PECULIAR HOMOMORPHISMS ON COMMUTATIVE BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS (Researches on isometries from various viewpoints)

PECULIAR HOMOMORPHISMS ON COMMUTATIVE BANACH ALGEBRAS OF VECTOR‐VALUED FUNCTIONS

OSAMUHATORI, SHIHO_OI,AND HIROYUKI TAKAGI

The unital commutative Banach

_algebra

of all

_{complex‐valued}

continuous functions on a _compact Hausdorff _space

K_{j}

is denoted

by

C(K_{j})

. A map

$\psi$

:

C(K_{1})

\rightarrow

C(K_{2})

is a

unital

_homomorphism

_(a

_homomorphism

which _preserves

_identity)

if and

_only

if there is a

continuous_{map $\varphi$}:

K_{2}\rightarrow K_{1}

such that

$\psi$(f)=f\mathrm{o} $\varphi$

for_every

f\in C(K_{1})

. In

general

Gelfand

theory

assertsthat aunital

homomorphism

between unital

semisimple

commutativeBanach

algebras

is

_{represented by}

a

composition

_operatorinduced

by

the associated continuous_map

between maximal ideal_spaces. Theconverseassertion isnot

_always

_true; there isarestriction onthe continuous_mapbetween maximal ideal_spaceswhich defines aunital

homomorphism.

There can be a continuous _map whose

composition

does not even define a _map between

underlying algebras.

For

_example

a _{map $\varphi$}:

\overline{D}\rightarrow\overline{D}

from the closed umit disk

\overline{D}

into itself

defineaunital

homomorphism

(composition operator)

from the disk

algebra

intoitself if and

only

if the_{map $\varphi$}is

_analytic

onthe _opendisk.

Let K bea_compact metric _space and Eaunital commutative Banach

algebra.

We _say

that a _map F : K \rightarrow E is a

Lipschitz

_map from K into E if the

Lipschitz

constant is

finite;

L(F)=\displaystyle \sup_{x\neq y}\frac{\Vert F(x)-F(y)\Vert_{E}}{d(x,y)}

<\infty, where d

)

denotes the metric on K. The

algebra

of all

_Lipschitz

_maps from K into E is denoted

_by

_Lip

_{(K, E)}

. Then

Lip

(K, E)

isa unital

commutative Banach

_algebra

with the norm

\Vert

.

\Vert_{L}

= L ₊

\Vert

\Vert_{\infty(K)}

. In this paper we

study

Banach

_algebras

between which a unital

homomorphism always

has a

special

form.

In

_particular,

we

study

the case of the

algebras

of

Lipschitz

_maps from _compact metric

spaces into unital

_semisimple

commutataive Banach

_algebras.

The maximal ideal space of

Lip

_{(K_{j}, E_{j})}

is

_homeomorphic

to

_{K_{j}\times M(E_{j})}

,where

M(E_{j})

isthe maximal idealspaceof

E_{j}.

If

_{E_{j}}

is

_semisimple,

then

_Lip

_{(K_{j}, E_{j})}

is

_semisimple,

andwe_{may suppose} that

Lip

_{(K_{j}, E_{j})\subset C(K_{j}, E_{j})}

.

Suppose

that

$\psi$

:

Lip

(K_{1}, E_{1})\rightarrow \mathrm{L}\mathrm{i}\mathrm{p}(K_{2}, E_{2})

is a unital

homomorphism.

Then there exists acontinuous _map $\Phi$ :

K_{2}\times M(E_{2})

\rightarrow K_{1}

\times

M(E_{1})

denoted

_by

_{$\Phi$(x, $\phi$)=($\varphi$_{1}(x, $\phi$), $\varphi$_{2}(x, $\phi$))}

such that

$\psi$(F)(x, $\phi$)=F($\varphi$_{1}(x, $\phi$), $\varphi$_{2}(x, $\phi$ \forall(x, $\phi$)\in K_{2}\times M (

E

_2).

In the case of

E_{j}

=

\mathrm{L}\mathrm{i}\mathrm{p}(L_{j}, \mathbb{C})

for _a compact metric _space

L_{j}

, the maximal ideal space

of

_Lip

_{(K_{j}, E_{j})}

is

_homeomorphic

to

_{K_{j}}

\times

_{L_{j}}

and the induced

_{composition operator}

de‐ fined

_by

_any

Lipschitz

map from

K_{2}

\times

L_{2}

into

_{K_{1}}

\times

_{L_{1}}

is a unital

homomorphism

from

Lip

(K_{1}, E_{1})

into

Lip

(K_{2}, E_{2})

. On the other

hand,

an

interesting

observationwas exhibited

Key words andphrases. vector‐valued functions, vector‐valued Lipschitz algebras, algebra homomor‐

by

Botelho and Jamison

_[1];

if

_{K_{2}}

isconnected and

_{E_{j}}

isthe

_algebra

of

_convergent

sequences or the

algebra

of bounded _sequences, then _{$\varphi$_{2}}

depends only

on

M(E_{2})

, not on

K_{2}

. Oi

[6]

generalized

their result

_{by proving}

that it is the case where

E_{j}

is a unital commutative

C^{*}

_‐algebra.

