Fréchet algebra

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In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A {\displaystyle A} {\displaystyle A} over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ( a , b ) ↦ a ∗ b {\displaystyle (a,b)\mapsto a*b} {\displaystyle (a,b)\mapsto a*b} for a , b ∈ A {\displaystyle a,b\in A} {\displaystyle a,b\in A} is required to be jointly continuous. If { ‖ ⋅ ‖ n } n = 0 ∞ {\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} {\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} is an increasing family[a] of seminorms for the topology of A {\displaystyle A} {\displaystyle A}, the joint continuity of multiplication is equivalent to there being a constant C n > 0 {\displaystyle C_{n}>0} {\displaystyle C_{n}>0} and integer m ≥ n {\displaystyle m\geq n} {\displaystyle m\geq n} for each n {\displaystyle n} {\displaystyle n} such that ‖ a b ‖ n ≤ C n ‖ a ‖ m ‖ b ‖ m {\displaystyle \left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}} {\displaystyle \left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}} for all a , b ∈ A {\displaystyle a,b\in A} {\displaystyle a,b\in A}.[b] Fréchet algebras are also called B0-algebras.[1]

A Fréchet algebra is m {\displaystyle m} {\displaystyle m}-convex if there exists such a family of semi-norms for which m = n {\displaystyle m=n} {\displaystyle m=n}. In that case, by rescaling the seminorms, we may also take C n = 1 {\displaystyle C_{n}=1} {\displaystyle C_{n}=1} for each n {\displaystyle n} {\displaystyle n} and the seminorms are said to be submultiplicative: ‖ a b ‖ n ≤ ‖ a ‖ n ‖ b ‖ n {\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}} {\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}} for all a , b ∈ A . {\displaystyle a,b\in A.} {\displaystyle a,b\in A.}[c] m {\displaystyle m} {\displaystyle m}-convex Fréchet algebras may also be called Fréchet algebras.[2]

A Fréchet algebra may or may not have an identity element 1 A {\displaystyle 1_{A}} {\displaystyle 1_{A}}. If A {\displaystyle A} {\displaystyle A} is unital, we do not require that ‖ 1 A ‖ n = 1 , {\displaystyle \|1_{A}\|_{n}=1,} {\displaystyle \|1_{A}\|_{n}=1,} as is often done for Banach algebras.

