topological convolution algebras - bgu

Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space

Topological convolution algebras

Guy Salomon

Joint work with Daniel Alpay

Department of MathematicsBen-Gurion University

Beer-Sheva, Israel

May 21st, 2011


Outline

Strong algebras: A new family of dF algebras A =⋃

Φ′pwhich satisfy ‖ab‖p ≤ Ap,q‖a‖q‖b‖p.Strong convolution algebras: A special case where A is aconvolution algebra.

Which convolution algebra is a strong one?Examples:

The algebra of germs of holomorphic functions in zero: Thelast theorem implies it is a strong algebra. In particular we willsee that ‖fg‖p ≤ (1− 4−(p−q))−

12 ‖f‖q‖g‖p.

Relation to Dirichlet series: e.g.∫∞1

(∫ x1

√byc⌊xy

⌋dyy

)2

dxxt+1 ≤ infr∈(1,t)

(ζ(t)ζ(r)t(t−r)r

).

A non commutative version of the Kondratiev space ofstochastic distributions: Together with the definition ofVoiculescu of the non commutative WNS, it will form anon-commutative counterpart to the Gelfand triplet of(S1,W,S−1).


L2(G, µ) is usually not an algebra

L2(G, µ) is usually not an algebra

Let G be a locally compact topological group with a Haar measureµ. The convolution of two measurable functions f, g is defined by

(f ∗ g)(x) =

∫Gf(y)g(y−1x)dµ(y).

It is well known that L1(G,µ) is a Banach algebra with theconvolution product, while L2(G,µ) is usually not closed underconvolution. More precisely,

Theorem (N.W. Rickert (1969))

For any locally compact group G, L2(G,µ) is closed underconvolution if and only if G is compact.

In case G is compact it holds that, ‖f ∗ g‖ ≤√µ(G)‖f‖‖g‖.

Thus, L2(G,µ) is a Banach algebra.


So what can be done if G is not compact, but we still want an ”Hilbert environment”?

So what can be done if G is not compact, but we stillwant an ”Hilbert environment”?

We will try to answer this question in the sequel.

In the meanwhile...


Definition

Strong algebras

Definition (D. Alpay & S (2012))

Let A =⋃p Φ′p be an algebra which is also a dual of a countably

normed space⋂p Φp. We call A a strong algebra if it satisfies

the property that there exists a constant d such that for any q andfor any p > q + d there exists a positive constant Ap,q such thatfor any a ∈ Φ′q and b ∈ Φ′p,

‖ab‖p ≤ Ap,q‖a‖q‖b‖p.

Let A be a strong algebra equipped with its strong topology, thatis, a neighborhood of zero is defined by means of any bounded setB ⊆

⋂p Φp and any number ε > 0, as the set of all a ∈ A for

whichsupb∈B|a(b)| < ε.

With this topology A is locally convex.


Definition

Strong algebras

In these settings, for any a ∈ Φ′q the multiplication operator

Ma : b 7→ ab

is bounded operator on Φ′p (p > q + d).

Moreover, we may easily consider power series. If∑∞

n=0 fnzn

converges in the open disk with radius R, then for any a ∈ Awith ‖a‖q < R

Ap,q(p > q + d), we obtain

∞∑n=0

|fn|‖an‖p ≤∞∑n=0

|fn|(Ap,q‖a‖q)n <∞,

and hence∑∞

n=0 fnan ∈ Φ′p ⊆ A.


They are topological algebras with continuous inverse

Strong algebras

Theorem (D. Alpay & S (2012))

Let A be a strong algebra.

(a) The multiplication is a continuous function A×A → A.Hence (A,+, ·) is a topological algebra.

(b) GL(A) is open, and the function a 7→ a−1 is continuous (thus,GL(A) is a topological group).

In view of the above, we can apply to unital strong algebras theresults of Naimark on (locally convex) topological algebraswith continuous inverse. In particular,

Theorem (M.A. Naimark)

The spectrum of any element in a topological algebra withcontinuous inverse is non-empty and closed.

But is it bounded?


