On Fréchet Algebras with the Dominating Norm Property

On Fr´echetalgebras with the dominating norm property

Tomasz Cia´s

Faculty of Mathematics and Computer Science Adam Mickiewicz University in Pozna´n Poland

Banach Algebras and Applications Oulu, July 3–11, 2017

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 1 / 17 Property (DN)

Deﬁnition

A Fr´echetspace E with a fundamental sequence (|| · ||q)q∈N of seminorms has the property (DN) if there is a continuous norm || · || on E such that

2 ∀q ∈ N ∃r ∈ N, C > 0 ∀x ∈ X ||x||q ≤ C||x|| ||x||r . Every norm || · || with this property is called a dominating norm on E.

Tomasz Cia´s (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA 2017 2 / 17 Example The Fréchetspace of rapidly decreasing sequences is defined as

∞ 1/2 N q X 2 2q s := ξ ∈ C : ∀q ∈ N0 |ξ|q := ||(ξj · j )j∈N||`2 = |ξj | j < ∞ , j=1

where the topology is determined by the sequence (| · |q)q∈N0 of norms. By the Cauchy-Schwartz inequality, the space s has the property (DN) and the norm

|| · ||`2 is a dominating norm on s.

Property (DN) – examples

Example Let H(C) be the Fr´echetspace of entire functions with the topology given by the sequence (|| · ||q)q∈N of norms, ||f ||q := sup|z|≤q |f (z)|. By Hadamard’s three circle theorem, the space H(C) has the property (DN) and the norm || · ||1 is already a dominating norm on H(C).

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 3 / 17 Property (DN) – examples

Example The Fr´echetspace of rapidly decreasing sequences is deﬁned as

∞ 1/2 N q X 2 2q s := ξ ∈ C : ∀q ∈ N0 |ξ|q := ||(ξj · j )j∈N||`2 = |ξj | j < ∞ , j=1

where the topology is determined by the sequence (| · |q)q∈N0 of norms. By the Cauchy-Schwartz inequality, the space s has the property (DN) and the norm

|| · ||`2 is a dominating norm on s.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 3 / 17 the space C ∞(M) of smooth functions on a compact smooth manifold M without boundary [Z. Ogrodzka 1967, M. Valdivia 1980]; the Schwartz space S(Rn) of smooth rapidly decreasing functions on Rn.

Property (DN) – examples

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 4 / 17 the Schwartz space S(Rn) of smooth rapidly decreasing functions on Rn.

Property (DN) – examples

The following Fr´echetspaces (over C) are isomorphic to s: the space C ∞(Ω) of smooth functions with uniformly continuous partial derivatives on open, bounded set Ω ⊂ Rn with Lipschitz boundary, e.g. C ∞[−1, 1], C ∞(D) [we use Sobolev extension operator constructed by E. Stein in 1966]; the space C ∞(M) of smooth functions on a compact smooth manifold M without boundary [Z. Ogrodzka 1967, M. Valdivia 1980];

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 4 / 17 Property (DN) – examples

Tomasz Cia´s (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA 2017 4 / 17 Theorem (Vogt 1977) For a Fréchetspace E the following assertions are equivalent: 1 E is nuclear and has the property (DN); 2 E is isomorphic to some closed subspace of s.

Closed subspaces of sN and s

Deﬁnition A Fr´echetspace E is called nuclear if every unconditionally convergent series in E is absolutely convergent.

Theorem (K¯omura-K¯omura1966) For a Fr´echetspace E the following assertions are equivalent: 1 E is nuclear; 2 E is isomorphic to some closed subspace of sN.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 5 / 17 Closed subspaces of sN and s

Deﬁnition A Fr´echetspace E is called nuclear if every unconditionally convergent series in E is absolutely convergent.

Theorem (K¯omura-K¯omura1966) For a Fr´echetspace E the following assertions are equivalent: 1 E is nuclear; 2 E is isomorphic to some closed subspace of sN.

