Universidad de Murcia - Multiplicatively semiprime algebras · 2012-04-19 · VIII Encuentro de la...

Multiplicatively semiprimealgebras

Juan Carlos Cabello PíñarVIII Encuentro de la Red de Análisis Funcional 2012.

La Manga, 19-21 abril 2012

Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 1 / 27

Results are contained in

Structure theory for multiplicatively semiprime algebras. J. of Algebra

(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto


Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra

(2004)

Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).

with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto



(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).

with M. Cabrera

ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto



(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabrera

ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto



(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto


algebraic concepts

Definition

An algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).

S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.

A subspace I of A is said to be an ideal (two-sided) if

IA, AI ⊆ I.


algebraic concepts

DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).

S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.

A subspace I of A is said to be an ideal (two-sided) if

IA, AI ⊆ I.


DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.

Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)

non-associative= not necessarily associative

DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.



Examplenot associative algebras

J. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)





Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).

P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)





Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).

S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)





Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)

The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)







DefinitionA non-associative algebra.

L(A), the algebra of all linear operators on A, isan associative algebra.





DefinitionA non-associative algebra. L(A),

the algebra of all linear operators on A, isan associative algebra.


Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra

Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by

La(x) = ax and Ra(x) = xa.

The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.

Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.


Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?

The answer is yes. The multiplication algebra










Is there a vehicle of information back and forth between the twoalgebras?

The answer is yes. The ε-closure.






Is there a vehicle of information back and forth between the twoalgebras? The answer is yes.

The ε-closure.


DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).

Every complete lattice has a greatest,1L, and a least element, 0L.

ExampleX vector space,SX = subspaces of X is a complete lattice with

∨ =

, ∧ = ∩, and 1SX= X and 0SX

= 0.

ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.




Example

X vector space,SX = subspaces of X is a complete lattice with

∨ =


= 0.






∨ =


= 0.






∨ =


= 0.

ExampleA algebra, IA = ideals ofA.

IA, is also a complete lattice with the same operations.





∨ =


= 0.



Closure Operations

DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:

x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.

x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).

The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .

ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S

||.||X

, is a complete lattice for the

supremum and infimum given by ∨Si = (

Si)||.||

and ∧ Si = ∩Si

X and 0 are respectively the largest and the smallest elements in S||.||X

.


Closure Operations

Definition

A map x → x from a complete lattice L into itself is called a closureoperation if:





||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations






||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations


x1 ≤ x2 ⇒ x1 ≤ x2,

x ≤ x , ∀x , x1, x2 ∈ L,

(∧xi)∼ = ∧xi , for every subset xi of L.




||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations


x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,

(∧xi)∼ = ∧xi , for every subset xi of L.




||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations






||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations





Example

X normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S

||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations





ExampleX normed space,

The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S

||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations





ExampleX normed space, The norm-closure is a closure operation in SX .

The set of all ||.||-closed subspaces of X , S||.||X



Si)||.||

and ∧ Si = ∩Si


.


Closure Operations






||.||X

, is a complete lattice

for the


Si)||.||

and ∧ Si = ∩Si


.


Closure Operations






||.||X



Si)||.||

and ∧ Si = ∩Si


.


Galois connexions

DefinitionA Galois connexion between two complete lattices L and M is a pair of maps

L

M,

satisfying:

x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗

iand (∨yi) = ∧y

i, for all subsets xi ⊆ L, yi ⊆ M.

(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.

The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.

Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete

lattice, and Lε

Mε is an order-reversing bijection pairing.


Galois connexions

Definition

A Galois connexion between two complete lattices L and M is a pair of mapsL

M,

satisfying:


iand (∨yi) = ∧y


(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions

DefinitionA Galois connexion between two complete lattices L and M

is a pair of mapsL

M,

satisfying:


iand (∨yi) = ∧y


(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions


L

M,

satisfying:


iand (∨yi) = ∧y


(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions


L

M,

satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.

x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.

(∨xi)∗ = ∧x∗i

and (∨yi) = ∧yi

, for all subsets xi ⊆ L, yi ⊆ M.(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions


L

M,

satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.

(∨xi)∗ = ∧x∗i

and (∨yi) = ∧yi

, for all subsets xi ⊆ L, yi ⊆ M.(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions


L

M,

satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.