The spectrum

Strong algebras


For any a ∈ A,

σ(a) ⊆ {z ∈ C : |z| ≤ inf{(p,q):p>q+d}

Ap,q‖a‖q}.

Thus, spectral theory can be easily done on strong algebras.

So what to strong algebras and our previous question?


Definition

Strong convolution algebras

Let G be a locally compact topological group with a Haar measureµ and let S ⊆ G be a Borel semi-group. Let (µp) be a sequence ofmeasures on G such that µ� µ1 � µ2 � · · · .

Definition⋃p L2(S, µp) is called a strong convolution algebra if there exists

a constant d such that for any p > q + d there exists a positiveconstant Ap,q such that for any f ∈ L2(µq) and g ∈ L2(µp),

‖f ∗ g‖L2(µp) ≤ Ap,q‖f‖L2(µq)‖g‖L2(µp)

(where (f ∗ g)(x) =∫S∩xS−1 f(x)g(y−1x)dµ(y)).

Due to the following theorem we can define a wide family of strongconvolution algebras.


How to construct examples



If for any x, y ∈ S and for any p ∈ N,

dµpdµ (xy) ≤ dµp

dµ (x)dµpdµ (y),

then for any f ∈ L2(S, µq) and g ∈ L2(S, µp) such that p > q,

‖f ∗ g‖p ≤(∫

Sdµpdµq

dµ) 1

2 ‖f‖q‖g‖p.

In particular, if there exists d such that∫Sdµpdµq

dµ <∞ for any

p > q + d, then⋃p L2(S, µp) is a strong convolution algebra.

The simplest example: If G is compact, and we take µp = µ forall p, we obtain that ‖f ∗ g‖ ≤

√µ(G)‖f‖‖g‖. In particular,

L2(G,µ) is a SCA iff G is compact.


An inverse theorem and a counter example


Proposition (The converse in a special case)

If dµp/dµ = (dµ1/dµ)p for any p ∈ N, then

‖f ∗ g‖p ≤ Ap,q‖f‖q‖g‖p.

implies

dµpdµ (xy) ≤ dµp

dµ (x)dµpdµ (y).

Example:Let s be the space of rapidly decreasing sequences (that is, thespace of all complex sequences (an) with supn |an(n+ 1)p| <∞for any p ∈ N). It can be shown that

s =⋂`2(n+1)2p

where `2(n+1)2p = {a ∈ CN0 :∑|an|2(n+ 1)2p <∞}.



Strong convolution algebras: a counter example

Therefore, its dual s′ can be viewed as

s′ =⋃p∈N

`2(n+1)−2p .

Thus s′ is of the form s′ =⋃p∈N L2(S, µp), where S = N0 and

µp(n) = (n+ 1)−2p.The convolution then becomes the standard one sided convolutionof sequences:

(a ∗ b)(n) =

n∑k=0

a(k)b(n− k).




One may ask whether s′ is a strong convolution algebra, i.e. ifthere exists a constant d such that for any p > q + d there existsAp,q such that for any a ∈ s′q and b ∈ s′p,

‖a ∗ b‖p ≤ Ap,q‖a‖q‖b‖p.

However since,

dµpdµ

(n) = (n+ 1)−2p =

(dµ1dµ

(n)

)pfor any p ∈ N,

and

dµpdµ

(n+m) = (n+m+1)2p 6≤ (n+1)2p(m+1)2p =dµpdµ

(n)dµpdµ

(m),




we conclude in view of the last theorem that the answer isnegative, that is,

Theorem

s′ is not a strong convolution algebra.

In a previous paper (2011, IDAQP), we replace the measures(n+ 1)−2p by 2−np, and obtain a strong convolution algebra thatcontains s′, and which can be identified as the dual of a space ofentire holomorphic functions that is included in the Schwartz spaceof complex-valued rapidly decreasing smooth functions.



Strong convolution algebras: the discrete case

Theorem (D. Alpay & S, IDAQP)

In the discrete case it holds that

A SCA is nuclear.