We call a unital

homomorphism represented

by

the

composition

_operator in‐

duced

_by

acontinuous map

$\Phi$(x, $\phi$)

=

($\varphi$_{1}(x, $\phi$), $\varphi$_{2}( $\phi$))

is of_typeBJ. Oi

[6]

in fact

proved

that _any unital

_{homomorphisms}

between

_algebras

of

Lipsichtz

maps on connected _compact

metric _spaces into unital commutative C^{*}

_‐algebras

areof

_type

BJ;

aunital

homomorphism

$\psi$

:

Lip

(K_{1}, C(K_{1}))\rightarrow \mathrm{L}\mathrm{i}\mathrm{p}(K_{2}, C(K_{2}))

is

represented by

theform of

$\psi$(F)(x, $\phi$)=F($\varphi$_{1}(x, $\phi$), $\varphi$_{2}( $\phi$)) , \forall(x, $\phi$)\in K_{2}\times K_{2}

if

_{K_{2}}

is connected. In this_paper,wefurther

study homomorphisms

of

_type

BJ. It is

interesting

to note that certain isometries between Banach

_algebras

of vector‐valued

_Lipschitz

maps

Lip

(K, E)

is oftypeBJ unless K is connected. We

_give

asufficient condition for admissible

quadruples

between which unital

_{homomorphisms}

are

always

of_type BJ.

1. ADMISSIBLE QUADRUPLES

An admissible

quadruple

is defined

_by

Nikou and O’Farrell in

_[5].

For a

_given

Banach

algebra

of

_{complex‐valued}

continuous

_functions,

the

_{corresponding}

admissible

_quadruple

is

a Banach

algebra

of vector‐valued continuous _maps of the same kind as

complex‐valued

continousfunctions in the

_give

Banach

_algebra.

Priortodefine anadmissible

quadruple,

we

defineavector‐valued function

algebra.

Definition 1. We_saythat A isaE‐valued function

algebra

on a_compactHausdorff_spaceX in the

_strong

sense if A isa

subalgebra

of

C(X, E)

foraunital commutative Banach

algebra

Esuch that the

_following.

conditionsaresatisfied.

(1)

A isaBanach

algebra

withsomenorm

\Vert\cdot\Vert_{A},

(2)

For_every a\in E theconstant map onX defined

by

x\mapsto a is in

A,

(3)

A_separates the

_points

of X,that

is,

forevery

pair

x andy of different

points

in

X,

there exists

_f

inA such that

_{f(x)\neq f(y)}

_,

(4)

for every x\in X the evaluationmap e_{x} : A\rightarrow E defined

by

f\mapsto f(x)

is continuous.

Note that a\mathbb{C}‐valued function

algebra

in the _strong sense is a\mathbb{C}‐valued function

algebra

in the sense of Nikou and O’Farrell. But E‐valued function

algebra

in the sense of Nikou

and O’Farrell neednot be in the

_strong

sense when E is of dimension 2or more. Note also

that if E is

_semisimple,

then the evaluation _{map e_{x}} : A\rightarrow E defined

by

e_{x}(f) =f(x)

for

f\in E

is

_{automatically}

continuous for _every x\in X

_by

a theorem of

Šilov

(cf. [7,

Theorem

3.1.11]).

The

_algebra

_{C(X, E)}

isaE‐valued function

algebra

onX in the

_strong

sensewith

the_{supremum norm;}

_{\displaystyle \Vert f\Vert_{\infty(X)}=\sup\{\Vert f(x)\Vert_{E} : x\in X\}}

. We calla\mathbb{C}‐valuedfunction

algebra

Ainthe_strongsenseisnatural if the_mapfrom X into

M(A)

defined

_by

_{x\mapsto e_{x}}is

_surjective,

to say

simply

X=M(A)

.

Let A be a \mathbb{C}‐valued function

algebra

on a _compactHausdorff_space in the _strongsense

and Eaunital commutative Banach

_algebra.

