lesson 4; the second uses norm continuity. Hence \(\Phi\) is isometric and its image \(D_0\) is closed.

Each evaluation \(D_0\to C\) is onto. At a specified \(s_0\), any continuous \(B\)-valued function \(g\) on \(\mathbb T\) can be split by a finite partition of unity into terms supported on arcs that lift to bounded intervals in \(\mathbb R\). On such a lift choose \(f(r)=e^{-2\pi is_0r}g([r])\), with that partition term understood. The lifted term vanishes at its interval endpoints, so extension by zero is in \(C_c(\mathbb R,B)\), and its periodization in (14) is exactly that term of \(g\). These compact convolution functions are dense in \(C\). Since evaluation on the closed \(C^*\)-algebra \(D_0\) has closed range, it is surjective.

The central multiplier \(u_1\) of \(B\rtimes\mathbb R\) acts on the family as \(e^{2\pi is}1\). It is central because \(\beta_1=\operatorname{id}\) and \(\mathbb R\) is abelian. Its continuous functional calculus therefore multiplies \(D_0\) by every continuous scalar function on the parameter circle. Given \(F\in M_\sigma\) and \(\varepsilon>0\), choose at every circle point a member of \(D_0\) agreeing with \(F\) in that fiber. Continuity makes the difference less than \(\varepsilon\) on a neighborhood, using the \(\sigma\)-twisted identification near the endpoint. A finite subordinate scalar partition of unity then produces a member of \(D_0\) within \(\varepsilon\) of \(F\) uniformly. Closedness gives \(D_0=M_\sigma\). \(\square\)

For the trivial action this becomes \(B\otimes C_0(\mathbb R)\cong M_\sigma\), with \(C\cong B\otimes c_0(\mathbb Z)\). In the Fourier coordinates \(\xi\in\mathbb R\), the fiber formula is \(h\mapsto(h(s+k))_{k\in\mathbb Z}\), and \(\sigma\) shifts that sequence to \((c_{k+1})_k\). Decay at infinity is uniform for \(s\in[0,1]\). This example displays how a noncompact spectrum is assembled from compact parameter space and nonunital fibers.

The hypothesis concerns period one. It does not hold for every flow on a compact space. For the Kronecker flow \(h_t(z,w)=(e^{2\pi it}z,e^{2\pi i\theta t}w)\) on \(\mathbb T^2\), irrational \(\theta\) makes \(h_1\) nontrivial. The action on functions is \(\alpha_t(f)=f\circ h_{-t}\), and \(\delta(z^mw^n)=-2\pi i(m+\theta n)z^mw^n\). The flow is free: a return requires \(t\in\mathbb Z\) and \(\theta t\in\mathbb Z\), hence \(t=0\). Its orbits are dense, since after fixing \(t=r\pmod1\), the second coordinate varies through the dense integer rotation. The relation of this flow to a rotation crossed product uses a transversal and Morita equivalence, rather than the periodic-flow hypothesis of Theorem 5.3.

Translation and the Wiener–Hopf extension

Proposition 5.4. For \((\tau_ta)(x)=a(x-t)\),

Proof. On \(L^2(\mathbb R)\), take multiplication \(M\) and translation \(\lambda\). For \(f(t)(x)\in C_c(\mathbb R^2)\), the integrated operator has kernel

The coordinate change \((t,x)\leftrightarrow(x,y=x-t)\) is a homeomorphism with Jacobian one. Thus all continuous compactly supported kernels occur. They give compact operators: uniform approximation on a fixed compact rectangle by finite sums of separate continuous functions gives Hilbert–Schmidt norm approximation by finite-rank kernels. They include every rank-one kernel \(\xi(x)\overline{\eta(y)}\) for \(\xi,\eta\in C_c(\mathbb R)\), and these rank-one operators span a dense subalgebra of \(\mathcal K\).