The tensor product (with respect to the π or ε topology) oftwo SCA’s is again a SCA.


Definitions

The space of germs of holomorphic functions in zero

Let H(C) be the space of entire holomorphic functions equippedwith the usual topology of convergence on compact sets. It can beshown that

H(C) =⋂p∈N

H2(2pD),

where H2(2pD) is the Hardy space of the disk 2pD. Thus,

H ′(C) =⋃p∈N

H2(2−pD),

and

‖f‖p = sup0<r<1

1

2π

∫|z|=r·2−p

|f(z)|2 dz =

∞∑n=0

|an|22−2np.


Definitions


For any two germs in H ′(C), f(z) =∑∞

n=0 anzn and

g(z) =∑∞

n=0 bnzn, we define

(fg)(z) =

∞∑n=0

(n∑

m=0

ambn−m

)zn.

Since for any n,m ∈ N0 and for any p ∈ N,dµpdµ (n+m) = 2−2(n+m)p = 2−2np2−2mp =

dµpdµ (n)

dµpdµ (m), and

since for any p > q∫N0

dµpdµq

dµ =

∞∑n=0

2−2np

2−2nq= (1− 4−(p−q))−1,


The algebra of germs of holomorphic functions in zero is strong


In particular we obtain that

Theorem (The algebra of germs of holomorphic functions in zero)

H ′(C) is a strong (convolution) algebra. More precisely, for anyf ∈ H2(2−qD) and g ∈ H2(2−pD) (p > q) fg belongs toH2(2−pD) and

‖fg‖p ≤ (1− 4−(p−q))−1/2‖f‖q‖g‖p.

We note that since

σ(f − f(0)) ⊆ {z ∈ C : |z| ≤ inf{(p,q):p>q+d}

Ap,q‖f − f(0)‖q}

⊆ {z ∈ C : |z| ≤ infq

(1− 4−1)−1/2∑

06=n∈N|an|22−2nq} = {0},

f ∈ H ′(C) is invertible if and only if f(0) 6= 0.


Definitions

The space P

Let P be the space of all functions f : [1,∞)→ R such that thereexists p ∈ N with ∫ ∞

1|f(x)|2 dx

xp+1<∞.

In particular, any restriction of a polynomial function into [1,∞)belongs to P. Thinking of [1,∞) as a semi-group with respect tothe multiplication of the real numbers, and since the Haar measureof ((0,∞), ·) is dx

x = d(ln(x)), we obtain the following convolution

(f ∗ g)(x) =

∫ x

1f(y)g

(x

y

)dy

y, ∀x ∈ [1,∞).


Definitions

The space P

We also note that for any p > q,∫ ∞1

dµpdµq

dµ =

∫ ∞1

x−(p+1)

x−(q+1)

dx

x=

1

p− q.

Thus, we obtain that

Theorem (The space P)

The space P is a strong convolution algebra. More preciesly, forany f ∈ L2([1,∞), dx

xq+1 ) and g ∈ L2([1,∞), dxxp+1 ), f ∗ g belongs

to L2([1,∞), dxxp+1 ) and

‖f ∗ g‖p ≤ (p− q)−1/2‖f‖q‖g‖p.


Mellin transform

The space P : Mellin transform

The last theorem can be expressed in terms of Mellin transform.

Definition

The one-sided Mellin transform is defined by

(Mf)(s) =

∫ ∞1

xsf(x)dx

x.

Thus, the inequality in the last theorem can be rewritten as

(M(f ∗ g)2)(−t) ≤ 1

t− r(Mf2)(−r)(Mg2)(−t) for any t > r.


Relations to Dirichlet series

The space P : relations to Dirichlet series

Now if we consider the Dirichlet series∞∑n=1

ann−s and

∞∑n=1

bnn−s.

Since for any s in the half-plane of absolute convergence,

∞∑n=1

ann−s = s

M∑n≤y

an

(−s),

we obtain

Corollary

For any r < t it holds that:

∫ ∞1

∫ x

1

√∑n≤y

an∑n≤ x

y

bndy

y

2

dx

xt+1≤ 1

t(t− r)r

∞∑n=1

ann−r

∞∑n=1

bnn−t.


