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RING HOMOMORPHISMS ON REAL BANACH ALGEBRAS
TAKESHI MIURA and SIN-EI TAKAHASI Received 26 February 2003
LetB be a strictly real commutative real Banach algebra with the carrier space ΦB. IfAis a commutative real Banach algebra, then we give a representation of a ring homomorphismρ:A→B, which needs not be linear nor continuous. IfA is a commutative complex Banach algebra, thenρ(A)is contained in the radical ofB.
2000 Mathematics Subject Classification: 46J10.
1. Introduction and results. Ring homomorphisms are mappings between two rings that preserve both addition and multiplication. In particular, we are concerned with ring homomorphisms between two commutative Banach al- gebras. IfRis the real number field, then the zero map and the identity are typical examples of ring homomorphisms onR. Furthermore, the converse is true: ifρis a nonzero ring homomorphism onR, thenρ(t)=tfor everyt∈R. For ifρis nonzero, thenρ(1)=ρ(1)2impliesρ(1)=1, and henceρpreserves every rational number. Suppose thata≥0. Then we haveρ(a)=ρ(√a)2≥0.
It follows thatρpreserves the order. Fixt∈Rand choose rational sequences {pn}and{qn}converging totsuch thatpn≤t≤qn. Sinceρpreserves both rational numbers and the orderpn≤ρ(t)≤qn, thusρ(t)=t.
LetCR(K)be the commutative real Banach algebra of all real-valued contin- uous functions on a compact Hausdorff spaceK. In the proof of [11, Theorem 3.1], Šemrl essentially gave a representation of a ring homomorphism onCR(X) intoCR(Y )which states that ring homomorphisms preserve scalar multiplica- tion automatically.
Theorem1.1(Šemrl [11]). Ifρ:CR(X)→CR(Y )is a ring homomorphism, then there exist a closed and open subset Y0⊂Y and a continuous map ϕ: Y\Y0→Xsuch that
ρ(f )(y)=
0, y∈Y0, f
ϕ(y)
, y∈Y\Y0, (1.1)
for everyf∈CR(X).
Recall that a commutative real Banach algebraAis said to be strictly real if φ(A)⊂Rfor allφ∈ΦA(cf. [4]), whereΦAdenotes the carrier space ofA. We generalize the above result as follows.
Theorem1.2. Suppose thatAis a commutative real Banach algebra with carrier spaceΦAand thatBis a commutative strictly real Banach algebra with carrier spaceΦB. Ifρis a ring homomorphism onAintoB, then there exist a closed subsetΦ0⊂ΦBand a continuous mapϕ:ΦB\Φ0→ΦAsuch that
ρ(a)ˆ(ψ)=
0, ψ∈Φ0,
aˆϕ(ψ), ψ∈ΦB\Φ0, (1.2) for everya∈A, whereˆ·denotes the Gelfand transform.
If, in addition,Ais unital, then the aboveΦ0is closed and open.
LetC(K)be the commutative complex Banach algebra of all complex-valued continuous functions on a compact Hausdorff spaceK. One might expect that a similar result holds for ring homomorphisms onC(X)intoC(Y ). Unfortu- nately, this is not the case. Indeed, there exists a nonzero ring homomorphism τ onCsuch thatτ is not the identity nor complex conjugate (cf. [6]); such a map is called nontrivial. More precisely, there exist 2cnontrivial ring homomor- phisms onC(cf. [2]), wherecdenotes the cardinal number of the continuum.
However, many authors treat ring homomorphisms between two complex Ba- nach algebras (cf. [1,3,5,7,8,9,10,11,12]).
On the other hand, it is easy to see that the zero map is the only ring homo- morphism onCintoR. This fact can be generalized as follows.
Theorem 1.3. Suppose that Ais a commutative complex Banach algebra and thatBis a commutative strictly real Banach algebra with carrier spaceΦB. Ifρ:A→Bis a ring homomorphism, thenρ(a)ˆ=0for alla∈A, or equivalently, Ais mapped into the radical ofB.
2. Proof of results. Suppose thatᏭ is a commutative algebra. We define Ꮽe=ᏭifᏭis unital; otherwise,Ꮽedenotes the commutative algebra adjoining a unit elementetoᏭ.
Lemma 2.1. If Ꮽ is a commutative algebra over F∈ {R,C}and if φ is a nonzero ring homomorphism onᏭintoC, thenφcan be extended to a unique ring homomorphismφ˜onᏭeintoC.