Theorem (Vogt 1977) For a Fr´echetspace E the following assertions are equivalent: 1 E is nuclear and has the property (DN); 2 E is isomorphic to some closed subspace of s.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 5 / 17 O∗-algebras

Deﬁnition (G. Lassner 1972) Let D be a dense linear subspace of a Hilbert spaces H. We deﬁne

L∗(D) := {unbounded operators x on H : D(x) = D, x(D) ⊂ D D ⊂ D(x ∗) and x ∗(D) ⊂ D}

∗ with the locally convex topology τ given by the seminorms (pn,B )a∈L∗(D),B∈B, ∗ pn,B := max sup ||axξ||, sup ||ax ξ|| . ξ∈B ξ∈B

D(x ∗) := {η ∈ H : ∃ζ ∈ H ∀ξ ∈ D hxξ, ηi = hξ, ζi}, x ∗η := ζ for all η ∈ D(x ∗) B is the class of all bounded subsets of D endowed with the graph topology given by the seminorms (|| · ||a)a∈L∗(D), ||ξ||a := ||aξ|| L∗(D) is called the maximal O∗-algebra on D and any ∗-subalgebra of L∗(D) is called an O∗-algebra. O∗-algebras were investigated extensively by K.-D. K¨ursten,G. Lassner, K. Schm¨udgen and others.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 6 / 17 Example

∗ ∗ ∗ L (s) := {unbdd ops x on `2 : D(x) = s, x(s) ⊂ s, s ⊂ D(x ) and x (s) ⊂ s} Here, the graph topology coincides with the Fr´echetspace topology on s, and thus the ∗ topology τ is given by the seminorms (pn,B )n∈N0,B∈B, ∗ pn,B := max sup |xξ|n, sup |x ξ|n . ξ∈B ξ∈B

B is the class of all bounded subsets of s 2 P∞ 2 2n |ξ|n := j=1 |ξj | j for ξ ∈ s

O∗-algebras

Example

∗ L (`2) = B(`2)

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 7 / 17 O∗-algebras

Example

∗ L (`2) = B(`2)

Example

B is the class of all bounded subsets of s 2 P∞ 2 2n |ξ|n := j=1 |ξj | j for ξ ∈ s

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 7 / 17 As a ∗-algebra with locally convex topology noncommutative topological ∗-algebra (multiplication is separately continuous, involution is continuous) the identity map is the unit not a Q-algebra (the set of invertible elements is not open) not locally m-convex (there is no fundamental system of multiplicative seminorms)

Properties of L∗(s)

As a topological vector space: locally convex, complete, nuclear, ultrabornological, PLS-space one may apply Hahn-Banach theorem, closed graph theorem, open mapping theorem, uniform boundedness principle

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 8 / 17 Properties of L∗(s)

As a ∗-algebra with locally convex topology noncommutative topological ∗-algebra (multiplication is separately continuous, involution is continuous) the identity map is the unit not a Q-algebra (the set of invertible elements is not open) not locally m-convex (there is no fundamental system of multiplicative seminorms)

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 8 / 17 Reperesentations of L∗(s)

Theorem (TC & K. Piszczek, 2017) The topological ∗-algebra L∗(s) is isomorphic to: 1 L(s) ∩ L(s0) 2 the multiplier algebra of L(s0, s) (formally deﬁned via the so-called double centralizers) 3 the matrix algebra

N N 2 X i j x = (x ) ∈ N : ∀N ∈ ∃n ∈ |x | max , < ∞ . ij C N N ij j n i n 2 i,j∈N

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 9 / 17 closed =∼ of topological ∗-algebras abstract ∗-subalgebras description??? of L∗(s)

∗ ∗ Closed -subalgebras of B(`2) and L (s)

isometric ∗-isomorphism ∗ ∗ closed -subalgebras of B(`2) separable C -algebras

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 10 / 17 ∗ ∗ Closed -subalgebras of B(`2) and L (s)

isometric ∗-isomorphism ∗ ∗ closed -subalgebras of B(`2) separable C -algebras

closed =∼ of topological ∗-algebras abstract ∗-subalgebras description??? of L∗(s)

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 10 / 17 closed list of commutative abstract topological ∗-subalgebras of description??? ∗-algebras??? L∗(s) with Id

∗ ∗ Closed commutative -subalgebras of B(`2) and L (s)

closed C(K), separable commutative K compact commutative ∗-subalgebras of Hausdorﬀ C ∗-algebras B(`2) with Id metrizable space with 1