Properties

  • Continuity of multiplication. Multiplication is separately continuous if a k b → a b {\displaystyle a_{k}b\to ab} {\displaystyle a_{k}b\to ab} and b a k → b a {\displaystyle ba_{k}\to ba} {\displaystyle ba_{k}\to ba} for every a , b ∈ A {\displaystyle a,b\in A} {\displaystyle a,b\in A} and sequence a k → a {\displaystyle a_{k}\to a} {\displaystyle a_{k}\to a} converging in the Fréchet topology of A {\displaystyle A} {\displaystyle A}. Multiplication is jointly continuous if a k → a {\displaystyle a_{k}\to a} {\displaystyle a_{k}\to a} and b k → b {\displaystyle b_{k}\to b} {\displaystyle b_{k}\to b} imply a k b k → a b {\displaystyle a_{k}b_{k}\to ab} {\displaystyle a_{k}b_{k}\to ab}. Joint continuity of multiplication is part of the definition of a Fréchet algebra. For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous.[3]
  • Group of invertible elements. If i n v A {\displaystyle invA} {\displaystyle invA} is the set of invertible elements of A {\displaystyle A} {\displaystyle A}, then the inverse map { i n v A → i n v A u ↦ u − 1 {\displaystyle {\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}} {\displaystyle {\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}} is continuous if and only if i n v A {\displaystyle invA} {\displaystyle invA} is a G δ {\displaystyle G_{\delta }} {\displaystyle G_{\delta }} set.[4] Unlike for Banach algebras, i n v A {\displaystyle invA} {\displaystyle invA} may not be an open set. If i n v A {\displaystyle invA} {\displaystyle invA} is open, then A {\displaystyle A} {\displaystyle A} is called a Q {\displaystyle Q} {\displaystyle Q}-algebra. (If A {\displaystyle A} {\displaystyle A} happens to be non-unital, then we may adjoin a unit to A {\displaystyle A} {\displaystyle A}[d] and work with i n v A + {\displaystyle invA^{+}} {\displaystyle invA^{+}}, or the set of quasi invertibles[e] may take the place of i n v A {\displaystyle invA} {\displaystyle invA}.)
  • Conditions for m {\displaystyle m} {\displaystyle m}-convexity. A Fréchet algebra is m {\displaystyle m} {\displaystyle m}-convex if and only if for every, if and only if for one, increasing family { ‖ ⋅ ‖ n } n = 0 ∞ {\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} {\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} of seminorms which topologize A {\displaystyle A} {\displaystyle A}, for each m ∈ N {\displaystyle m\in \mathbb {N} } {\displaystyle m\in \mathbb {N} } there exists p ≥ m {\displaystyle p\geq m} {\displaystyle p\geq m} and C m > 0 {\displaystyle C_{m}>0} {\displaystyle C_{m}>0} such that ‖ a 1 a 2 ⋯ a n ‖ m ≤ C m n ‖ a 1 ‖ p ‖ a 2 ‖ p ⋯ ‖ a n ‖ p , {\displaystyle \|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},} {\displaystyle \|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},} for all a 1 , a 2 , … , a n ∈ A {\displaystyle a_{1},a_{2},\dots ,a_{n}\in A} {\displaystyle a_{1},a_{2},\dots ,a_{n}\in A} and n ∈ N {\displaystyle n\in \mathbb {N} } {\displaystyle n\in \mathbb {N} }.[5] A commutative Fréchet Q {\displaystyle Q} {\displaystyle Q}-algebra is m {\displaystyle m} {\displaystyle m}-convex,[6] but there exist examples of non-commutative Fréchet Q {\displaystyle Q} {\displaystyle Q}-algebras which are not m {\displaystyle m} {\displaystyle m}-convex.[7]
  • Properties of m {\displaystyle m} {\displaystyle m}-convex Fréchet algebras. A Fréchet algebra is m {\displaystyle m} {\displaystyle m}-convex if and only if it is a countable projective limit of Banach algebras.[8] An element of A {\displaystyle A} {\displaystyle A} is invertible if and only if its image in each Banach algebra of the projective limit is invertible.[f][9][10]