Proof. Choosea∈Ꮽso thatφ(a)≠0. If we define ˜φ:Ꮽe→Cby φ(f ,λ)˜ def=φ(f )+φ(λa)
φ(a) , (f ,λ)∈Ꮽe, (2.1) it is trivial to verify that ˜φ is additive. Identifyingf with (f ,0), we obtain φ˜|Ꮽ =φ. We show that ˜φ is multiplicative. For everyν,λ,µ∈Fandh∈Ꮽ,
we have
φ(νh)=φ(h)φ(νa)
φ(a) , φ(λµa)
φ(a) =φ(λa) φ(a)
φ(µa)
φ(a) . (2.2) Hence
φ˜
(f ,λ)(g,µ)
=φ(f g˜ +µf+λg,λµ)
=φ(f )φ(g)+φ(µf )+φ(λg)+φ(λµa) φ(a)
=
φ(f )+φ(λa) φ(a)
φ(g)+φ(µa) φ(a)
=φ(f ,λ)˜ φ(g,µ)˜
(2.3)
whenever(f ,λ),(g,µ)∈Ꮽe, and thus ˜φis multiplicative.
We have now proved that there exists an extension ˜φofφonᏭe.
It remains to prove that ˜φ=φ˜whenever ˜φis a ring homomorphism which extendsφonᏭe. So, fix(f ,λ)∈Ꮽe. Since
φ(λa)=φ˜(λa)=φ˜(λe)φ(a), (2.4) it follows that
φ˜(f ,λ)=φ˜(f )+φ˜(λe)=φ(f )+φ(λa)
φ(a) =φ(f ,λ),˜ (2.5) and the uniqueness is proved.
Definition2.2. LetA be a commutative Banach algebra overF∈ {R,C}
and letBbe a commutative real or complex Banach algebra with carrier space ΦB. Ifρis a ring homomorphism onAintoB, then the formula
ρψ(f )def=ρ(f )ˆ(ψ), f∈A, (2.6) assigns to eachψ∈ΦBa ring homomorphismρψ:A→C.
Ifρψ is nonzero, then there is a unique extensionρψofρψ onAe(Lemma 2.1). We define a ring homomorphismσψ:F→Cby
σψ(λ)=ρψ(λe), λ∈F. (2.7)
It follows from this definition that
ρψ(λf )=ρψ(λf )=ρψ(λe)ρψ(f )=σψ(λ)ρψ(f ) (2.8) wheneverρψis nonzero,λ∈F, andf∈A.
Proof ofTheorem1.2. PutΦ0def
= {ψ∈ΦB: kerρψ=A}(possibly empty), and let{ψα} ⊂Φ0be a net converging toψ0∈ΦB. Fixa∈A. Sinceρ(a)ˆis a continuous function onΦB, then 0=ρ(a)ˆ(ψα)→ρ(a)ˆ(ψ0). Sinceawas arbi- trary, it follows thatψ0∈Φ0, and hence,Φ0is a closed subset ofΦB. Moreover, ifAhas a unite, thenρψ(e)=0 or 1, and hence,
Φ0= ψ∈ΦB:ρψ(e)=0
= ψ∈ΦB:ρψ(e) <2−1
. (2.9)
It follows thatΦ0is closed and open wheneverAis unital.
Pickψ∈ΦB\Φ0. SinceBis strictly real, the mapσψas inDefinition 2.2is a nonzero ring homomorphism onRintoRso thatσψis the identity map onR. It follows from (2.8) that
ρψ(ta)=σψ(t)ρψ(a)=tρψ(a), t∈R, a∈A, (2.10) proving thatρψ∈ΦAfor everyψ∈ΦB\Φ0. Letϕ:ΦB\Φ0→ΦA be the map defined byϕ(ψ)def=ρψ. Then we have (1.2) for everya∈A. Finally, we show the continuity ofϕ. Suppose that{ψβ} ⊂ΦB\Φ0is a net converging toψ1∈ΦB\Φ0. Then (1.2) gives
aˆϕψβ
=ρ(a)ˆψβ
→ρ(a)ˆψ1
=aˆϕψ1
(2.11) for every a∈A. Henceϕ(ψβ) converges to ϕ(ψ1). This implies that ϕ is continuous onΦB\Φ0.
Proof ofTheorem1.3. Pick a∈ A and ψ∈ ΦB. If ρ(a)ˆ(ψ) ≠ 0, then σψ(i)= ±i, and hence,ρ(ia)ˆ(ψ) would be a nonzero pure imaginary num- ber by (2.8), in contradiction toBbeing strictly real.
Acknowledgment. The authors are partially supported by the Grants-in- Aid for Scientific Research, Japan Society for the Promotion of Science.
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Takeshi Miura: Department of Basic Technology, Faculty of Engineering, Yamagata University, Yonezawa 992-8510, Japan
E-mail address:[email protected]
Sin-Ei Takahasi: Department of Basic Technology, Faculty of Engineering, Yamagata University, Yonezawa 992-8510, Japan
E-mail address:[email protected]