(∨xi)∗ = ∧x∗i

and (∨yi) = ∧yi

, for all subsets xi ⊆ L, yi ⊆ M.

(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Galois connexions


L

M,

satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗

iand (∨yi) = ∧y


(0L)∗ = 1M and (0M) = 1L.

L

M Galois connexion.



lattice, and Lε



Example

H Hilbert space,

the pairing SH

⊥⊥

SH , where

S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH

is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:

S||.||

= (S⊥)⊥ ∀S ∈ SH

The map S → S⊥ is an order-reversing bijection from SH

||.|| onto itself.


Example

H Hilbert space, the pairing SH

⊥⊥

SH , where

S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH


S||.||

= (S⊥)⊥ ∀S ∈ SH


||.|| onto itself.


Example


⊥⊥

SH , where

S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH

is a Galois connexion.

The ||.||-closure is the closure operation determined bythis Galois connexion:

S||.||

= (S⊥)⊥ ∀S ∈ SH


||.|| onto itself.


Example


⊥⊥

SH , where

S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH


S||.||

= (S⊥)⊥ ∀S ∈ SH


||.|| onto itself.


Example

X normed space and X ∗ its topological dual space.

The pairing SX

(.)0

(.)0

SX∗ defined by

S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are

ε(S) = (S0)0 = Sσ(X ,X∗)

and ε(V ) = (V0)0 = Vσ(X∗,X)

.

Sσ(X ,X∗)X

and Sσ(X∗,X)X∗ are complete lattice, and

Sσ(X ,X∗)X

(.)0

(.)0

Sσ(X∗,X)X∗

is an order-reversing bijective pairing.


ExampleX normed space and X ∗ its topological dual space.

The pairing SX

(.)0

(.)0

SX∗ defined by


ε(S) = (S0)0 = Sσ(X ,X∗)

and ε(V ) = (V0)0 = Vσ(X∗,X)

.

Sσ(X ,X∗)X


Sσ(X ,X∗)X

(.)0

(.)0

Sσ(X∗,X)X∗




The pairing SX

(.)0

(.)0

SX∗ defined by

S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion

and the closure operations in SX and SX∗ areε(S) = (S0)0 = S

σ(X ,X∗)and ε(V ) = (V0)0 = V

σ(X∗,X).

Sσ(X ,X∗)X


Sσ(X ,X∗)X

(.)0

(.)0

Sσ(X∗,X)X∗




The pairing SX

(.)0

(.)0

SX∗ defined by


ε(S) = (S0)0 = Sσ(X ,X∗)

and ε(V ) = (V0)0 = Vσ(X∗,X)

.

Sσ(X ,X∗)X


Sσ(X ,X∗)X

(.)0

(.)0

Sσ(X∗,X)X∗



algebraic closures: π-closure and ε-closure

I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J

of A satisfying the conditions IJ = JI = 0.

Definition

The pairing IA

Ann(.)

Ann(.)IA is a Galois connexion.

The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).

IπA

= π-closed ideals of A is a complete lattice.



I ideal of an algebra A,

the annihilator of I in A, Ann(I), is the largest ideal J


Definition

The pairing IA

Ann(.)



IπA






Definition

The pairing IA

Ann(.)



IπA






Definition

The pairing IA

Ann(.)


The π-closure is the closure associated to this Galois connexion, that is,

I = Ann(Ann(I)).

IπA






Definition

The pairing IA

Ann(.)



IπA



algebraic closures:

π-closure and ε-closure



Definition

The pairing IA

Ann(.)



IπA






Definition

The pairing IA

Ann(.)



IπA



The π-closure determine some algebraic properties.

DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.

PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ

A= 0, A.

DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.

PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)




Proposition

A algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ

A= 0, A.






PropositionA algebra, A2 = 0. (non null)

A is prime ⇐⇒ IπA

= 0, A.







A= 0, A.







A= 0, A.


Proposition

A semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)





A= 0, A.




ε-closure

A algebra, S subspace of A.

Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.

N subspace of M(A),

Nann = a ∈ A : F (a) = 0 for each F ∈ N.

DefinitionA algebra. The pairing

IA

(.)ann

(.)ann

IM(A)

is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,

I∧ = (Iann)ann I ∈ IA.