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 11 / 17 ∗ ∗ Closed commutative -subalgebras of B(`2) and L (s)

closed C(K), separable commutative K compact commutative ∗-subalgebras of Hausdorﬀ C ∗-algebras B(`2) with Id metrizable space with 1

closed list of commutative abstract topological ∗-subalgebras of description??? ∗-algebras??? L∗(s) with Id

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 11 / 17 Hilbert algebras

Deﬁnition A ∗-algebra E with unit and a Hilbert norm || · || := p(·, ·) is called a left Hilbert algebra if (xy, z) = (y, x ∗z) for all x, y, z ∈ E and for all x ∈ E there is C > 0 such that

||xy|| ≤ C||y||

for all y ∈ E, i.e. the left multiplication maps mx :(E, || · ||) → (E, || · ||), mx (y) := xy, are continuous. A left Hilbert algebra (E, || · ||) is called a Hilbert algebra if

(y ∗, x ∗) = (x, y)

for all x, y ∈ E.

Hilbert algebras are considered in the context of von Neumann algebras.

Tomasz Cia´s (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA 2017 12 / 17 commutative complemented com- ∞ mutative Fréchet C (Ω) where DN-algebras n isomorphic as ∗-subalgebras of =∼ Ω ⊂ R open, =∼ L∗(s) with Id and bounded with Fréchetspaces Schauder basis, Lipschitz to complemented contained in B(` ) boundary,. . . subspaces of s with 2 Schauder basis

DN-algebras

Deﬁnition A Fr´echet ∗-algebra E with unit and a dominating Hilbert norm || · || is called a DN-algebra if (E, || · ||) is a Hilbert algebra.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 13 / 17 DN-algebras

Deﬁnition A Fr´echet ∗-algebra E with unit and a dominating Hilbert norm || · || is called a DN-algebra if (E, || · ||) is a Hilbert algebra.

commutative complemented com- ∞ mutative Fr´echet C (Ω) where DN-algebras n isomorphic as ∗-subalgebras of =∼ Ω ⊂ R open, =∼ L∗(s) with Id and bounded with Fr´echetspaces Schauder basis, Lipschitz to complemented contained in B(` ) boundary,. . . subspaces of s with 2 Schauder basis

Tomasz Cia´s (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA 2017 13 / 17 Theorem (TC 2017) Let E be a not necessarily commutative Fréchet ∗-algebra with unit and isomorphic as a Fréchet space to a complemented subspace of s with Schauder basis. If (E, || · ||) is a DN-algebra for some norm || · ||, then E is isomorphic to a ∗ ∗ complemented -subalgebra of L (s) consisting of bounded operators on `2.

DN-algebras and subalgebras of L∗(s)

Theorem (TC 2017) Let E be a commutative Fr´echet ∗-algebra with unit and isomorphic as a Fr´echet space to a complemented subspace of s with Schauder basis. Then TFAE: 1 E is isomorphic to a complemented ∗-subalgebra of L∗(s) consisting of bounded operators on `2; 2 (E, || · ||) is a DN-algebra for some norm || · ||.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 14 / 17 DN-algebras and subalgebras of L∗(s)

Theorem (TC 2017) Let E be a not necessarily commutative Fr´echet ∗-algebra with unit and isomorphic as a Fr´echet space to a complemented subspace of s with Schauder basis. If (E, || · ||) is a DN-algebra for some norm || · ||, then E is isomorphic to a ∗ ∗ complemented -subalgebra of L (s) consisting of bounded operators on `2.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 14 / 17 If H is the completion of the pre-Hilbert space (E, || · ||) then

L∗(E) := {unbounded operators x on H : ...}.

Φ: L∗(E) → L∗(s), Φx := uxu−1 L∗(E) =∼ L∗(s) as topological ∗-algebras Aim: E is isomorphic to a complemented ∗-subalgebra of L∗(E).

−1 x7→mx ∗ x7→uxu ∗ E / {mx }x∈E / L (E) / L (s)

−1 Since mx :(E, || · ||) → (E, || · ||) are continuous, umx u ∈ B(`2).

