Dual and annihilator topological algebras in H …pcma12/talks/Charalabidou.pdf · Dual and annihilator topological algebras in H-structures Marina Haralampidou University of Athens,

Dual and annihilator topologicalalgebras in H-structures

Marina Haralampidou

University of Athens, HELLAS

e-mail:[email protected]

PATRAS, May 18-19, 2012 () Dual and annihilator topological algebras in H-structures 1 / 22

This presentation contains a part of the re-sults, announced in the First International Sym-posium on Algebra, Topology and Topologi-cal algebras (ICATTA 2011), organized by Uni-versidad Autonoma Metropolitana Iztapalapa,in Veracruz, Mexico. Publication: “Interre-lations between annihilator, dual and pseudo-H-algebras”, Communications in Mathematicsand Applications. Special Issue on Algebra,Topology and Topological Algebras, Veracruz,3(2012), no 1, 25-38.


ABSTRACTThe annihilator operators play an important role in Wedderburndecompositions of pseudo-H-algebras. Through them the notions ofannihilator and dual topological algebra are defined. A dual algebra isannihilator. The converse is not in general true. Our aim is topresent conditions that through H-(topological) structures, thenotions dual and annihilator coincide.


Basic Definitions

Throughout all vector spaces and algebras are taken over the field ofcomplexes.

Definition

A pseudo H-space is a vector space E equipped with a family(<,>α)α∈A of positive semi-definite (:pseudo-) inner productssuch that the induced topology makes E into a locally convexspace.A pseudo-H-algebra is a pseudo H-space and an algebra(which is locally convex) with separately continuousmultiplication (or even m-convex).


The topology of a pseudo-H-algebra E is defined by a family (pα)α∈Aof seminorms so that pα(x) =< x , x >

1/2α for every x ∈ E . The

“m-convex” case will be referred each time it is used, otherwise, weshall always employ the locally convex case for the termpseudo-H-algebra. Such a topological algebra will be denoted by

(E , <,>α)α∈A.

The orthogonal S⊥ of a non-empty subset S in E is defined by

S⊥ = {x ∈ E :< x , y >α= 0 for every y ∈ S , α ∈ A} (1)

In the sequel, we denote by Ml(E ) the set of all closed maximalregular left ideals of a topological algebra E (separately continuousmultiplication).


Definition

A pseudo-H-algebra E is called

(i) left modular complemented H-algebra if it satisfies theconditions:

Any left or right ideal I in E with I⊥ = (0) is dense in E (2)

(the density property).

⋂M∈Ml (E)

M = (0), and M⊥is a left ideal for each M ∈Ml(E )

(3)(the intersection property).

(ii) properly left precomplemented H-algebra if

E = M⊕⊥M⊥for every maximal regular left ideal M of E . (4)

(iii) anti-properly left precomplemented H-algebra if

E = I ⊕⊥ I⊥for every minimal left ideal I of E . (5)


Peirce property

We shall say that a pseudo-H-algebra E has the Peirce property(on the left (right)) if it satisfies the condition:

If x is a right (left) unit for E modulo

a maximal regular left (right) ideal M of E ,

then x ∈ M⊥, and M⊥ is a left (right) ideal .

The last terminology is justified by the fact that the Peirce propertyleads to the Peirce decomposition for Hausdorff pseudo-H-algebrasE. Namely, E = E (1− x)⊕ Ex (resp. E = (1− x)E ⊕ xE ), whereE (1− x) = {y − yx : y ∈ E} (resp. (1− x)E = {y − xy : y ∈ E}).


Annihilator and dualtopological algebrasLet E be an algebra. If (∅ 6=)S ⊆ E , Al(S) (resp. Ar (S)) denotesthe left (right) annihilator of S . Further, Ll (resp. Lr ) denotes theset of all closed left (right) ideals in a topological algebra E .

DefinitionA topological algebra E is said to be an annihilator algebra, if it ispreannihilator (viz. Al(E ) = Ar (E ) = (0)) with Ar (I ) 6= (0) forevery I ∈ Ll , I 6= E and Al(J) 6= (0) for every J ∈ Lr , J 6= E .

A topological algebra E satisfying Al(Ar (I )) = I for all I ∈ Ll andAr (Al(J)) = J for all J ∈ Lr is called a dual algebra.