For

f\in A

and

_{b\in E,}

_{f\otimes b}

denotes the_map in

_{C(X, E)}

such that

_{(f\otimes b)(x)=f(x)b}

for x\in X. We denote

where \mathbb{N} is the set of all

_{positive integers.}

We _say that \mathbb{C}‐valued function

algebra

on X in

the

_strong

senseisauniform

algebra

onX if it is

uniformly

closed. See

[2]

for

general theory

of uniform

_algebras.

Note that the

_terminology

“

afunction

algebra”

in

[2]

means auniform

algebra.

An admissible

quadruple

is a vector‐valued version ofa

given

function

algebra.

It

wasdefined

by

Nikou and O’Farrell in

[5].

The

following

isan

essentially

thesamedefinition

astheone

given

in

[5].

Definition2.

_By

anadmissible

quadruple

we mean a

quadruple

(X,

E, B,

B where

(1)

X is a_compactHausdorff_space,

(2)

E is aunital commutative Banach

_algebra,

(3) B\subset C(X)

isa natural \mathbb{C}‐valued function

algebra

on

X,

(4)

\overline{B}\subset C(X, E)

isan E‐valued function

algebra

onX inthe

_strong

_sense,

(5)

B\otimes E\subset\tilde{B}

and

(6)

\{ $\lambda$\circ f:f\in\tilde{B}, $\lambda$\in M(E)\}\subset B.

Fora

_compact

metric_spaceK andaunitalcommutative Banach

algebra E,

(

K,

E,

Lip

(K), \mathrm{L}\mathrm{i}\mathrm{p}(K, E) )

isan admissible

quadruple.

Definition3. Let

(X, E, B,\tilde{B})

bean admissible

quadruple.

Let $\pi$ :

X\times M(E)\rightarrow M(\tilde{B})

be

given by

$\pi$(x, $\phi$)

= $\phi$\circ e_{x}

, where

$\phi$ \mathrm{o}e_{x}(F) = $\phi$(F(x))

for every F \in

\tilde{B}

. Then

by

a routine

argument

$\pi$ is a continuous

injection.

We _say that an admissible

quadruple

(X, E, B,\tilde{B})

is natural if the associated _map $\pi$ is

bijective.

In this case $\pi$ is a

homeomorphsims

from

X\times M(E)

onto

\{ $\phi$\circ e_{x} : (x, $\phi$)\in X\times M(E)\}=M(\overline{B})

.

Suppose

that

(X, E, B,\overline{B})

is

_semisimple

and

_natural;

$\pi$ :

X\times M(E)\rightarrow M(\tilde{B})

is

surjection.

Thenwe_{may suppose}that

(1.1)

\tilde{B}\subset C(X\times M(E))

.

Proposition

4. Let

(X, \underline{E}, B,\overline{B})

be an admissible

quadruple. Suppose

that B is dense in

C(X)

.

Suppose

also that B is

inverse‐closed;

F\in B with

$\Gamma$_{\overline{B}}(F)( $\phi$\circ e_{x})\neq 0

for

every

pair

x\in X and

_{$\phi$\in M(E)}

_implies

F^{-1}\in\tilde{B}

. Then

(X,

E,

B

,

Ẽ)

is natural.

By Proposition

4 we

easily

seethat

(K, E, \mathrm{L}\mathrm{i}\mathrm{p}(K, \mathbb{C}), \mathrm{L}\mathrm{i}\mathrm{p}(K, E))

is a natural admissible

quadruple.

Proposition

5. An admissible

quadruple

(X, E, B,\tilde{B})

is

semisimple

if

and

_only

_if

E w

semisimple.

If E is

_semisimple,

then

_{(K, E, \mathrm{L}\mathrm{i}\mathrm{p}(K, \mathbb{C}), \mathrm{L}\mathrm{i}\mathrm{p}(K, E))}

is

_semisimple

and natural. Hencewe may supposethat

(1.2)

Lip

(K, E)\subset C(K\times M(E))

2. ALGEBRA HOMOMORPHISMS

In this section weshow that aunital

homomorphism

between admissible

quadruples

has a

peculiar

form under certain

topological

assumptions

on maximal ideal _spaces. Just for

simplicity

weassmethat acommutativeBanach

algebra

E_{j}

is

semisimple;

see

[3]

fora

genral

Theorem 6.

_Suppose

that

_{E_{j}}

is

_semisimple

and

(X_{j}, E_{j}\underline{B}_{j},,\overline{B_{j}})

is natural.

_Suppose

that

\overline{B_{1}}\subset\overline{B_{1}\otimes E_{1}}

, where

-denotes the

_uniform

closure on

M(B_{1})

.