∼ ∗ (E, || · ||) is a DN-algebra, E = s ⇒ E ,→ L (s) ∩ B(`2)

Theorem (Vogt 2013) Let E be a Fr´echetspace isomorphic to s. Then for every dominating Hilbert norm || · ||

on E there is an isomorphism u : E → s such that ||uξ||`2 = ||ξ|| for all ξ ∈ E.

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 15 / 17 −1 x7→mx ∗ x7→uxu ∗ E / {mx }x∈E / L (E) / L (s)

−1 Since mx :(E, || · ||) → (E, || · ||) are continuous, umx u ∈ B(`2).

∼ ∗ (E, || · ||) is a DN-algebra, E = s ⇒ E ,→ L (s) ∩ B(`2)

Theorem (Vogt 2013) Let E be a Fr´echetspace isomorphic to s. Then for every dominating Hilbert norm || · ||

on E there is an isomorphism u : E → s such that ||uξ||`2 = ||ξ|| for all ξ ∈ E.

If H is the completion of the pre-Hilbert space (E, || · ||) then

L∗(E) := {unbounded operators x on H : ...}.

Φ: L∗(E) → L∗(s), Φx := uxu−1 L∗(E) =∼ L∗(s) as topological ∗-algebras Aim: E is isomorphic to a complemented ∗-subalgebra of L∗(E).

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 15 / 17 −1 Since mx :(E, || · ||) → (E, || · ||) are continuous, umx u ∈ B(`2).

∼ ∗ (E, || · ||) is a DN-algebra, E = s ⇒ E ,→ L (s) ∩ B(`2)

Theorem (Vogt 2013) Let E be a Fr´echetspace isomorphic to s. Then for every dominating Hilbert norm || · ||

on E there is an isomorphism u : E → s such that ||uξ||`2 = ||ξ|| for all ξ ∈ E.

If H is the completion of the pre-Hilbert space (E, || · ||) then

L∗(E) := {unbounded operators x on H : ...}.

Φ: L∗(E) → L∗(s), Φx := uxu−1 L∗(E) =∼ L∗(s) as topological ∗-algebras Aim: E is isomorphic to a complemented ∗-subalgebra of L∗(E).

−1 x7→mx ∗ x7→uxu ∗ E / {mx }x∈E / L (E) / L (s)

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 15 / 17 ∼ ∗ (E, || · ||) is a DN-algebra, E = s ⇒ E ,→ L (s) ∩ B(`2)

Theorem (Vogt 2013) Let E be a Fr´echetspace isomorphic to s. Then for every dominating Hilbert norm || · ||

on E there is an isomorphism u : E → s such that ||uξ||`2 = ||ξ|| for all ξ ∈ E.

If H is the completion of the pre-Hilbert space (E, || · ||) then

L∗(E) := {unbounded operators x on H : ...}.

Φ: L∗(E) → L∗(s), Φx := uxu−1 L∗(E) =∼ L∗(s) as topological ∗-algebras Aim: E is isomorphic to a complemented ∗-subalgebra of L∗(E).

−1 x7→mx ∗ x7→uxu ∗ E / {mx }x∈E / L (E) / L (s)

−1 Since mx :(E, || · ||) → (E, || · ||) are continuous, umx u ∈ B(`2).

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 15 / 17 Examples of DN-algebras which can be embedded into ∗ ∗ L (s) as complemented -subalgebras contained in B(`2)

∞ 2 R 2 n C (Ω), ||f || := Ω |f (x)| dx, where Ω ⊂ R open, bounded with Lipschitz boundary ∞ 2 R 2 C (M), ||f || := M |f (x)| dV , where M compact smooth manifold without boundary and dV is a volume form associated to a ﬁxed Riemannian metric π 2 R 2 2 S(R) ⊕ C1, ||f + λ|| := π |f (tan x) + λ| dx − 2 2 P∞ 2 −2 s ⊕ C1, ||x + λ|| := j=1 |xj + λ| j

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 16 / 17 Kiitos huomiostanne!

Thank for your attention!

Tomasz Cia´s (A. Mickiewicz University in Pozna´n) On Fr´echet algebras with (DN) BAA 2017 17 / 17