ε-closure



N subspace of M(A),


Definition

A algebra. The pairing

IA

(.)ann

(.)ann

IM(A)




ε-closure



N subspace of M(A),


DefinitionA algebra.

The pairing

IA

(.)ann

(.)ann

IM(A)




ε-closure



N subspace of M(A),



IA

(.)ann

(.)ann

IM(A)




ε-closure



N subspace of M(A),



IA

(.)ann

(.)ann

IM(A)

is a Galois connexion.

The ε-closure is the closure in IA associated to thisGalois connexion,



ε-closure



N subspace of M(A),



IA

(.)ann

(.)ann

IM(A)




Properties of the ε-closure

PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)

Continuity property

F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.

Relationships between closuresA algebra, I ideal de A.

I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then

I ⊆ I||.|| ⊆ I ⊆ I.




Continuity property

F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.


I ⊆ I;

Ann(I)∧ = Ann(I ∧) = Ann(I);

If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then

I ⊆ I||.|| ⊆ I ⊆ I.




Continuity property

F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.


I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);

If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then

I ⊆ I||.|| ⊆ I ⊆ I.




Continuity property

F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.


I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then

I ⊆ I||.|| ⊆ I ⊆ I.


Relationships between the semiprimeness of A andM(A)

A semiprime ⇒ M(A) semiprime

Example: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:

u2 = 1, uv = v2 = v , vu = 0.

M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra

DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.



A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)

The algebra generated by 1, u, v, whose product is defined by:u2 = 1, uv = v2 = v , vu = 0.





A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:

u2 = 1, uv = v2 = v , vu = 0.






u2 = 1, uv = v2 = v , vu = 0.

M(A) semiprime ⇒ A semiprime

Example: A quotient algebra of the associative free algebra





u2 = 1, uv = v2 = v , vu = 0.






u2 = 1, uv = v2 = v , vu = 0.


Definition

An algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.




u2 = 1, uv = v2 = v , vu = 0.




The class of multiplicatively semiprime algebras is quite large

Example

nondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set

TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).

TheoremA algebra, I ∈ IA.

A m.s.p. =⇒ I m.s.p.A/I m.s.p.



Examplenondegenerate alternative algebras

(semiprime associative algebras)

nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set






Examplenondegenerate alternative algebras (semiprime associative algebras)

nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set






Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebras

free nonassociative algebra generated by a nonempty set






Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set







Theorem

A algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).






TheoremA algebra.

A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).






TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π

⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).







Theorem

A algebra, I ∈ IA.







A m.s.p. =⇒ I m.s.p.

A/I m.s.p.


subspaces complemented

DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that

X = Y ⊕ Z .

Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.



Definition

X Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that

X = Y ⊕ Z .




DefinitionX Banach space.

A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that

X = Y ⊕ Z .




DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X

if there is a||.||-closed subspace Z of X such that

X = Y ⊕ Z .




DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X

such that

X = Y ⊕ Z .





X = Y ⊕ Z .





X = Y ⊕ Z .

Theorem of Lindenstrauss, Tzafriri

X Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.




X = Y ⊕ Z .

Theorem of Lindenstrauss, TzafririX Banach space.

Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.




X = Y ⊕ Z .

Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X

if, and only if, X isisomorphic to a Hilbert space.




X = Y ⊕ Z .



Complemented algebra

A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that

A = I ⊕ J.

A is a complemented algebra if, every ideal I of A is complemented.

DefinitionA non null algebra A is simple lacks proper ideals.

TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.



A algebra, I ∈ IA.

I is complemented if there exists J ∈ IA such that

A = I ⊕ J.







A = I ⊕ J.







A = I ⊕ J.


Definition

A non null algebra A is simple lacks proper ideals.





A = I ⊕ J.







A = I ⊕ J.



Theorem

A normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.




A = I ⊕ J.



TheoremA normed algebra with zero annihilator.

A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.




A = I ⊕ J.



TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒

A is an algebra isomorphic to a direct sum of simplealgebras.




A = I ⊕ J.





∼-Complemented algebra

A algebra,

∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that

A = I ⊕ J.

A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)

minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.

PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A

semiprime



A algebra, ∼ closure operation on IA.

A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that

A = I ⊕ J.




semiprime



A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A

such thatA = I ⊕ J.




semiprime



A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that

A = I ⊕ J.




semiprime




A = I ⊕ J.