Suppose

that

X_{2}\dot{u}

connected

with _respect to the relative

_topology

induced

_by

the metric inherited

from

the dual space

of

B_{2}

and that

_{M(E_{1})}

is

_totally

disconnected with

_respect

to the relative

_topology

induced

_by

the metricinherited

_from

the dual_space

_{ofE_{1}}

. Let

$\psi$

:

B_{1}\rightarrow\overline{B_{2}}

beaunital

homomorphism.

Then

thereexistsa_{continuous map} $\tau$ :

M(E_{2})\rightarrow M(E_{1})

andacontinuous_{map $\varphi$} :

X_{2}\times M(E_{2})\rightarrow

X_{1}

which

_satisfies

that

$\psi$(x, $\phi$)=F( $\varphi$(x, $\phi$), $\tau$( $\phi$)) , (x, $\phi$)\in X_{2}\times M(E_{2})

for

every

F\in\overline{B_{1)}}\cdot $\psi$

is

of

type

BJ.

Theorem 7.

_Suppose

that

_{E_{j}}

\dot{\uparrow}s

_semisimple

and

(X_{j}, E_{j}\underline{B}_{j},,\overline{B_{j}})

is natural.

_Suppose

that

\overline{B_{1}}\subset\overline{B_{1}\otimes E_{1}}

, where

-denotes the

_uniform

closure onM

(B1).

Suppose

that

X_{2}

is connected

and

_{M(E_{1})}

is

_totally

disconnected. Let

_$\psi$

:

\overline{B_{1}}\rightarrow\overline{B_{2}}

be a unital

homomorphism.

Then there

exists acontinuous _map $\tau$ :

M(E_{2})\rightarrow M(E_{1})

anda continuous_{map $\varphi$} :

X_{2}\times M(E_{2})\rightarrow X_{1}

which

_satisfies

that

$\psi$(F)(x, $\phi$)=F( $\varphi$(x, $\phi$), $\tau$( $\phi$)) , (x, $\phi$)\in X_{2}\times M(E_{2})

for

every

F\in\overline{B_{1}}; $\psi$

ts

of

type

BJ.

3. THECASE OF ALGEBRAS OFVECTOR VALUED LIPSCHITZ MAPS

If E is

_semisimple

wehave

Lip

(K, \mathbb{C})\otimes E\subset \mathrm{L}\mathrm{i}\mathrm{p}(K, E)\subset\overline{\mathrm{L}\mathrm{i}\mathrm{p}(K,\mathbb{C})\otimes E}.

Hence we have the

following

as a

corollary

of Theorems 6 and 7. Note that the

original

topology

onK,the Gelfand

topology

induced

by

Lip

(K;\mathbb{C})

,and the relative

topology

induced

by

the metric induced

by

operatornorm

topology

onthe dual_spaceof

Lip

(K, \mathbb{C})

all coincide

witheach other

Corollary

8. Let

_{K_{j}}

be a _compact metric space and

E_{j}

a unital

semisimple

commutative

Banach

_{algebra for}

_j

= 1

,2.

Suppose

that

K_{2}

is connected.

Suppose

that

M(E_{1})

is

totally

disconnected with

_respect

to either the

_{Gelfand topology}

_(the

_{original topology}

asthe maximal

ideal

_space)

or the relative

_topology

induced

_by

the metric inherited

from

the dual space

of

E_{1}

. Let

$\psi$

:

Lip

(K_{1}, E_{1})

\rightarrow

\mathrm{L}\mathrm{i}\mathrm{p}(K_{2}, E_{2})

be a unital

homomorphism.

Then there exists a

continuous _map $\tau$ :

M(E_{2})

\rightarrow

M(E_{1})

and a continuous _{map $\varphi$} :

K_{2}\times M(E_{2})

\rightarrow

_{K_{1}}

such that the _{map $\varphi$}

_$\phi$

₎

:

K_{2}\rightarrow K_{1}

is a

Lipschitz

_map

for

each

$\phi$\in M(E_{2})

, which

satisfies

that

( $\psi$(F))(x, $\phi$)=F( $\varphi$(x, $\phi$), $\tau$( $\phi$)) , (x, $\phi$)\in K_{2}\times M(E_{2})

for

every

F\in \mathrm{L}\mathrm{i}\mathrm{p}(K_{1}, E_{1})_{f}\cdot $\psi$

is

of

type

BJ,

We show several

_examples

of unital

_semisimple

commutative Banach

_algebras

E such that the maximal ideal spaces are

totally

disconnecte with _respect to

_{corresponding topologies}

desicribedin

_Cororally

8.