It remains to prove the integrated representation is faithful. The representation \(M\) of \(C_0(\mathbb R)\) is faithful and nondegenerate. By absorption, its induced regular representation is equivalent to \((M\otimes1,\lambda\otimes\lambda)\). On \(L^2(\mathbb R^2)\) the unitary

preserves Lebesgue measure, fixes the coefficient multiplication, and conjugates diagonal translation to translation in \(x\) alone:

Its integrated operator is therefore \((M\rtimes\lambda)(f)\otimes1\). This gives the regular norm of \(f\), and amenability identifies it with the full norm. The extension of the kernel map is isometric with image \(\mathcal K\), proving (19). The functions in \(C_c(\mathbb R^2)\) used above are dense in \(C_c(\mathbb R,C_0(\mathbb R))\) in \(L^1\), by cutting off \(x\) uniformly on the compact coefficient image. \(\square\)

Let \(Y=\mathbb R\cup\{+\infty\}\), with neighborhoods of the added point containing a positive tail. A member of \(C_0(Y)\) vanishes as \(x\to-\infty\) and has a finite continuous limit at \(+\infty\). Translation fixes the added point and acts strongly continuously. Evaluation there gives an equivariant exact sequence

Surjectivity follows by choosing a continuous scalar function equal to one on a positive tail and zero on a negative tail. Exactness from Lesson 1 and (19) give

For any flow \(\alpha\) on \(A\), put \(\mathcal W_\alpha=(C_0(Y)\otimes_{\min}A)\rtimes_{\tau\otimes\alpha}\mathbb R\). Its coefficient algebra is \(C_0(Y,A)\). On the ideal \(C_0(\mathbb R,A)\), the isometric automorphism

conjugates the diagonal action to translation with trivial coefficient action:

The minimal trivial-factor tensor formula of Lesson 2, amenability, and (19) identify its crossed product with \(\mathcal K\otimes_{\min}A\). Evaluation at \(+\infty\) has quotient action \(\alpha\), so

No nuclearity of \(A\) is needed. Formula (23) is asserted on the ideal: it need not extend to a function with a limit at \(+\infty\). The boundary maps of (24), after stability, will be the Thom maps; proving they are isomorphisms requires the work in Lessons 10–12.

Exercises with complete solutions

Exercise 1 (basic). Determine the gauge-fixed algebra and all gauge spectral subspaces of \(A\rtimes_\alpha\mathbb Z\), allowing nonunital \(A\).

Solution. The contractive projection \(P_n\) in (2) maps polynomials into \(Au^n\), which is closed because multiplication by the unitary multiplier \(u^n\) is isometric. Therefore its entire range lies there. Every member of \(Au^n\) has gauge character \(z^n\), so the range is exactly that spectral subspace. Taking \(n=0\) gives the fixed algebra \(A\). For a fixed element \(x\), the integral defining \(P_0(x)\) equals \(x\), completing both inclusions. None of this requires \(u\) to lie in the algebra.

Exercise 2 (intermediate). Prove the real-group identification in (5) assuming Plancherel, and explain the functional-calculus normalization.

Solution. Plancherel takes the regular integrated operator to \(M_{\mathcal F_+g}\), with norm equal to the continuous function's supremum norm. Integration by parts and \(L^1\) approximation put these functions in \(C_0(\mathbb R)\). Their self-adjoint algebra separates frequencies and vanishes at no frequency, using concentrated bumps as in Proposition 5.2; Stone–Weierstrass gives all of \(C_0(\mathbb R)\). Amenability gives the same full norm. For \(U_t=e^{2\pi itD}\), the spectral theorem gives \(\int g(t)U_t\,dt=\mathcal F_+g(D)\). With \(U_t=e^{itD'}\), the generator would instead be \(D'=2\pi D\). The sign and factor are part of the identification.

Exercise 3 (intermediate). Verify the unitary computation in (15)–(17), including translation across the edge of \([0,1)\).

Solution. Fix \(r\) and write \(t-r=t'+k\). In the Fourier sum, replace \(n+k\) by \(m\):

Multiplication by \(U\) on the output and \(U^*\) on the input adds the factor \(e^{2\pi is(t-t')}\). The total is \(e^{2\pi is(t-t'-k)}=e^{2\pi isr}\), proving (17). This includes negative \(k\), which is exactly the edge-crossing contribution. The coefficient \(\rho(\beta_{-(t+n)}b)\) equals \(\rho(\beta_{-t}b)\) by periodicity. The calculation starts on compactly supported elementary functions, where finite sums suffice, and extends by the two unitaries and boundedness. It is a fiber representation of \(B\rtimes\mathbb T\), not of \(B\rtimes\mathbb Z\).

