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
Plan The Main Result Motivation Preliminaries The Counter Example
1 The Main Result
2 Motivation
3 Preliminaries
4 The Counter Example
Plan The Main Result Motivation Preliminaries The Counter Example
1 The Main Result
2 Motivation
3 Preliminaries
4 The Counter Example
Plan The Main Result Motivation Preliminaries The Counter Example
1 The Main Result
2 Motivation
3 Preliminaries
4 The Counter Example
Plan The Main Result Motivation Preliminaries The Counter Example
1 The Main Result
2 Motivation
3 Preliminaries
4 The Counter Example
Plan The Main Result Motivation Preliminaries 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?
Plan The Main Result Motivation Preliminaries The Counter Example
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).
Plan The Main Result Motivation Preliminaries The Counter Example
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 ))).
Plan The Main Result Motivation Preliminaries The Counter Example
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
Plan The Main Result Motivation Preliminaries The Counter Example
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).
Plan The Main Result Motivation Preliminaries The Counter Example
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.
Plan The Main Result Motivation Preliminaries The Counter Example
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.
Plan The Main Result Motivation Preliminaries The Counter Example
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} .
Plan The Main Result Motivation Preliminaries The Counter Example
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.
Plan The Main Result Motivation Preliminaries The Counter Example
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)].
Plan The Main Result Motivation Preliminaries The Counter Example
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)).
Plan The Main Result Motivation Preliminaries The Counter Example
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.
Plan The Main Result Motivation Preliminaries The Counter Example
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 :
Plan The Main Result Motivation Preliminaries The Counter Example
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∗}
Plan The Main Result Motivation Preliminaries The Counter Example
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)].
Plan The Main Result Motivation Preliminaries The Counter Example
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.
Plan The Main Result Motivation Preliminaries The Counter Example
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).
Plan The Main Result Motivation Preliminaries The Counter Example
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