Zero products preserving maps

Zero products preserving maps
´
Juana Sanchez-Ortega
´
Universidad de Malaga
(Spain)
CIMPA School on “Operator Theory and the Principles of Quantum
Mechanics”
` Morocco
Meknes,
September 16, 2014
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
1 / 13
Some definitions I
M. B URGOS , J. S. O.: On mappings preserving zero products. Linear
and Multilinear Algebra 1 (2012), 1–13.
Let A, B be two algebras and T : A → B a linear map.
1
T is zero product preserving if T (a)T (b) = 0 whenever ab = 0.
2
T preserves zero product in both directions if it satisfies
T (a)T (b) = 0 iff ab = 0.
3
T is Jordan zero product preserving T (a) ◦ T (b) = 0 whenever
a ◦ b = 0.
F “P OPULAR ” PROBLEM : to characterize zero product preserving linear
maps on Banach algebras.
Note! In several cases, a map preserves zero products iff it is a central
element multiple of an homomorphism.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
2 / 13
Some definitions II
Chebotar, Ke, Lee, Zhang, Hou, Zhao
F “M OST RECENT ” PROBLEM : to describe Jordan homomorphisms
through the action on Jordan zero products.
DEF. A linear map T : A → B is a Jordan homomorphism if
T (a ◦ b) = T (a) ◦ T (b), for all a, b ∈ A.
Note! A Jordan homomorphism multiplied by a central element
preservers Jordan products.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
3 / 13
Recent results
2003 Araujo, Jarosz: Every bijective linear map between unital standard
operator algebras, which preserves zero products in both directions, is a
scalar multiple of an algebra isomorphism.
2003 Chebotar, Fe, Lee, Wong: Describe zero product preserving maps for
rings generated by idempotents.
2004 Chebotar, Fe, Lee: Describe zero product preserving maps for prime
rings containing nontrivial idempotents.
2005 Hou, Zhao: Study surjective additive maps preserving Jordan zero
products in both directions on the von Neumann algebra of all bounded
linear operators on an infinite dimensional complex Hilbert space.
2006 Hou, Zhao: Every surjective bounded linear map preserving Jordan
zero products between von Neumann algebras or C∗ -algebras is a
Jordan homomorphism multiplied by an invertible central element.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
4 / 13
Our goal
F O UR PROBLEM : to describe (Jordan) zero product preserving maps
on Banach algebras having enough minimal idempotents.
For A, B unital complex algebras, and T : A → B a linear map, we
consider the following conditions:
C1
ab = 0 ⇒ T (a)T (b) = 0
C2
a ◦ b = 0 ⇒ T (a) ◦ T (b) = 0
Note! A will denote a semisimple unital complex Banach algebra with
identity element 1, and h = T (1).
We will assume that A has nonzero socle.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
5 / 13
About the socle ...
DEF. The socle of A is the sum of all minimal left (or right) ideals of A, if they
exists; otherwise it is zero.
Note! Every minimal left ideal of A is of the form Ae for some minimal
idempotent e, that is, a nonzero idempotent with eAe = Ce.
S1 Every element of the socle is a finite sum of rank-one elements.
The socle coincides with the set of all finite rank elements.
rank-one elements
S2 Every element of the socle is von Neumann regular, i.e., for every
a ∈ Soc(A) there exists x ∈ A such that a = axa.
S3 Every element of the socle of a semisimple Banach algebra is a
linear combination of minimal idempotents.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
6 / 13
About the socle ...
DEF. The socle of A is the sum of all minimal left (or right) ideals of A, if they
exists; otherwise it is zero.
Note! Every minimal left ideal of A is of the form Ae for some minimal
idempotent e, that is, a nonzero idempotent with eAe = Ce.
S1 Every element of the socle is a finite sum of rank-one elements.
The socle coincides with the set of all finite rank elements.
rank-one elements
S2 Every element of the socle is von Neumann regular, i.e., for every
a ∈ Soc(A) there exists x ∈ A such that a = axa.
S3 Every element of the socle of a semisimple Banach algebra is a
linear combination of minimal idempotents.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
6 / 13
Zero product preserving linear maps
C1
ab = 0 ⇒ T (a)T (b) = 0
THM. (Burgos, S.O.) Let T : A → B be a bijective linear map satisfying (C1).
Assume that h is not a zero divisor. Then
T −1 (T (ab)h − T (a)T (b))Soc(A) = 0,
∀ a, b ∈ A.