Examples

  • Zero multiplication. If E {\displaystyle E} {\displaystyle E} is any Fréchet space, we can make a Fréchet algebra structure by setting e ∗ f = 0 {\displaystyle e*f=0} {\displaystyle e*f=0} for all e , f ∈ E {\displaystyle e,f\in E} {\displaystyle e,f\in E}.
  • Smooth functions on the circle. Let S 1 {\displaystyle S^{1}} {\displaystyle S^{1}} be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let A = C ∞ ( S 1 ) {\displaystyle A=C^{\infty }(S^{1})} {\displaystyle A=C^{\infty }(S^{1})} be the set of infinitely differentiable complex-valued functions on S 1 {\displaystyle S^{1}} {\displaystyle S^{1}}. This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation.) It is commutative, and the constant function 1 {\displaystyle 1} {\displaystyle 1} acts as an identity. Define a countable set of seminorms on A {\displaystyle A} {\displaystyle A} by ‖ φ ‖ n = ‖ φ ( n ) ‖ ∞ , φ ∈ A , {\displaystyle \left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,} {\displaystyle \left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,} where ‖ φ ( n ) ‖ ∞ = sup x ∈ S 1 | φ ( n ) ( x ) | {\displaystyle \left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|} {\displaystyle \left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|} denotes the supremum of the absolute value of the n {\displaystyle n} {\displaystyle n}th derivative φ ( n ) {\displaystyle \varphi ^{(n)}} {\displaystyle \varphi ^{(n)}}.[g] Then, by the product rule for differentiation, we have ‖ φ ψ ‖ n = ‖ ∑ i = 0 n ( n i ) φ ( i ) ψ ( n − i ) ‖ ∞ ≤ ∑ i = 0 n ( n i ) ‖ φ ‖ i ‖ ψ ‖ n − i ≤ ∑ i = 0 n ( n i ) ‖ φ ‖ n ′ ‖ ψ ‖ n ′ = 2 n ‖ φ ‖ n ′ ‖ ψ ‖ n ′ , {\displaystyle {\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}} {\displaystyle {\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}} where ( n i ) = n ! i ! ( n − i ) ! , {\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}},} {\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}},} denotes the binomial coefficient and ‖ ⋅ ‖ n ′ = max k ≤ n ‖ ⋅ ‖ k . {\displaystyle \|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.} {\displaystyle \|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.} The primed seminorms are submultiplicative after re-scaling by C n = 2 n {\displaystyle C_{n}=2^{n}} {\displaystyle C_{n}=2^{n}}.
  • Sequences on N {\displaystyle \mathbb {N} } {\displaystyle \mathbb {N} }. Let C N {\displaystyle \mathbb {C} ^{\mathbb {N} }} {\displaystyle \mathbb {C} ^{\mathbb {N} }} be the space of complex-valued sequences on the natural numbers N {\displaystyle \mathbb {N} } {\displaystyle \mathbb {N} }. Define an increasing family of seminorms on C N {\displaystyle \mathbb {C} ^{\mathbb {N} }} {\displaystyle \mathbb {C} ^{\mathbb {N} }} by ‖ φ ‖ n = max k ≤ n | φ ( k ) | . {\displaystyle \|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.} {\displaystyle \|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.} With pointwise multiplication, C N {\displaystyle \mathbb {C} ^{\mathbb {N} }} {\displaystyle \mathbb {C} ^{\mathbb {N} }} is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative ‖ φ ψ ‖ n ≤ ‖ φ ‖ n ‖ ψ ‖ n {\displaystyle \|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}} {\displaystyle \|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}} for φ , ψ ∈ A {\displaystyle \varphi ,\psi \in A} {\displaystyle \varphi ,\psi \in A}. This m {\displaystyle m} {\displaystyle m}-convex Fréchet algebra is unital, since the constant sequence 1 ( k ) = 1 , k ∈ N {\displaystyle 1(k)=1,k\in \mathbb {N} } {\displaystyle 1(k)=1,k\in \mathbb {N} } is in A {\displaystyle A} {\displaystyle A}.
  • Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, C ( C ) {\displaystyle C(\mathbb {C} )} {\displaystyle C(\mathbb {C} )}, the algebra of all continuous functions on the complex plane C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }, or to the algebra H o l ( C ) {\displaystyle \mathrm {Hol} (\mathbb {C} )} {\displaystyle \mathrm {Hol} (\mathbb {C} )} of holomorphic functions on C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }.
  • Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let G {\displaystyle G} {\displaystyle G} be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements U = { g 1 , … , g n } ⊆ G {\displaystyle U=\{g_{1},\dots ,g_{n}\}\subseteq G} {\displaystyle U=\{g_{1},\dots ,g_{n}\}\subseteq G} such that: ⋃ n = 0 ∞ U n = G . {\displaystyle \bigcup _{n=0}^{\infty }U^{n}=G.} {\displaystyle \bigcup _{n=0}^{\infty }U^{n}=G.} Without loss of generality, we may also assume that the identity element e {\displaystyle e} {\displaystyle e} of G {\displaystyle G} {\displaystyle G} is contained in U {\displaystyle U} {\displaystyle U}. Define a function ℓ : G → [ 0 , ∞ ) {\displaystyle \ell :G\to [0,\infty )} {\displaystyle \ell :G\to [0,\infty )} by ℓ ( g ) = min { n ∣ g ∈ U n } . {\displaystyle \ell (g)=\min\{n\mid g\in U^{n}\}.} {\displaystyle \ell (g)=\min\{n\mid g\in U^{n}\}.} Then ℓ ( g h ) ≤ ℓ ( g ) + ℓ ( h ) {\displaystyle \ell (gh)\leq \ell (g)+\ell (h)} {\displaystyle \ell (gh)\leq \ell (g)+\ell (h)}, and ℓ ( e ) = 0 {\displaystyle \ell (e)=0} {\displaystyle \ell (e)=0}, since we define U 0 = { e } {\displaystyle U^{0}=\{e\}} {\displaystyle U^{0}=\{e\}}.[h] Let A {\displaystyle A} {\displaystyle A} be the C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }-vector space S ( G ) = { φ : G → C | ‖ φ ‖ d < ∞ , d = 0 , 1 , 2 , … } , {\displaystyle S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},} {\displaystyle S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},} where the seminorms ‖ ⋅ ‖ d {\displaystyle \|\cdot \|_{d}} {\displaystyle \|\cdot \|_{d}} are defined by ‖ φ ‖ d = ‖ ℓ d φ ‖ 1 = ∑ g ∈ G ℓ ( g ) d | φ ( g ) | . {\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.} {\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.}[i] A {\displaystyle A} {\displaystyle A} is an m {\displaystyle m} {\displaystyle m}-convex Fréchet algebra for the convolution multiplication φ ∗ ψ ( g ) = ∑ h ∈ G φ ( h ) ψ ( h − 1 g ) , {\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),} {\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),}[j] A {\displaystyle A} {\displaystyle A} is unital because G {\displaystyle G} {\displaystyle G} is discrete, and A {\displaystyle A} {\displaystyle A} is commutative if and only if G {\displaystyle G} {\displaystyle G} is Abelian.
  • Non m {\displaystyle m} {\displaystyle m}-convex Fréchet algebras. The Aren's algebra A = L ω [ 0 , 1 ] = ⋂ p ≥ 1 L p [ 0 , 1 ] {\displaystyle A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]} {\displaystyle A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]} is an example of a commutative non- m {\displaystyle m} {\displaystyle m}-convex Fréchet algebra with discontinuous inversion. The topology is given by L p {\displaystyle L^{p}} {\displaystyle L^{p}} norms ‖ f ‖ p = ( ∫ 0 1 | f ( t ) | p d t ) 1 / p , f ∈ A , {\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,} {\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,} and multiplication is given by convolution of functions with respect to Lebesgue measure on [ 0 , 1 ] {\displaystyle [0,1]} {\displaystyle [0,1]}.[11]

