The Compactum of a Commutative Semisimple Banach Algebrasfsr.um5.ac.ma/icart2018/pdf/talks/Hadder.pdf · 2018. 7. 14. · the question whether all Banach spaces have the approximation

Plan The Main Result Motivation Preliminaries The Counter Example

The Compactum of a Commutative SemisimpleBanach Algebras

YOUNESS HADDER

CRMEF FesE-mail : [email protected]

ICART Rabat, July 2018


1 The Main Result

2 Motivation

3 Preliminaries

4 The Counter Example


The study of the relationship between certain usual classes ofelements in a given semisimple Banach algebra constitutes animportant research topic.

Notation

Let A be a semisimple complex Banach algebra.Let soc(A) be the socle of A.Let K(A) be the compactum of A.

Examples

A =Mn(C);B(H);B(X ); C0(K ); C ∗-algebra ...If A =Mn(C) then soc(A) =Mn(C) .If A = B(X ) then soc(A) = F(X );If A = B(X ) then K(A) = K(X ).

General Problem

For a given class of semisimple complex Banach algebra ”A”, isthe equality K(A) = soc(A) true?


In this talk we will show that

Proposition (Y.H)

There exists a commutative complex unital semisimple Banachalgebra A such that soc(A) 6= K(A).


In the context of our study [N. Boudi and Y. H (2009)] of a linearmaps that preserve generalized invertibility in the commutativecase, we have the following question:

Question

For a given commutative semisimple complex Banach algebra ”A”,is the equality soc(A) = K(A) true?

Theorem

Every compact operator on a Hilbert space H can be approximatedin the norm topology for operators by finite rank operators, ieK(H) = F(H), (ie K(B(H)) = soc(B(H))).

Definition (Per Enflo 1973)

A Banach space X is said to have the approximation property ifK(X ) = F(X ), ie (K(B(X )) = soc(B(X ))).


The Original Problem

The classical approximation problem is

Question (PER ENFLO (1973))

the question whether all Banach spaces have the approximationproperty.

The following is a negative answer to this question

Theorem (PER ENFLO (1973))

There is a Banach space which does not have the approximationproperty, ie there exists a Banach space X such thatK(X ) 6= F(X ), ie (K(B(X )) 6= soc(B(X ))).

Let’s now examine this equality in the context of a ”general”Banach algebra


General Problem

Theorem (Theorem C ∗.1.3; B. A. Barnes, G. J. Murphy, M. R. F.Smyth and T. T. West (1982))

If A is a C ∗-algebra then K(A) = soc(A).

A.H.Al-Moajil proves that in fact, this result is still true for anycommutative complex semisimple Banach algebra :

Theorem (A.H.Al-Moajil (1984))

If A is a semisimple commutative complex Banach algebra. ThenK(A) = Soc(A).

But his argument is not rigorous, knowing that

Remarque

This article is one of the bibliography of the book : T. W. Palmer.Banach Algebras and The General Theory of *-Algebras, Vol. I,Algebras and Banach Algebras, Encyclopedia of Math. Appl. 49,Cambridge Univ. Press, (1994).


Along this talk the letter A denotes a semisimple complex unitalBanach algebra.

Semisimple Algebras

Definition

A is said to be semisimple if rad(A) = {0}.

Remarque

rad(A) = {q ∈ A; q +A−1 ⊂ A−1}.

Examples

The algebra B(X ) of all bounded linear operators on aBanach space X is semisimple.Every C ∗-algebra is semisimple. Then

The commutative algebra C0(K ) of complex-valuedcontinuous functions which vanish at infinity on K issemisimple, where K is a locally compact Hausdorff space.


Socle

Definition (Rickart)

If A contains minimal left ideals, then the sum of all minimalleft ideals is called the socle of A and is denoted by soc(A). IfA does not have minimal left-ideals we define soc(A) = {0}.soc(A) is a two-sided ideal of A and coincides with the sumof all minimal right ideals of A.A minimal idempotent of A is a non-zero idempotent e ∈ Asuch that eAe = Ce . The set of all minimal idempotents ofA denoted by Im(A).

Proposition (Rickart)

A left ideal J of A is minimal if and only if there exists a minimalidempotent e ∈ A such that J = Ae.


Socle

Examples

If A = B(X ) then soc(B(X )) = F(X ) which is the ideal ofrank finite operators on X .If A is, additionally, commutative then soc(A) = {u ∈ A;u =

∑k=nk=1 λkek ; n ∈ N∗; (ek)n≥k≥1 ⊂ Im(A), (λk)n≥k≥1 ⊂

C} .