Moreover, if Soc(A) is essential then T is an isomorphism multiplied by a
central invertible element.
essential ideals
Lemma. Let T : A → B be a linear map satisfying (C1). Then
(i) T (ax)T (b) = T (a)T (xb),
(ii) T (ax)h = T (a)T (x),
(iii) hT (xa) = T (x)T (a),
(iv) hT (x) = T (x)h,
for all a, b ∈ A, and x ∈ Soc(A).
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
7 / 13
Zero product preserving linear maps
C1
ab = 0 ⇒ T (a)T (b) = 0
THM. (Burgos, S.O.) Let T : A → B be a bijective linear map satisfying (C1).
Assume that h is not a zero divisor. Then
T −1 (T (ab)h − T (a)T (b))Soc(A) = 0,
∀ a, b ∈ A.
Moreover, if Soc(A) is essential then T is an isomorphism multiplied by a
central invertible element.
essential ideals
Lemma. Let T : A → B be a linear map satisfying (C1). Then
(i) T (ax)T (b) = T (a)T (xb),
(ii) T (ax)h = T (a)T (x),
(iii) hT (xa) = T (x)T (a),
(iv) hT (x) = T (x)h,
for all a, b ∈ A, and x ∈ Soc(A).
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
7 / 13
Some nice consequences
Recall T preserves zero product in both directions if T (a)T (b) = 0 ⇔ ab = 0.
1
Let T : A → B be a surjective linear map preserving zero products in
both directions. If Soc(A) is essential, then T is an isomorphism
multiplied by a central invertible element.
2
Let T : A → B be a bijective linear map satisfying (C1). Assume that h is
an idempotent of B, and Soc(A) is essential. Then T is an
isomorphism multiplied by a central invertible element.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
8 / 13
Jordan zero product preservers
C2
a ◦ b = 0 ⇒ T (a) ◦ T (b) = 0
THM. (Burgos, S.O.) Let T : A → B be a bijective linear map satisfying (C2).
Assume that h is not a Jordan zero divisor. Then
T −1 (hT (a)−T (a)h)Soc(A) = 0, T −1 (hT (a2 )−T (a)2 )Soc(A) = 0, ∀ a ∈ A.
Moreover, if Soc(A) is essential then T is a Jordan isomorphism
multiplied by a central invertible element.
Lemma. Let T : A → B be a linear map satisfying (C2) and 1 ∈ T (A).
For every a ∈ A and x, y , z ∈ Soc(A), the following hold:
(i) T (a ◦ x) ◦ h = T (a) ◦ T (x),
(ii) hT (x) = T (x)h,
(iii) {T (x), T (y ), T (z)} = T ({x, y , z})h2 .
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
9 / 13
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
10 / 13
References
J. A RAUJO, K. J AROSZ: Biseparating maps between operator
algebras. J. Math. Anal. Appl. 282 (2003) 48–55.
M. A. C HEBOTAR , W.F. K E , P.H. L EE , N. C. W ONG: Mappings
preserving zero products. Studia Math. 155 (2003), 77–94.
M. A. C HEBOTAR , W.F. K E , P.H. L EE: Maps characterized by action
on zero products. Pacific J. Math 216 (2004), 217–228.
J. H OU, L. Z HAO: Zero-product preserving additive maps on
symmetric operator spaces and self-adjoint operator spaces. Linear
Algebra Appl. 399 (2005) 235–244.
J. H OU, L. Z HAO: Jordan zero-product preserving additive maps on
operator algebras. J. Math. Anal. Appl. 314 (2006) 689–700.
´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
11 / 13
Rank-one elements
Let A be a semisimple complex Banach algebra with unit 1.
DEF. An element u ∈ A is of rank-one if it satisfies one of the following
equivalent conditions:
u 6= 0, and u belongs to some minimal left ideal of A.
uAu = Cu 6= 0.
u 6= 0, and |σ(xu)| \ {0} ≤ 1, for all x ∈ A.
u 6= 0, and |σ(ux)| \ {0} ≤ 1, for all x ∈ A.
Given a ∈ A, we write σ(a) to denote the spectrum of a
σ(a) = {λ ∈ C : a − λ1 is not invertible in A}.
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´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
12 / 13
Essential ideals
DEF. An ideal of an algebra A is called essential if it has nonzero intersection
with every nonzero ideal of A.
F Assume that A is a semisimple Banach algebra. Then:
I is essential if and only if aI = 0 ⇒ a = 0,
∀ a ∈ A.
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´
J. Sanchez-Ortega
(UMA)
Zero products preserving maps
September 16, 2014
13 / 13

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