Relations to Dirichlet series

The space P : Dirichlet series

For example, taking the zeta function of Riemann,ζ(s) =

∑∞n=1 n

−s, which converges for Re s > 1, one obtains∫ ∞1

(∫ x

1

√byc⌊x

y

⌋dy

y

)2

dx

xt+1≤ ζ(t)ζ(r)

t(t− r)rfor any 1 < r < t.

Hence,∫ ∞1

(∫ x

1

√byc⌊x

y

⌋dy

y

)2

dx

xt+1≤ inf

r∈(1,t)

(ζ(t)ζ(r)

t(t− r)r

)for any t > 1.

In a similar way, if we take ϕ(n) to be the phi Euler function, weobtain∫ ∞1

∫ x

1

√∑n≤y

ϕ(n)∑n≤x

y

ϕ(n)dy

y

2

dx

xt+1≤ inf

r∈(1,t)

(ζ(t− 1)ζ(r − 1)

tζ(t)(t− r)rζ(r)

).


Definitions

The Kondratiev space of stochastic distributions

Let H be an Hilbert space and let H◦n be the closed subspace ofH⊗n generated by all the vectors of the form

u1 ◦ · · · ◦ un =1

n!

∑σ∈Sn

uσ(1) ⊗ · · · ⊗ uσ(n).

Definition

The full Fock space associated to H is defined by

Γ(H) := ⊕∞n=0H⊗n = C⊕H⊕ (H⊗H)⊕ · · · .

The symmetric Fock space associated to H is defined by

Γ◦(H) := ⊕∞n=0H◦n = C⊕H⊕ (H ◦H)⊕ · · · .


Definitions


The (commutative) white noise space is defined by

W = Γ◦(H0),

where it is conventional to fix H0 = L2(R). Letting (en) be anorthonormal basis of L2(R) (for example, the Hermite functions),the associated orthogonal basis of W can be viewed as follows.Denote by ` is the free commutative monoid generated by N, thatis

` = N(N)0 =

{α ∈ NN

0 : supp(α) is finite}

= ⊕n∈NN0en,

Then for α = (α1, α2, α3, . . .) ∈ `

eα := e◦α11 ◦ e◦α2

2 ◦ e◦α33 ◦ · · ·

is an orthogonal basis of W with ‖eα‖ = α! = α1!α2!α3! · · · .Thus, W is isometrically isomorphic to L2(`, ν), where ν(α) = α!.


Definitions


The Wick product is defined by (f, g) 7→ f ◦ g. In terms of thesecond representation we obtain that

f ◦ g =

(∑α∈`

fαeα

)◦

(∑α∈`

gαeα

)=∑α∈`

∑β≤α

fβgα−β

eα.

Since ` is not compact, in view of Rickert theorem W = L2(`, ν)cannot be closed under the Wick product. Kondratiev was lookingfor an algebra which includes W, he defined

Definition (Y. Kondratiev, 1978)

S−1 = {∑

α∈` fαeα :∑

α∈` |fα|2(2N)−αp <∞ for some p ∈ N}is called the Kondratiev space of stochastic distributions, where(2N)α = 2α1 · 4α2 · 6α3 · · · .

He also showed that this space is closed under the Wick product.We note that S−1 =

⋃p∈N L2(`, µp), where µp(α) = (2N)−αp.


Definitions


In 1992, Zhang showed that

Lemma (Zhang, 1992)∑α∈`(2N)−αd <∞ iff d > 1.

So actually, he showed that∫`dµpdµq

dµ <∞ iff p > q + 1.In 1996, Vage showed that

Theorem (Vage, 1996)

In the space S−1 it holds that,

‖f ◦ g‖p ≤ Ap−q‖f‖q‖g‖p

for any p > q + 1, and for any f ∈ L2(`, µq), g ∈ L2(`, µp), whereA2p−q =

∑α∈`(2N)−α(p−q) is finite due to Zhang.


