Bounded distance preserving surjective mappings on block triangular matrix algebras

Bounded distance preserving surjective mappings on block triangular matrix algebras

Chooi Wai Leong

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Abstract

Let Mn be the algebra of n x n square matrices. Let T and U be block triangular matrix subalgebras of Mn and Mm, respectively. Let r be an integer such that 1 ~r

<

min

{rttp 1 ' rmtll}.

In this talk, we show that every surjective mappings 'lj; : T -t U satisfying

rank (A - B) ~ r ^rank ('lj;(A) -1/;(B)) ~ r are bijective mappings preserving adjacency in both directions.

Keywords: Block triangular matrix, adjacency preserving mapping, bounded distance pre- serving mapping, geometry of matrices, rank.

Joint work with M.H. Lim (University of Malaya, Malaysia)

E-mail address:wlchooi@um.edu.my

e

c:

'oQI 1Il

....

_o

~'"

:; 'iii u >,

"'.!!!

LL. '"

.- vi ~

o QI _

o g ~

...c:::.!!! .!!!

u~~

...,iCii ...._{, u}

0

~.~.~

~ ~ ... _

'" c:

~::::>

'0

...QI :::l

.~

.s

ii,

<f rv

,

~;

...-i

",

...-i M

0 _M

N 0

«

^{N t,}

0:: 10

U ^MI

0

~

VI M

ro

~

...

..c :J 0

~

^-,

« ~

ro

"'0

~

c:

~

ro

c:

o

~

c:

~ ~

c:o U roc:

'':;

o

roc:

Q)

...

+"

c:

o

VI (;

Q)

c::

o

.!!!

Q)u c::ro

.~

"C

4-

o .l1

c::

'0

ro

c..

E o

c::

~ E

<l)

<l) VI

iii<l)

.;:u

...

nJ

E

10 0\

0\...

~

o a.

C1l bOc:

\f)

<l)

.iii>

c ...

<l) 0

~~ o

c 0

nJ..c

&-E

c:

~ X N

•

n

."!:::! ."!:::!^Q) c:

<+=

.~

c:

Q)

..cQ)

•

•

•

'-'

•

_cr

,

....

be ^!II ..c c:: E ...

~ ·x

⁰

.iii ^!II ..c

In E

.~

!II

U ^..c... ^Q)u

.~

·3

^c::_!II ^\'J

""0 ^In ...

Q)

...

.!!!

...

^c:: ""0

Q) Q)

0

""0 E ""0

.iii_c:: Q) ^Q)x

0 V ;.;::::

U '+- vi

c:: ^{0 CII}

Q) In

Q)

.=

V

..c ^!II

Q) a.

::

> ^{be In}

!II c:: ^!II

..c

.~

_C1i

.!!!In ^Q)In ^uc::

""0 ^Q) CII ::J

... ...

t:

^a.In :0^In c:: ^be ""0

0 c:: _Q)

..j:i a. ""0

!II a. ^c::

::J !II ::J

c:: E ⁰

..j:i ..c

c:: Q) vi

0 > C1i ^c::

u ^..j:i u ⁰

Q) u c:: ^..j:i

E ^.;:>Q)

...

^{!II UQ)}

0 ::J .!!!

.=

In In ""0 ""0

•

.'

o

.. -

>.

W

E

('Q

c:

VI .!!!

~

"C

c:

!1l

c:

o

'B

c:

..2

.s:c:

!1l

...

!1l

<11 .-

E

E ~

~ E

<11

-2~ • •

o •

,I

o

·!:

• •

n

,~,

~

(f

r.

~;

Q) c::

~

^...c -lc ^..s::... ""C

,~

...

_{Q) ..c}0 Q)

,~

""C

.b ""C _U ""C

<1l ^Q) c:: c:: ^t: _t:

1:

^>0 ...^ro _N ^:::I0 ^nJ

.,

...

^{._ ,!!!} ... .c ^VI

0 c.. -0 -lc bO ^Q)u

~

""C -0 Q) t: ''::: lrl

....

_Q)

1:

^t:_{nJ -0}c::^{Q) U}_c::ro

e

_Q)

...

^nJ

E

...c ::J

...

0 c.. 0 ,!!! ^VI_Q) o

~

^{nJ._ ..c -0 ._}_{c.. ,~}^t:

bO ^{-0 -0}

Q) Q) Q) VI :t!

":S

^NM ^Q)U ;.;:::^X ;.;:::^{x bO} _E

"1J ^N t: t: ._

I ro ro ^{,_ Q)}

c: .-I nJ §::c

<1l N

...

bO bO

III N ,!:Q c:: c nJ

..t::: ...-... ""C _{'~ '~} E t:₀

0..0

...

._

~ .-I ""C t: ^{Q) Q) Q)} Q)

Q) 0

'" '"

c.. bO~ ^{Q) Q) Q)} > u

0

... c.