Exercise 4 (advanced). Suppose an integer action \(\beta\) on \(B\) satisfies \(\beta_n=\operatorname{id}\) for some positive integer \(n\). Construct and prove a mapping-torus description of \(B\rtimes_\beta\mathbb Z\).

Solution. Set \(C=B\rtimes_\beta(\mathbb Z/n\mathbb Z)\), and write \(v\) for its canonical generator, so \(v^n=1\) in \(M(C)\). The dual generator \(\sigma\) fixes \(B\) and sends \(v\) to \(e^{2\pi i/n}v\). Define a family of covariant pairs by \(b\mapsto i_B(b)\), \(u\mapsto e^{2\pi is/n}v\). On polynomials its integrated map is

It has the endpoint relation \(\Phi(x)(s+1)=\sigma(\Phi(x)(s))\). To prove its norm, use a faithful \(\rho\) and the regular representation on \(\ell^2(\mathbb Z,H)\). The unitary transform

is Fourier-series Plancherel along each coset. If \(j-1=j'+nk_0\) with \(0\leq j'<n\), the group shift acquires the factor \(e^{-2\pi ik_0s}\). Conjugate by multiplication by \(e^{2\pi isj/n}\). The remaining factor is

and the coefficient action is \(\rho(\beta_{-j}b)\). These are the faithful finite-group regular models of (25). Their norms are continuous in \(s\), so the regular integer norm is \(\sup_{[0,1]}\|\Phi(x)(s)\|\); amenability makes it the full norm. Thus the image is closed.

At any \(s_0\), the monomials \(bu^j\) map to scalar multiples of \(bv^j\). The latter span the finite-group crossed product densely, so evaluation is onto by closed range. The central multiplier \(u^n\) maps to \(e^{2\pi is}1\). Its functional calculus supplies every scalar multiplier on the parameter circle. The partition-of-unity approximation in the last paragraph of Theorem 5.3 therefore proves surjectivity onto \(M_\sigma\). This includes \(n=1\), when the action is trivial, \(C=B\), and the mapping torus is \(C(\mathbb T,B)\).

What this lesson does not prove

The periodic mapping-torus theorem, its discrete analogue, the translation compact-operator isomorphism, the smooth convolution construction, and all four solutions were proved here. Scalar Plancherel, Fourier inversion and the integer Fourier-series unitary use the written programme lessons linked in the Fourier section: the series case is Corollary 4.1 of The Plancherel theorem. Stone's theorem is proved in the programme lesson Holomorphy in Banach spaces, Stone’s theorem and resolvent convergence. Stone–Weierstrass, Banach-valued integration and locally compact partitions of unity are the functional-analysis and measure-theory prerequisites.

The multiplier, regular-representation, exactness, tensor-factor and amenability results are the proved prerequisites in Lessons 1, 2 and 4. We have constructed the Wiener–Hopf extension, without yet proving its boundary maps invertible. No holomorphic functional calculus or \(K\)-theory equivalence of the Schwartz algebra in (7) is claimed.

References

• [Blackadar 1998] Bruce Blackadar, K-Theory for Operator Algebras, second edition, MSRI Publications 5, Cambridge University Press, 1998. Definition 10.3.1, Propositions 10.3.2–10.3.3, and §10.9. Author's second edition.
• [Blackadar 2006] Bruce Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer, 2006. II.10.2.7 and II.10.4.12. Author's revised edition, 2017.
• [Emerson 2024] Heath Emerson, An Introduction to C*-Algebras and Noncommutative Geometry, Birkhäuser, 2024. §1.4; Theorem 1.9.9 and Corollary 1.9.15.
• [Schulz-Baldes–Stoiber 2022] Hermann Schulz-Baldes and Tom Stoiber, Harmonic Analysis in Operator Algebras and its Applications to Index Theory and Topological Solid State Systems, Springer, 2022. §1.7, Spaces of differentiable elements. Preprint.
• Frequency calculus for an action of Euclidean space Proposition 1.1, the unital Euclidean smooth convolution and density theorem. The estimates here cover nonunital \(A\) in one dimension.