Khsocle

Definition

kAhA(soc(A)) (or kh(soc(A))) denotes the intersection of allprimitive ideals of A containing soc(A) and is said to be the idealof inessential elements of A.

Remarques (Rickart; Barnes)

kh(soc(A)) = π−1(rad(A/soc(A))) where π is the canonicalquotient map of A onto A/soc(A). Sokh(soc(A)) = {u ∈ A; u + soc(A)) ∈ rad(A/soc(A))}If A is commutative thenkh(soc(A)) = {u ∈ A; rA/soc(A)(u + soc(A)) = 0}Every element x ∈ kh(soc(A)) is inessential, ie, its spectrumis either finite or a sequence converging to zero.


Khsocle

Examples

If A = B(X ) then kh(soc(A)) = In(X ) which is the ideal ofinessential operators on X .If A = B(H) then F(H) = K(H) = In(H).If A is a C ∗-algebra then kh(soc(A)) = soc(A) [LemmaC ∗.2.4; B. A. Barnes, G. J. Murphy, M. R. F. Smyth and T.T. West (1982)].


Compact Elements

Definition ( Alexander (1968))

An element a ∈ A is compact if the operator aTa defined byaTa(x) = axa for all x ∈ A is compact on A . The set of allcompact elements of A is said to be the compactum of A andis denoted by K(A).A is said to be compact if A = K(A).

Remarques

K(A) is not additive in general [ Alexander (1968)];K(A) is closed set of A;AK(A) ⊂ K(A) and K(A)A ⊂ K(A).

Theorem (B. Aupetit and T. Mouton (1994))

soc(A) ⊂ K(A) ⊂ kAhA(soc(A)).


Compact Elements

Examples

If A = B(H), K(B(H)) = K(H) which is the closed ideal ofcompact operators on a Hilbert space H [Vala (1964)].If A = B(X ) then K(B(X )) = K(X ) which is the closed idealof compact operators on a Banach space X [Vala (1964)]. .If A is a C ∗-algebras then K(A) = soc(A) [B. A. Barnes, G.J. Murphy, M. R. F. Smyth and T. T. West (1982)].

In particular if A = C0(K ) then K(C0(K )) = soc(C0(K )).K(X ) is a compact Banach algebra.


Regarding the commutative case a natural question arises

General Problem

For a given commutative semisimple complex Banach algebra A, isthe equality K(A) = soc(A) true?

The answer is negative , indeed :


General Problem

The following is the main result of this talk

Proposition (Y.Hadder)

There exists a commutative complex unital semisimple Banachalgebra A such that soc(A) 6= K(A).

Indeed. In [Smyth (1975)] the author gave the following algebraB = {x = (xn)n≥1; xn ∈ C, ∀n ∈ N∗; sup{n|xn|; n ∈ N∗}


General Problem

Question

Is it kh(soc(A)) = K(A) for every commutative complex unitalsemisimple Banach algebra?

Remarques

We have already seen that this equality is true in the C *-algebracontext.However it is known that this equality is false in the nocommutative context [G.Androulakis and T.Schlumprecht (2001)].


Alexander, J.C., Compact Banach algebras, P.L.M.S. (3) 18,1968.

A. H. Al-Moajil, The compactum of a semi-simplecommutative Banach algebra, Int. J. Math. Sci. 7 (1984),821-822.

G.Androulakis and T.Schlumprecht, Strictly singular,non-compact operators exist on the space of Gowers andMaurey, J. London Math. Soc. 64 (2001) 655-674.

B. Aupetit and T. Mouton, Spectrum preserving linearmappings in Banach algebras, Studia Math 109 (1994),91-100.

Barnes, B.A., A generalised Fredholm theory for certain mapsin the regular representation of an algebra, Can. J. Math. 20,1968.


B. A. Barnes, G. J. Murphy, M. R. F. Smyth and T. T. West,Riesz and Fredholm theory in Banach algebras, Boston :Pitman, 1982.

N. Boudi and Y. Hadder, On linear maps preservinggeneralized invertibility on commutative algebras, J. Math.Anal. Appl. 345 (2009) 20-25.

Rickart, C.E., General Theory of Banach Algebras, (VanNostrand) Princeton, 1960.

M.R. F. Smyth, Riesz theory in Banach algebras, Math. Z.145 (1975), 145-155.

K. Vala, Sur les e’le’ments compacts d’une algebre norme’e,Ann. Acad. Sci. Fenn. Ser. A I 407 (1967).


THANK YOU

The Main ResultMotivationPreliminariesThe Counter Example

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The Compactum of a Commutative Semisimple Banach Algebrasfsr.um5.ac.ma/icart2018/pdf/talks/Hadder.pdf · 2018. 7. 14. · the question whether all Banach spaces have the approximation