Example

Consider the normed algebra AP(G ) of all almost periodic functionson a topological group G the underlying linear space being apre-Hilbert space. Then every (not necessarily closed) 2-sided ideal inAP(G ) is a dual algebra.

Example

The (Arens) algebra Lω is a real unital commutative semisimpleannihilator (topological) algebra.


Characterizations of annihilator algebrasover certain properly and anti-properlyprecomplemented H−algebras

Let (E , <,>α)α∈A be a pseudo-H-algebra. An element x l is a leftadjoint of x ∈ E if < xy , z >α=< y , x lz >α for all y , z ∈ E , α ∈ A.x l is unique, if there exists. If the algebra E is preannihilator andx 6= 0, then x l 6= 0. A right adjoint is defined, analogously.

Theorem

Let E be a (Jacobson) semisimple Hausdorff anti-properly, andproperly precomplemented H-algebra with continuous quasi-inversion,satisfying the density property. Then the following are equivalent:(1) E is an annihilator algebra.(2) E is a left annihilator algebra.(3) Every element in the socle SE of E has a left adjoint.(4) Every nonzero right ideal of E contains an element with leftadjoint.


Steps of the proof

(1)⇒ (2) It is obvious by the very definitions.(2)⇒ (3):

If I is a minimal right ideal of E , I⊥ is a closed maximal rightideal.Al(I⊥) 6= (0) thenI⊥ = (1− x)E ≡ {y − xy : y ∈ E} with x a minimal idempotentelement in E .I = xE .The elements in xE have left adjoints, a f o r t i o r ithis will be true for the elements in SE .


Steps of the proof, cont.

(3)⇒ (4): Let J be a right ideal of E .

It is proved E = SE henceJ contains an element with a left adjoint.

(4)⇒ (2): Let I be a proper closed right ideal of E .

By the density property, I⊥ 6= (0).By hypothesis, I⊥ has a nonzero element, say x , with a leftadjoint x l . This leads toAl(I ) 6= (0). Actually, x l I = (0).



(2)⇒ (1): The annihilation property is proved on the right.

Consider a minimal left ideal J of E ; by a result mentionedbefore any element of J has a right adjoint.The same is true for any element in SE , which by the firstWedderburn structure theorem for properly precomplementedH-algebras, is dense in E .The continuity of the pseudo-inner product in both variables, theseparate continuity of multiplication in E and the density of thesocle imply < Ex , z >α= {0} for all z in a proper closed leftideal I of E and 0 6= x ∈ I⊥. Finally,0 6= x r ∈ Ar (I ), where x r is the right adjoint of x . Hence theassertion.


Every dual algebra is an annihilator one

The converse is not in general true even in the normed case;However, we do know special cases for which the converse is alsotrue. For no-normed contexts, we have proved thatA Hausdorff locally C ∗-algebra is annihilator if and only if, it is dual,if and only if, it is complemented with left (right) complementor⊥l = ∗ ◦ Ar . (resp. ⊥r = ∗ ◦ Al).

In the sequel, we shall give one more no-normed framework in whichthe two classes of “annihilator” and “dual” algebras coincide.


In the sequel, we employ a class of algebras we introduced in 1994, inconnection with annihilator (non-normed) topological algebras. Wecalled them deep algebras.A deep algebra is an algebra in which every non-zero left (right)ideal contains a minimal left (right) ideal.

Example

The locally C ∗-algebra Cc(X ) of all C-valued continuous functions ona discrete space X , with the standard algebraic-topological structure,is a deep algebra.


A (left, right) precomplemented H-algebra is apseudo-H-algebra E , such that

E = I ⊕ I⊥for every closed left (resp. right) ideal I of E .

Theorem

Let E be a semisimple Hausdorff precomplemented, and anti-properlyprecomplemented H-algebra with continuous quasi-inversion.Moreover, suppose that E has the Peirce and the density properties.Then the following are equivalent:(1) E is a dual algebra.(2) E is a annihilator algebra.


Steps of the proof

Any semisimple Hausdorff pseudo H-algebra, under the Peirceproperty, is a properly precomplemented H-algebra, and modularannihilator one.We only have to prove that (2) ⇒ (1):By the first structure theorem, E has a dense socle and it is aQ ′-algebra (viz. any maximal regular left (right) ideal is closed).E as a semisimple annihilator Q ′-algebra is a deep one.Therefore, for a proper closed left ideal I , I⊥ contains a minimalleft ideal, say L.We have I ⊆ L⊥ and L⊥ is a closed maximal regular left ideal.