Example

9

_{(cf. [3]).}

₍₁₎

Let M be a _compact Hausdorff _space. The Banach

algebra

C(M)

of all

_{complex‐valued}

continuousfunctions on M. Then M is

homeomorphic

to the maximal ideal _space of

_C(M)

.

By

the

Urysohn’s

lemma we infer that M is

discrete with respect to the relative

_topology

induced

_by

the metric inherited from the dualspace of

C(M)

.

(2)

Let $\Gamma$ be the unit circle in the

_{complex plane.}

Recall that the Wiener

_algebra

is the

_algebra

of all

_{complex‐valued}

continuous functions on $\Gamma$ which have absolute

converging

Fourier

_series;

_{W( $\Gamma$)}

=

\displaystyle \{f \in C( $\Gamma$) : \sum|\hat{f}(n)| < \infty\}

with the norm

\Vert f\Vert_{W}

=

\displaystyle \sum_{m}|\hat{f}(m)|

for

f

_\in

W( $\Gamma$)

. The maximal ideal space of

W( $\Gamma$)

is homeo‐

morphic

to $\Gamma$.

By

a

simple

calculationwe seethat $\Gamma$ is discretewith respect to the

relative

_topology

induced

_by

the metric inherited from the dualspaceof

W( $\Gamma$)

.

(3)

Let A be a uniform

algebra

such that the maximal ideal _space coincides with the

Choquet boundary.

The

_Choquet

_boundary

for auniform

algebra

A is discrete with

respect tothe relative

_{topology.induced}

_by

the metric inheritedfrom the dualspace of A. It is known asthe Cole’scounter

example

tothe

peak point conjecture

[2]

that

sucha uniform

algebra

which isnot a

C’‐algebra

exists.

(4)

LetG bea_compactAbelian_groupand $\Gamma$ its dual_group.

Suppose

that $\Gamma$ isadiscrete groupof bounded order. Then G isa

totally

disconnected_compactAbelian_group

[8,

Example

2.5.7.

_(iii)].

The_group

_algebra

_A(G)

of all Fourier transforms of functions in

_{L^{1}( $\Gamma$)}

is a unital

semisimple

commutative Banach

algebra

whose maximal ideal spaceis G. See thepaper of Katznelson and Rudin

[4]

and a bookof Rudin

[8]

for

further

examples

and informations.

REFERENCES

[1]

F.Botelho andJ._{Jamison, Homomorphisms}on adassof $\omega$mmutativeBanachalgebras, RockyMoun‐

tainJ._Math.,43

_(2013),

395−416

[2]

A._Browder, IntroductiontoFunction_Algebras,W. A._{Benjamin, Inc.,}NewYork‐Amsterdam1969

[3]

O.Hatori,S. Oiand H.Takagi, Peculiarhomomorphismsonalgebras ofvector‐valuedfunctions,preprint

[4]

Y. Katznelson and W._Rudin, The Stone‐WeierstrasspropertyinBanach_algebras,Pacific J._Math.,11

(1961),

253−265

[5]

A.Nikou andA. G.O’Farrell, Banach_{algebras of}vector‐valuedfunctions, GlasgowMath.J.,56

_(2014),

419−426

[6]

S. Oi, Homomorphisms between _{algebras ofLipschitzfunctions} with the values infunction algebras, Jour. Math. Anal.Appl.,444_(2016),210−229

[7]

T. W. _Palmer, Banach _algebras and the _{general theory of* ‐algebras.} Vol. I. _Algebras and Banach

algebras, Encyclopediaof Mathematicsandits_{Applications,}49._{Cambridge University Press, Cambridge,} 1994.

[S]

W._Rudin, Fourier_analysison_{gropus,Reprint}of the 1962_{original. Wiley}Classics_Library.A_Wiley‐

DEPARTMENT OF MATHEMATICS, FACULTY OF _SCIENCE, NỉIGATA _UNIVERSITY, NIIGATA 950‐2181,

JAPAN

E‐mail address: hatoriQmath.sc._niigata-\mathrm{u}.ac.jp

NIIGATA PREFECTURAL NAGAOKA HIGH_SCHOOL,3‐14‐1_GAKKO‐CHO,NAGAOKA_CITY,NIIGATA PRE‐

FECTURE940‐0041, JAPAN.

E‐mailaddress: shiho._{[email protected]}

DEPARTMENT \mathrm{O} $\Gamma$ MATHEMATICAL SCIENCES, FACULTY OF _SCIENCE, SHINSHU _UNIVERSITY, MAT‐ SUMOTO _390‐8621, JAPAN