'~ t:

nJ Q)

...

bO ^{nJ c,} U nJ

c.. _u

M ^,~ E ^{bO bO Q)}

...

c: ^nJ :::I ',...., VI

,!:Q M c:: c:: ._ ,_

<1l

o:t ^4- c- ^,!!! _'a'

'a.

^:::I ""C

...c _,!!!

....

₀ Q) ..s:: _VI

Cl. Cl. Cl. ""C

t: "1J"1J 0..Cl.

...

c.. ._Q) ^Q)nJ 0

...

E roE roE -0^{Q) Q)X}

00

⁰ <! ^u VI ^{N ;.;:::} vi

,~

Q) Q) '':::

t:

15

^{t: t: > vi > Q) VI Q)}

nJ ro

...

0 0 '~ c:: '':;

...

ro u

:::I ..c u

'€

^{..s:: u ,Q u u '':::}

:c

^bb_c:: ^Q)_bO ^{Q) Cl. Q) ... Q) nJ Q)}

...

Q) ro ^';::> u ';::> vi ._

3:

^nJ

a.:

^ro::J

«

^...c

...

^VIVI

...

^{bO ::JV')}

.=

^{Q) ::J}

'"

^c::0 ^{..t:ro E}

_j I

...

_ro -0_Q) ro ro ro-O ro '':; U ^VIro U

bO ",..s:: u Q) '':::

0.: ^Q)c:: ^u t: ^,!!! ,- ... ^{,!!! Q)} ..t: ^Q)

...

0 :::I

~_g ...

_u Q)

.-! ..J ::i -0 _'~

~ ~

^-0

r::

^{t: E}

0

2

^ro

N_t:

• ...

_t:

.2 • • •

^..t:f-^{Q) ,!:Q}""C

...

^EVI>.

e.

c

t;'

,'I

-0

.'

V) V

V -0

.~ ~

^C_:::J

~ 0 ~

ro CO .

~ bO

0

~ c:

ro

:::J

~

^V)

bO

^V

c

c

^V)_V

.2

ro ~ ~

~ o, u

l-

V) ^V

~

~ bOO

u c

0 ._ ...c::

o..~

CO

^0..0

4-

roOO

0 ~ c

>.

^Q)

~ .~

^V

~

V

~ u

E u

^Q)

c ro

0

-0

.~ ~

V C :::J ^V)

~ ro r..nO

"

o

""'0 t:

!1l Q)

o

t:

.!:!!

t:

o

'';;

!1l

b

t:

Q)

E o

1/1

"C

Q) Q)

t:

~

.j..J

z-

t:

Q)

..s::

.j..J

..-...

.

...,I

t:

o

'Vi

1/1 ::Ju .!:!!

"C ^Q)U

!1la.

VI

.s:Q) .j..J

.!:!!

...

::J

o

4-

o

j!

!1l 1/1 Q)

..s::

.j..J

J: ... • •

...

!1l ::J

bO t:

!1l ''::

.j..J

~

u .Q..c ..s::E

U Q)

!1l .j..J

Q) 0

... t:

~~

.j..J

..s::!1l

.j..J

..s::

U ::J 1/1

t:

VI

.

^...,

VI

..-...

t:

--.,;

X

!1l

E

•

o

~I

. ...,

...I

+

_t:

Li.J":;

o

~---....

b.O

00 >:l

~ 'a e

...

~ Q)

~

cQ$

S ~

S1$

L1]^v

~ ~

^0-

:a

^>:l ...

... ~ g

1~

... b.O

~

^{vI ..0:~}

as

cQ"O

~

^{0 It]}

I-< 1=1 ^In

0

~ ... .... .~ ~

I-< 0- ^1"11

~~

...._" ~...;

§' S +

^{~ ;::l}

In cQo:l

U')]

LU'

^..c1

~

^U')

+' ...-.

X _Q) ^In

.... ~

^Q)

«f'

U')..o ^{r1) 0}

II

'S~

^I

>:l

v ^{Q) ....~}

~ i

^{N ~ I-< 0}

~

r1)

..g ...

...._" • .;.>

i~

_oq::

+ ~-~UJ .._..M ...Q)I-<

Q)~

II

>:l"O

...c::~ "7:;; .~ ..c1

~ Q)

..-.. ~;aO

.E~

^'-"'oq::

II

^>:l..o

o . ~

>:l:sl

o

1=1

Q) (l)

oq:: ~ ...

"O~

OgC

~~

^{(l)'>~(l) 1=1}

~

_Q) ^{k Q) .;13 ~}

~ ~

k ~ ...,..

0

<B~~

VI

...

><

I ...__.l-

~

"

^c:C'Il

II

n

,.-...

t:

'--'><

tV

E V

~

VI

.J::

II

~ I CO

'--'

~

...

VI

:§ s:

.~ ... ~

"' E

QJ ..Q

~

~

liJ CO

'c::(

... "'t:I

QJ t:

-.J "'

...

t: t:

'-" '--'

"'t:I "'t:I

t: t:

"' "'

N M

oq--

II II

1\\

.J:: .J:: .J::

• • •

flp

~,

c

11)

II

....

v ..c

~

SO