On Fréchet algebras with the dominating norm property

Transcription

1 On Fréchet algebras with the dominating norm property Tomasz Ciaś Faculty of Mathematics and Computer Science Adam Mickiewicz University in Poznań Poland Banach Algebras and Applications Oulu, July 3 11, 2017 Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

2 Property (DN) Definition A Fréchet space E with a fundamental sequence ( q ) q N of seminorms has the property (DN) if there is a continuous norm on E such that q N r N, C > 0 x X x 2 q C x x r. Every norm with this property is called a dominating norm on E. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

3 Property (DN) examples Example Let H(C) be the Fréchet space 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

4 Property (DN) examples Example Let H(C) be the Fréchet space 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). Example The Fréchet space of rapidly decreasing sequences is defined as s := { ( ) 1/2 } ξ C N : q N 0 ξ q := (ξ j j q ) j N l2 = ξ j 2 j 2q <, 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 l2 is a dominating norm on s. j=1 Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

5 Property (DN) examples The following Fréchet spaces (over C) are isomorphic to s: the space C (Ω) of smooth functions with uniformly continuous partial derivatives on open, bounded set Ω R n with Lipschitz boundary, e.g. C [ 1, 1], C (D) [we use Sobolev extension operator constructed by E. Stein in 1966]; Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

6 Property (DN) examples The following Fréchet spaces (over C) are isomorphic to s: the space C (Ω) of smooth functions with uniformly continuous partial derivatives on open, bounded set Ω R n 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

7 Property (DN) examples The following Fréchet spaces (over C) are isomorphic to s: the space C (Ω) of smooth functions with uniformly continuous partial derivatives on open, bounded set Ω R n 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]; the Schwartz space S(R n ) of smooth rapidly decreasing functions on R n. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

8 Closed subspaces of s N and s Definition A Fréchet space E is called nuclear if every unconditionally convergent series in E is absolutely convergent. Theorem (Kōmura-Kōmura 1966) For a Fréchet space E the following assertions are equivalent: 1 E is nuclear; 2 E is isomorphic to some closed subspace of s N. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

9 Closed subspaces of s N and s Definition A Fréchet space E is called nuclear if every unconditionally convergent series in E is absolutely convergent. Theorem (Kōmura-Kōmura 1966) For a Fréchet space E the following assertions are equivalent: 1 E is nuclear; 2 E is isomorphic to some closed subspace of s N. Theorem (Vogt 1977) For a Fréchet space 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

10 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 (p n,b ) a L (D),B B, { p n,b := max sup ξ B } axξ, sup ax ξ. ξ B D(x ) := {η H : ζ H ξ D xξ, η = ξ, ζ }, 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. Kürsten, G. Lassner, K. Schmüdgen and others. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

11 O -algebras Example L (l 2) = B(l 2) Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

12 O -algebras Example L (l 2) = B(l 2) Example L (s) := {unbdd ops x on l 2 : D(x) = s, x(s) s, s D(x ) and x (s) s} Here, the graph topology coincides with the Fréchet space topology on s, and thus the topology τ is given by the seminorms (p n,b ) n N0,B B, { p n,b := max sup xξ n, sup x ξ n }. ξ B ξ B B is the class of all bounded subsets of s ξ 2 n := j=1 ξ j 2 j 2n for ξ s Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

13 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

14 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 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

15 Reperesentations of L (s) Theorem (TC & K. Piszczek, 2017) The topological -algebra L (s) is isomorphic to: 1 L(s) L(s ) 2 the multiplier algebra of L(s, s) (formally defined via the so-called double centralizers) 3 the matrix algebra { x = (x ij ) C N2 : N N n N i,j N 2 x ij max { i N j n, j N } } i n <. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

16 Closed -subalgebras of B(l 2 ) and L (s) closed -subalgebras of B(l 2 ) isometric -isomorphism separable C -algebras Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

17 Closed -subalgebras of B(l 2 ) and L (s) closed -subalgebras of B(l 2 ) isometric -isomorphism separable C -algebras closed -subalgebras of L (s) = of topological -algebras abstract description??? Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