Generalizations

We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space[12] or an F-space.[13]

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).[14] A complete LMC algebra is called an Arens-Michael algebra.[15]

Michael's Conjecture

The question of whether all linear multiplicative functionals on an m {\displaystyle m} {\displaystyle m}-convex Frechet algebra are continuous is known as Michael's Conjecture.[16] This conjecture is perhaps the most famous open problem in the theory of topological algebras.

Notes

  1. An increasing family means that for each a ∈ A , {\displaystyle a\in A,} {\displaystyle a\in A,}
    ‖ a ‖ 0 ≤ ‖ a ‖ 1 ≤ ⋯ ≤ ‖ a ‖ n ≤ ⋯ {\displaystyle \|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots } {\displaystyle \|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots }.
  2. Joint continuity of multiplication means that for every absolutely convex neighborhood V {\displaystyle V} {\displaystyle V} of zero, there is an absolutely convex neighborhood U {\displaystyle U} {\displaystyle U} of zero for which U 2 ⊆ V , {\displaystyle U^{2}\subseteq V,} {\displaystyle U^{2}\subseteq V,} from which the seminorm inequality follows. Conversely,
    ‖ a k b k − a b ‖ n = ‖ a k b k − a b k + a b k − a b ‖ n ≤ ‖ a k b k − a b k ‖ n + ‖ a b k − a b ‖ n ≤ C n ( ‖ a k − a ‖ m ‖ b k ‖ m + ‖ a ‖ m ‖ b k − b ‖ m ) ≤ C n ( ‖ a k − a ‖ m ‖ b ‖ m + ‖ a k − a ‖ m ‖ b k − b ‖ m + ‖ a ‖ m ‖ b k − b ‖ m ) . {\displaystyle {\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}} {\displaystyle {\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}}
  3. In other words, an m {\displaystyle m} {\displaystyle m}-convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: p ( f g ) ≤ p ( f ) p ( g ) , {\displaystyle p(fg)\leq p(f)p(g),} {\displaystyle p(fg)\leq p(f)p(g),} and the algebra is complete.
  4. If A {\displaystyle A} {\displaystyle A} is an algebra over a field k {\displaystyle k} {\displaystyle k}, the unitization A + {\displaystyle A^{+}} {\displaystyle A^{+}} of A {\displaystyle A} {\displaystyle A} is the direct sum A ⊕ k 1 {\displaystyle A\oplus k1} {\displaystyle A\oplus k1}, with multiplication defined as ( a + μ 1 ) ( b + λ 1 ) = a b + μ b + λ a + μ λ 1. {\displaystyle (a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.} {\displaystyle (a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.}
  5. If a ∈ A {\displaystyle a\in A} {\displaystyle a\in A}, then b ∈ A {\displaystyle b\in A} {\displaystyle b\in A} is a quasi-inverse for a {\displaystyle a} {\displaystyle a} if a + b − a b = 0 {\displaystyle a+b-ab=0} {\displaystyle a+b-ab=0}.
  6. If A {\displaystyle A} {\displaystyle A} is non-unital, replace invertible with quasi-invertible.
  7. To see the completeness, let φ k {\displaystyle \varphi _{k}} {\displaystyle \varphi _{k}} be a Cauchy sequence. Then each derivative φ k ( l ) {\displaystyle \varphi _{k}^{(l)}} {\displaystyle \varphi _{k}^{(l)}} is a Cauchy sequence in the sup norm on S 1 {\displaystyle S^{1}} {\displaystyle S^{1}}, and hence converges uniformly to a continuous function ψ l {\displaystyle \psi _{l}} {\displaystyle \psi _{l}} on S 1 {\displaystyle S^{1}} {\displaystyle S^{1}}. It suffices to check that ψ l {\displaystyle \psi _{l}} {\displaystyle \psi _{l}} is the l {\displaystyle l} {\displaystyle l}th derivative of ψ 0 {\displaystyle \psi _{0}} {\displaystyle \psi _{0}}. But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
    ψ l ( x ) − ψ l ( x 0 ) = lim k → ∞ ( φ k ( l ) ( x ) − φ k ( l ) ( x 0 ) ) = lim k → ∞ ∫ x 0 x φ k ( l + 1 ) ( t ) d t = ∫ x 0 x ψ l + 1 ( t ) d t . {\displaystyle {\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}} {\displaystyle {\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}}
  8. We can replace the generating set U {\displaystyle U} {\displaystyle U} with U ∪ U − 1 {\displaystyle U\cup U^{-1}} {\displaystyle U\cup U^{-1}}, so that U = U − 1 {\displaystyle U=U^{-1}} {\displaystyle U=U^{-1}}. Then ℓ {\displaystyle \ell } {\displaystyle \ell } satisfies the additional property ℓ ( g − 1 ) = ℓ ( g ) {\displaystyle \ell (g^{-1})=\ell (g)} {\displaystyle \ell (g^{-1})=\ell (g)}, and is a length function on G {\displaystyle G} {\displaystyle G}.