Put S =⋂

M , the intersection is taken over all closed maximalregular left ideals of E containing I , which by the previousargument is non-empty. Then I ⊆ S .Using among others the precomplementation, the deepness of Eand the Peirce property, we get I = S = ∩M ⊆ M .Each M has the form M = E (1− x), x ∈ Id(E ), from whichM = Al(Ar (M)).I ⊆ Al(Ar (I )) ⊆ Al(Ar (M)) = M for all M ’s.Thus, I ⊆ Al(Ar (I )) ⊆ ∩M = S = I ,and I = Al(Ar (I )).Similarly, E is a right dual algebra.


A class of pseudo-Hilbert algebras, whichare dual algebras

We introduce a generalization of Hilbert algebras in the followingsense.

Definition

An algebra E is called a pseudo-Hilbert algebra if it is apseudo-H-space equipped with an involution x → x∗ having theproperties:

< xy , z >α=< y , x∗z >α for all x , y , z ∈ E .< x , y >α=< y ∗, x∗ >α for all x , y ∈ E .The left multiplication is continuous.The set {xy , x , y ∈ E} is dense in E .


Theorem

Every Hausdorff orthocomplemented pseudo-Hilbert algebra E is adual algebra.

Its proof is based on the following facts:

Any pseudo-Hilbert algebra is proper.Proposition.- Let E be a proper algebra and I a left ideal of Esuch that there exists a left ideal I ′ of E with E = I ⊕ I ′. IfEx ⊆ I (resp. Ex ⊆ I ′) for some x ∈ E , then x ∈ I (resp.x ∈ I ′).Take a (closed) left ideal I which is orthocomplemented, then weprove that Ex ⊆ I (resp. Ey ⊆ I⊥) for some x ∈ E (resp.y ∈ E ), and thus x ∈ I (resp. y ∈ I⊥).For any Hausdorff pseudo-Hilbert algebra E , and any (closed)left (resp. right) orthocomplemented ideal I (resp. J) of E , weget Ar (I ) = (I⊥)∗ (resp. Al(J) = (J⊥)∗).


Example

Let I be an arbitrary set of elements. Consider the set CI×I of allcomplex-valued functions a on I × I , such that

∑i ,j |a(i , j)|2 ∈ R+.

The latter, endowed with “point-wise” defined operations becomes avector space and an algebra with “matrix” multiplication

(ab)(i , j) =∑k

a(i , k)b(k , j),

for all a, b ∈ CI×I . Take a family of real numbers (tα)α∈Λ, such thattα ≥ 1. For each α ∈ Λ, the mapping 〈·, ·〉α : CI×I ×CI×I → C givenby

〈a, b〉α = tα∑i ,j

a(i , j)b(i , j)

defines a pseudo-inner product on CI×I , where “−” denotes complexconjugation. Thus A ≡ (CI×I , (〈·, ·〉α)α∈Λ) becomes a pseudo-Hilbertalgebra with an involution given by a∗(i , j) = a(j , i).


References

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[4] M. Haralampidou, On complementing topological algebra.J. Math. Sci. 96(1999), 3722-3734.

[5] M. Haralampidou, Dual complementors in topologicalalgebras, Banach Center Publications, Institute of Math. PolishAcademy of Sci. 67(2005), 219-233.

[6] T. Husain and P.K. Wong, Modular complemented andannihilator algebras. Proc. Amer. Math. Soc. No 2 34(1972),457-462.

[7] A. Inoue, Locally C ∗-algebras. Mem. Faculty Sci. KyushuUniv. (Ser. A) 25(1971), 197–235.

[8] E.A. Michael, Locally multiplicatively-convex topologicalalgebras, Mem. Amer. Math. Soc., 11(1952). (Reprinted 1968).

[9] P.P. Saworotnow, On the imbedding of a rightcomplemented algebra into Ambrose’s H∗-algebra, Proc. Amer.Math. Soc. 8(1957), 56-62.[10] B. Yood, Ideals in topological rings. Can. J. Math.16(1964), 28-45.

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Dual and annihilator topological algebras in H …pcma12/talks/Charalabidou.pdf · Dual and annihilator topological algebras in H-structures Marina Haralampidou University of Athens,