18 Closed commutative -subalgebras of B(l 2 ) and L (s) closed commutative -subalgebras of B(l 2 ) with Id C(K), K compact Hausdorff metrizable space separable commutative C -algebras with 1 Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

19 Closed commutative -subalgebras of B(l 2 ) and L (s) closed commutative -subalgebras of B(l 2 ) with Id closed commutative -subalgebras of L (s) with Id C(K), K compact Hausdorff metrizable space list of topological -algebras??? separable commutative C -algebras with 1 abstract description??? Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

20 Hilbert algebras Definition A -algebra E with unit and a Hilbert norm := (, ) 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 m x : (E, ) (E, ), m x (y) := xy, are continuous. A left Hilbert algebra (E, ) is called a Hilbert algebra if for all x, y E. (y, x ) = (x, y) Hilbert algebras are considered in the context of von Neumann algebras. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

21 DN-algebras Definition A Fréchet -algebra E with unit and a dominating Hilbert norm is called a DN-algebra if (E, ) is a Hilbert algebra. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

22 DN-algebras Definition A Fréchet -algebra E with unit and a dominating Hilbert norm is called a DN-algebra if (E, ) is a Hilbert algebra. complemented commutative Fréchet -subalgebras of L (s) with Id and Schauder basis, contained in B(l 2 ) = C (Ω) where Ω R n open, bounded with Lipschitz boundary,... = commutative DN-algebras isomorphic as Fréchet spaces to complemented subspaces of s with Schauder basis Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

23 DN-algebras and subalgebras of L (s) Theorem (TC 2017) Let E be a commutative Fréchet -algebra with unit and isomorphic as a Fréchet 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 l 2 ; 2 (E, ) is a DN-algebra for some norm. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

24 DN-algebras and subalgebras of L (s) Theorem (TC 2017) Let E be a commutative Fréchet -algebra with unit and isomorphic as a Fréchet 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 l 2 ; 2 (E, ) is a DN-algebra for some norm. 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 l 2. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

25 (E, ) is a DN-algebra, E = s E L (s) B(l 2 ) Theorem (Vogt 2013) Let E be a Fréchet space isomorphic to s. Then for every dominating Hilbert norm on E there is an isomorphism u : E s such that uξ l2 = ξ for all ξ E. Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

26 (E, ) is a DN-algebra, E = s E L (s) B(l 2 ) Theorem (Vogt 2013) Let E be a Fréchet space isomorphic to s. Then for every dominating Hilbert norm on E there is an isomorphism u : E s such that uξ l2 = 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 Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

27 (E, ) is a DN-algebra, E = s E L (s) B(l 2 ) Theorem (Vogt 2013) Let E be a Fréchet space isomorphic to s. Then for every dominating Hilbert norm on E there is an isomorphism u : E s such that uξ l2 = 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). E x m x {m x} x E L (E) x uxu 1 L (s) Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

28 (E, ) is a DN-algebra, E = s E L (s) B(l 2 ) Theorem (Vogt 2013) Let E be a Fréchet space isomorphic to s. Then for every dominating Hilbert norm on E there is an isomorphism u : E s such that uξ l2 = 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). E x m x {m x} x E L (E) x uxu 1 L (s) Since m x : (E, ) (E, ) are continuous, um xu 1 B(l 2). Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

29 Examples of DN-algebras which can be embedded into L (s) as complemented -subalgebras contained in B(l 2 ) C (Ω), f 2 := Ω f (x) 2 dx, where Ω R n open, bounded with Lipschitz boundary C (M), f 2 := M f (x) 2 dv, where M compact smooth manifold without boundary and dv is a volume form associated to a fixed Riemannian metric S(R) C1, f + λ 2 := π 2 f (tan x) + λ 2 dx π 2 s C1, x + λ 2 := j=1 x j + λ 2 j 2 Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17

30 Kiitos huomiostanne! Thank for your attention! Tomasz Ciaś (A. Mickiewicz University in Poznań) On Fréchet algebras with (DN) BAA / 17