  9. To see that A {\displaystyle A} {\displaystyle A} is Fréchet space, let φ n {\displaystyle \varphi _{n}} {\displaystyle \varphi _{n}} be a Cauchy sequence. Then for each g ∈ G {\displaystyle g\in G} {\displaystyle g\in G}, φ n ( g ) {\displaystyle \varphi _{n}(g)} {\displaystyle \varphi _{n}(g)} is a Cauchy sequence in C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }. Define φ ( g ) {\displaystyle \varphi (g)} {\displaystyle \varphi (g)} to be the limit. Then
    ∑ g ∈ S ℓ ( g ) d | φ n ( g ) − φ ( g ) | ≤ ∑ g ∈ S ℓ ( g ) d | φ n ( g ) − φ m ( g ) | + ∑ g ∈ S ℓ ( g ) d | φ m ( g ) − φ ( g ) | ≤ ‖ φ n − φ m ‖ d + ∑ g ∈ S ℓ ( g ) d | φ m ( g ) − φ ( g ) | , {\displaystyle {\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}} {\displaystyle {\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}}
    where the sum ranges over any finite subset S {\displaystyle S} {\displaystyle S} of G {\displaystyle G} {\displaystyle G}. Let ϵ > 0 {\displaystyle \epsilon >0} {\displaystyle \epsilon >0}, and let K ϵ > 0 {\displaystyle K_{\epsilon }>0} {\displaystyle K_{\epsilon }>0} be such that ‖ φ n − φ m ‖ d < ϵ {\displaystyle \|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon } {\displaystyle \|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon } for m , n ≥ K ϵ {\displaystyle m,n\geq K_{\epsilon }} {\displaystyle m,n\geq K_{\epsilon }}. By letting m {\displaystyle m} {\displaystyle m} run, we have
    ∑ g ∈ S ℓ ( g ) d | φ n ( g ) − φ ( g ) | < ϵ {\displaystyle \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon } {\displaystyle \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon }
    for n ≥ K ϵ {\displaystyle n\geq K_{\epsilon }} {\displaystyle n\geq K_{\epsilon }}. Summing over all of G {\displaystyle G} {\displaystyle G}, we therefore have ‖ φ n − φ ‖ d < ϵ {\displaystyle \left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon } {\displaystyle \left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon } for n ≥ K ϵ {\displaystyle n\geq K_{\epsilon }} {\displaystyle n\geq K_{\epsilon }}. By the estimate
    ∑ g ∈ S ℓ ( g ) d | φ ( g ) | ≤ ∑ g ∈ S ℓ ( g ) d | φ n ( g ) − φ ( g ) | + ∑ g ∈ S ℓ ( g ) d | φ n ( g ) | ≤ ‖ φ n − φ ‖ d + ‖ φ n ‖ d , {\displaystyle {\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}} {\displaystyle {\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}}
    we obtain ‖ φ ‖ d < ∞ {\displaystyle \|\varphi \|_{d}<\infty } {\displaystyle \|\varphi \|_{d}<\infty }. Since this holds for each d ∈ N {\displaystyle d\in \mathbb {N} } {\displaystyle d\in \mathbb {N} }, we have φ ∈ A {\displaystyle \varphi \in A} {\displaystyle \varphi \in A} and φ n → φ {\displaystyle \varphi _{n}\to \varphi } {\displaystyle \varphi _{n}\to \varphi } in the Fréchet topology, so A {\displaystyle A} {\displaystyle A} is complete.
  10. ‖ φ ∗ ψ ‖ d ≤ ∑ g ∈ G ( ∑ h ∈ G ℓ ( g ) d | φ ( h ) | | ψ ( h − 1 g ) | ) ≤ ∑ g , h ∈ G ( ℓ ( h ) + ℓ ( h − 1 g ) ) d | φ ( h ) | | ψ ( h − 1 g ) | = ∑ i = 0 d ( d i ) ( ∑ g , h ∈ G | ℓ i φ ( h ) | | ℓ d − i ψ ( h − 1 g ) | ) = ∑ i = 0 d ( d i ) ( ∑ h ∈ G | ℓ i φ ( h ) | ) ( ∑ g ∈ G | ℓ d − i ψ ( g ) | ) = ∑ i = 0 d ( d i ) ‖ φ ‖ i ‖ ψ ‖ d − i ≤ 2 d ‖ φ ‖ d ′ ‖ ψ ‖ d ′ {\displaystyle {\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}} {\displaystyle {\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}}

Citations

  1. Mitiagin, Rolewicz & Żelazko 1962; Żelazko 2001.
  2. Husain 1991; Żelazko 2001.
  3. Waelbroeck 1971, Chapter VII, Proposition 1; Palmer 1994, § {\displaystyle \S } {\displaystyle \S } 2.9.
  4. Waelbroeck 1971, Chapter VII, Proposition 2.
  5. Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2.
  6. Żelazko 1965, Theorem 13.17.
  7. Żelazko 1994, pp. 283–290.
  8. Michael 1952, Theorem 5.1.
  9. Michael 1952, Theorem 5.2.
  10. See also Palmer 1994, Theorem 2.9.6.
  11. Fragoulopoulou 2005, Example 6.13 (2).
  12. Waelbroeck 1971.
  13. Rudin 1973, 1.8(e).
  14. Michael 1952; Husain 1991.
  15. Fragoulopoulou 2005, Chapter 1.
  16. Michael 1952, § {\displaystyle \S } {\displaystyle \S } 12, Question 1; Palmer 1994, § {\displaystyle \S } {\displaystyle \S } 3.1

Sources