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We rst show: (7) Ea h matrix -algebra A is equivalent to a dire t sum of a matrix -algebra ontaining the identity matrix and a zero algebra

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(1)

A matrix -algebra is a nonempty olle tion A C nn

losed under sums, s alar and

matrix multipli ation, and takingthe adjoint. Call A;A 0

C nn

equivalentifthere exists

a unitary matrixU 2C nn

su hthat

(1) A

0

=fU



MU jM 2Ag:

Formatri esM

1

andM

2

,the dire tsum is

(2) M

1

M

2 :=



M

1 0

0 M

2



:

The iterateddire tsum of M

1

;:::;M

n

isdenoted by

(3)

n

M

i=1 M

i :

Wewrite

(4) t M :=

t

M

i=1 M:

Call Abasi if

(5) A=t C mm

:=ft M jM 2C mm

g

forsomet andm.

Thedire tsum ofA and A 0

is

(6) AA

0

:=fMM 0

jM 2A;M 0

2A 0

g:

A is alled azero algebra ifAonly onsistsof thezero matrix.

Theorem. Ea h matrix-algebrais equivalent toa dire tsum of basi algebrasand a zero

algebra.

Proof. We rst show:

(7) Ea h matrix -algebra A is equivalent to a dire t sum of a matrix -algebra

ontaining the identity matrix and a zero algebra. In parti ular, A ontains a

unit.

Let N be a matrix inA of maximum rank. Then the row spa e rowN of N ontains the

row spa eof ea h matrixinA. Forlet M 2A. ThenkerM kerN,sin e

(2)

(8) x2ker(M M+N N) () (M M +N N)x=0 () x (M M +N N)x=

0 () x



M



Mx=0 and x



N



Nx=0 () Mx=0 and Nx=0 ()

x2kerM\kerN.

By themaximalityof therank ofN,ker(M



M +N



N)=kerN,hen e ker(M)kerN.

So the row spa e of ea h matrix in A is ontained in rowN = row(N



N). We an

assumethatN



N isa diagonalmatrix. (Repla eAbyfU



MU jM 2Agforsomeunitary

matrix U.) So A is a dire t sum of a zero algebra and a matrix -algebra ontaining a

nonsingular diagonal matrix . Then I is a linear ombination of ; 2

; 3

;::: (by the

theory ofVandermonde matri es).

Thisproves (7). Hen e toprove thetheorem, we anassumethat I 2A.

LetC

A

bethe enterof A;thatis,

(9) C

A

:=fC2AjCM =MC forallM 2Ag.

As C

A

is a ommutative matrix -algebra, the matri es in C

A

an be simultaneously di-

agonalized by some unitary matrix U. That is, fU



MU j M 2 C

A

g onsists of diagonal

matri es. So we an assume that C

A

onsists of diagonal matri es only. Then C

A is the

linear hull of ertaindiagonal 0,1matri es E

1

;:::;E

t

,withE

i E

j

=0ifi6=j.

Thenforea hM 2Aandea hionehasE

i

M =ME

i

. SoM is0inpositions(k;l)with

(E

i )

k;k 6=(E

i )

l:l

=0. So A is the dire t sum of matrix-algebras ea h with the property

that the s alar multiples of the identity matrix (of appropriate dimension) are the only

matri es ommuting withall matri esin thesubalgebra.

Hen eit suÆ estoshow

(10) ifAis amatrix -algebrawithC

A

=CI,then Ais basi .

Let B be a maximal subset of A with the property that A



= A and AB = BA for all

A;B 2B. ThisimpliesthatB isa ommutative-subalgebra ofAandthatI 2B. We an

assumethatB onsistsofdiagonalmatri esonly. LetE

1

;:::;E

t

betheminimalidempotents

of B. So B=CE

1

++CE

t andE

1

++E

t

=I.

Nowforea h i:

(11) Forea h M 2Athere is a 2C withE

i ME

i

=E

i

. Inother words, E

i AE

i

=

CE

i .

Suppose not. We an assume that M



= M, sin e we an repla e M by M +M



or

iM iM



. Thenwe anadd E

i ME

i

toB,sin eit ommuteswithea h ofE

1

;:::;E

t . This

ontradi tsthe maximalityofB. This proves(11).

Moreover,

(12) forall ithere isan M

i

2AwithE

1 M

i E

i 6=0.

Tosee this, onsider thelinear spa e I generated byAE

1

A. Thisis a-algebra. LetP be

(3)

PM =PMP =MP. SoP 2C

A

. Hen e bytheassumption in(10),P =I. So I =A,and

thereforeE

i

2I,and hen eE

i

isasumofmatri esinAE

1

A,hen einAE

1 AE

i

. Hen e we

have (12).

We anassume that M

i

=E

1 M

i E

i

(byresetting M

i := E

1 M

i E

i

). NowM



i M

i

belongs

toE

i AE

i

,hen e by(11) toCE

i

. Similarly, M

i M



i 2CE

1

. By s aling we anassume that

M



i M

i

=E

i

forea h i. This implies that M

i M



i

= E

1

for ea h i, sin e M

i M



i

= E

1 for

some2C. Then

(13) 

2

E

1

=M

i M



i M

i M



i

=M

i E

i M



i

=M

i M



i

=E

1

;

and hen e =1. Sin e M

1 2E

1 AE

1

,wehave by(11) thatM

1

=E

1

forsome2C. As

M

1 M



1

=E

1

,we know=1. Hen erepla ingM

1 by 

1

E

1

,weobtain M

1

=E

1 .

So rank(E

1

) = rank(M

i M



i

) = rank(M



i M

i

) =rank(E

i

) for ea h i. Hen e all E

i have

the same number of 1's. ForA 2 A and i;j =1;:::;t,let A

i;j

be the submatrix indu ed

bytherowswhereE

i

has1'sand the olumnswhereE

j

has1's. So E

i AE

j

arisesfrom A

i;j

byadding all-zerorows and olumns.

So for ea h i, the matrix (M

i )

1;i

is unitary. Let U

i

:= (M

i )



1;i

. Let U be the unitary

matrix with U

i;i

= U

i

forea h i, and U

i;j

=0 ifi 6=j. Repla ing A byU



AU,we obtain

that (M

i )

1;i

=I forea h i.

Nowfori;j=1;:::;t,let N

i;j

be be thematrixwithE

i N

i;j E

j

=N

i;j

and (N

i;j )

i;j

=I.

So N

i;i

= E

i

and N

1;i

= M

i

for ea h i = 1;:::;t. So ea h N

1;i

belongs to A. Sin e

N

i;j

=N



1;i N

1;j

forall i;j,it follows that N

i;j

2A forall i;j.

We nallyshow:

(14) forea h M 2A and i;j=1;:::;t,E

i ME

j

=N

i;j

forsome2C.

Indeed, using (11),

(15) E

i ME

j

=E

i ME

j N

j;i E

i N

i;j 2E

i AE

i N

i;j

=CE

i N

i;j

=CN

i;j

;

asrequired. So Ais basi .

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