Definitions


So actually he showed that S−1 is a strong algebra (this could have

been done directly from our theorem, sincedµpdµ are multiplicative).

The fact that S−1 is a strong algebra is a key tool:

- In the theory of stochastic partial differential equations, see:

H. Holden, B. Øksendal, J. Ubøe, and T. Zhang. Stochastic partialdifferential equations. Probability and its Applications, 1996.

- In the theory of stochastic linear systems, see:

D. Alpay and D. Levanony. Linear stochastic systems: a white noiseapproach. Acta Applicandae Mathematicae, 2010.

D. Alpay, D. Levanony, and A. Pinhas. Linear stochastic state spacetheory in the white noise space setting. SIAM Journal of Control andOptimization, 2010.

- In the theory of interpolations, see:

D. Alpay and H. Attia. An interpolation problem for functions with valuesin a commutative ring. Operators Theory: Advances and Applications,2011.


A non commutative version

The Kondratiev space: a non commutative version

The non-commutative white noise space is defined by

W = Γ(H0),

(where again, it’s conventional to fix H0 = L2(R)).

In the same manner as before, if we denote by ˜ the free(non-commutative) monoid generated by N, we obtain that W isisometrically isomorphic to L2(˜, ν), where ν is now the countingmeasure (the α! has now disappeared since we are no longer in thesymmetric case).

The Wick product is now defined by (f, g) 7→ f ⊗ g, and again canbe viewed as a convolution of functions over ˜. Again, since ˜ isnot compact we obtain in view of Rickert theorem that W is notclosed under the Wick product.




When we try to construct the non commutative version of theKondratiev space, we wish that a similar inequality will hold.

Definition (D. Alpay & S, 2012)

The non commutative Kondratiev space of stochastic distributionsis defined by

S1 =⋃p∈N

Γ(Hp),

where Hp = {∑fnen :

∑|fn|2(2n)−p <∞} ⊇ H0 and (en) is an

orthonormal basis of H0 = L2(R).

In the same manner as before, if for any α = zα1i1zα2i2· · · zαnin ∈ ˜ we

denote µp(α) = (2N)α =∏nk=1(2ik)

αk . We obtain, that

S−1 =⋃p∈N

L2(˜, µp).




Proposition (Non commutative Zhang lemma, D. Alpay & S)∑α∈˜(2N)−αd <∞ iff

∑(2n)−d < 1 (i.e. iff d > 1.4649...).

Thus, for any 2 integers p > q + 1 we obtain∫˜dµpdµq

dµ <∞.

Moreover, we note thatdµpdµ are multiplicative. Thus, we conclude

Corollary

S−1 is a strong (convolution) algebra. More precisely,

‖f ⊗ g‖p ≤ Bp−q‖f‖q‖g‖p

for any p > q + 1, and for any f ∈ L2(`, µq), g ∈ L2(`, µp), whereB2p−q =

∑α∈˜(2N)−α(p−q) <∞.




More generally, in case S is discrete, anddµpdµ = (dµ1dµ )p, it holds

that

Theorem (D. Alpay & S, 2012)⋃p∈N Γ(L2(S, µp)) with the multiplication ⊗ is a strong algebra if

and only if⋃p∈N L2(S, µp) is nuclear.

(In the case of S−1, S = N, µp(n) = (2n)p,⋃p∈N L2(S, µp) = s′

which is nuclear, and thus S−1 =⋃p∈N Γ(L2(S, µp)) is a SA).

The fact that the non commutative version of the Kondratievspace of stochastic distributions is a strong algebra, gives rise to atheory of non commutative stochastic linear systems, and to noncommutative versions of the other applications which weredescribed before.

But this will be done in a future work...



Further reading

For further reading on strong algebras, see:

D. Alpay and G. Salomon. A new family of C-algebras withapplications in linear systems. Infinite Dimensional Analysis,Quantum Probability and Related Topics, 2012.

D. Alpay and G. Salomon. Topological convolution algebraspreprint on arXiv, 2012.

Thank you very much!!!

topological convolution algebras - bgu

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