A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.

∼ is additive if I + J = I + J (I, J ∈ IA)



semiprime




A = I ⊕ J.




semiprime




A = I ⊕ J.


minimal caracter of the π-complementation

Every ∼-complemented algebra is π-complemented.


semiprime




A = I ⊕ J.




semiprime


ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.

π-complemented algebra not ε-complementedAlgebra of Albert

PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.

theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive

corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive


ε-complemented algebra not complemented

The algebra ∞ of all bounded complex sequences.







π-complemented algebra not ε-complemented

Algebra of Albert







PropositionA algebra with Ann(A) = 0.

A ε-complementend ⇐⇒ A m.s.p. and π-complemented.








corollaryA algebra with zero annihilator.

A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive


Decomposable algebras

DefinitionA algebra, ∼ closure operation on IA, m∼

A= minimal elements of I∼

A.

A is a ∼-decomposable algebra whenever A = B∈m∼

A

B

M∼A

= maximal elements ofI∼A,

A is a ∼-radical algebra whenever M∼A

= ∅.

TheoremA algebra. T.F.A.E.

A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.





A.


A

B

M∼A



= ∅.


A is m.s.p.

A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.





A.


A

B

M∼A



= ∅.


A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.


DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.

DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras

Theorem.A algebra with zero annihilator,

A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.




Theorem.

A algebra with zero annihilator,A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.

Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.





A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.

Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.


A normed m.s.p. algebra

||.||-atomic =⇒ ε-atomic

The converse is not true

ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm

||p|| = Max|p(t)|; t ∈ [0, 1],

then lacks minimal ||.||-closed ideals.


A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic



||p|| = Max|p(t)|; t ∈ [0, 1],





ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but

if P is endowed with the norm

||p|| = Max|p(t)|; t ∈ [0, 1],






||p|| = Max|p(t)|; t ∈ [0, 1],



TheoremA normed m.s.p. algebra.

A ||.||-atomic.

A ε-atomic and mεA

= I; I ∈ m||.||A

TheoremA normed algebra with zero annihilator. T.F.A.E.

A ||.||-decomposable

A ||.||-atomic and A = I ⊕ Ann(I)||.||



A ||.||-atomic.


= I; I ∈ m||.||A

Theorem

A normed algebra with zero annihilator. T.F.A.E.





A ||.||-atomic.


= I; I ∈ m||.||A

TheoremA normed algebra with zero annihilator. T.F.A.E.




Proposition

Ω locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.

Corollary

The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.


PropositionΩ locally compact topological space.

C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.

Corollary



PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic

⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.

Corollary



PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic

⇐⇒ The set of all isolated points ofΩ is dense in Ω.

Corollary



PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.

Corollary




CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1]

is a non-ε-atomic m.s.p. algebra.

The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.



CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences,

and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.

The algebra c0 of all null complex sequences is ||.||-decomposable.



CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences

are ε-decomposable algebras.




CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.




CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.


Finite dimension

Jacobson´s TheoremA algebra. T.F.A.E.

A is finite dimensional and M(A) is semiprime;A = ⊕n

i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.

CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.

TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.


Finite dimension

Jacobson´s Theorem

A algebra. T.F.A.E.






Finite dimension







Finite dimension


A is finite dimensional and M(A) is semiprime;

A = ⊕n

i=0Bi is a direct sum of ideals,

one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.




Finite dimension



i=0Bi is a direct sum of ideals,

one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.




Finite dimension



i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra

and the others are finite dimensional simplealgebras.




Finite dimension







Finite dimension





Theorem

A finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.


Finite dimension





TheoremA finite dimensional algebra.

A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.


Finite dimension





TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒

M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.


Finite dimension





TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒

M(A) π-complemented ⇐⇒ M(A) complemented.


Finite dimension







Finite dimensional ideals

TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.

Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.

TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.



Theorem

Every finite dimensional ideal of an m.s.p. algebra is complemented.






Lee-WongA prime associative algebra.

A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.





Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal

is finite dimensional and simple.






Theorem

A nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.





TheoremA nonzero m.p. algebra

If A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.





TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal,

then A is simple and finitedimensional.


Universidad de Murcia - Multiplicatively semiprime algebras · 2012-04-19 · VIII Encuentro de la...

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