Fourier stuff

J. M. Egger (joint work with G. Lukacs) 16 June 2012

Fourier transforms (general case)

G: (arbitrary) abelian locally compact group γ_{G}: its Haar measure Ĝ: its Pontryagin dual Φ_{G}: homomorphism of (not necessarily unital) Banach ∗-algebras

\xymatrix{L^{1}(G, γ_{G}) \ar[r] & C_{0}(Ĝ)}

Examples:

[figure: L^{1}(𝕋); Φ_{𝕋}; C_{0}(ℤ); f ↦(\frac{1}{2π} ∫_{−π}^{π} f (e^{iϑ})e^{−inϑ} dϑ)_{n∈ℤ}; ℓ^{1}(ℤ); Φ_{ℤ}; C(𝕋); (a_{n})_{n∈ℤ} ↦[e^{iϑ} ↦ ∑_{n∈ℤ} a_{n}e^{−inϑ}]]

Fourier transforms as a natural transformation?

Problem: neither G ↦ L^{1}(G, γ_{G}) nor G ↦ C_{0}(Ĝ) extend to functors—naturality does not even parse!

[figure: G; ω; L^{1}(G, γ_{G}); Φ_{G}; ?; C_{0}(Ĝ); ?; Ĝ; H; L^{1}(H, γ_{H}); Φ_{H}; C_{0}(Ĥ); Ĥ; ω̂]

Solution: restrict attention to lluf subcategory of CtsGrp _{abLT_{2}}?

Three Gelfand dualities

Categorists’ favourite version:

\xymatrix{(Cts_{KT_{2}})^{op} \ar[rr] & ≃ & ^{†}Alg_{ucC^{∗}} \ar[ll]}

Intermediate version:

\xymatrix{(1\Cts_{KT_{2}})^{op} = (Cts_{KT_{2}})^{op}/1 \ar[rr] & ≃ & (^{†}Alg_{ucC^{∗}})/ℂ ≃ ^{†}Alg_{cC^{∗}} \ar[ll]}

1\Cts_{KT_{2}} ≃ PartPropCts _{LT_{2}} (locally compact T_{2}-spaces and partial proper continuous functions with open domain). Analysts’ favourite version:

\xymatrix{(PropCts _{LT_{2}})^{op} \ar[rr] & ≃ & ^{†}Alg_{aucC^{∗}} \ar[ll]}

†Alg_{aucC^{∗}} lluf subcategory of “approximate unit”-preserving maps.

Three Pontryagin dualities

Analysts’ favourite version:

\xymatrix{(CtsGrp _{abLT_{2}})^{op} \ar[rr] & ≃ & CtsGrp _{abLT_{2}} \ar[ll]}

Intermediate version: (new?—surely not…)

\xymatrix{(PropCtsGrp _{abLT_{2}})^{op} \ar[rr] & ≃ & OpenCtsGrp _{abLT_{2}} \ar[ll]}

Categorists’ favourite version:

[figure: (CtsGrp _{abKT_{2}})^{op}; ≃; Grp_{ab}; (PropCtsGrp _{abLT_{2}}/0)^{op}; 0\OpenCtsGrp _{abLT_{2}}]

Functoriality of L^{1}

Preceding slides show that G ↦ C_{0}(Ĝ) does extend to a functor OpenCtsGrp _{abLT_{2}} → ^{†}Alg_{cC^{∗}} ⊆ Ban^{†}Alg _{c}. What about G ↦ L^{1}(G, γ_{G})? Theorem (E., Lukacs): A continuous group homomorphism ω : G → H is open if and only if

γ_{H}^{§}(B) = 0 ⟹ γ_{G}^{§}(ω^{∗}B) = 0

even for not necessarily abelian locally compact groups G, H. (Here γ_{G}^{§}, γ_{H}^{§} are Segalifications of Haar measure).

Functoriality of L^{1}

Preceding slides show that G ↦ C_{0}(Ĝ) does extend to a functor OpenCtsGrp _{abLT_{2}} → ^{†}Alg_{cC^{∗}} ⊆ Ban^{†}Alg _{c}. What about G ↦ L^{1}(G, γ_{G})? Theorem (E., Lukacs): A continuous group homomorphism ω : G → H is open if and only if

γ_{H}^{§}(B) = 0 ⟹ γ_{G}^{§}(ω^{∗}B) = 0

even for not necessarily abelian locally compact groups G, H. (Here γ_{G}^{§}, γ_{H}^{§} are Segalifications of Haar measure).

So … ?

[figure: OpenCtsGrp _{LT_{2}}; Wendt; Ban; G; ↦; (G, γ_{G}^{§}); ↦; ι_{G}; L^{1}(G, γ_{G}^{§}); ↦; (G, γ_{G}); L^{1}(G, γ_{G})]

are strong symmetric involutive monoidal functors; therefore also

\xymatrix{OpenCtsGrp _{abLT_{2}} & ⊆ & ^{†}Sgrp_{ab}(OpenCtsGrp _{LT_{2}}) \ar[d] \\  &  & ^{†}Sgrp_{ab}(Ban) & = & _{Ban^{†}Alg _{c}}}

(in fact, more)