On the Nonexistence of Nontrivial Involutive n-Homomorphisms of
  C*-algebras
TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000–000
S 0002-9947(XX)0000-0
ON THE NONEXISTENCE OF NONTRIVIAL INVOLUTIVE
n-HOMOMORPHISMS OF C⋆-ALGEBRAS
EFTON PARK AND JODY TROUT
Abstract. An n-homomorphism between algebras is a linear map φ : A → B
such that φ(a1 · · · an) = φ(a1) · · ·φ(an) for all elements a1, . . . , an ∈ A. Every
homomorphism is an n-homomorphism, for all n ≥ 2, but the converse is false,
in general. Hejazian et al. [7] ask: Is every ∗-preserving n-homomorphism
between C⋆-algebras continuous? We answer their question in the affirmative,
but the even and odd n arguments are surprisingly disjoint. We then use these
results to prove stronger ones: If n > 2 is even, then φ is just an ordinary
∗-homomorphism. If n ≥ 3 is odd, then φ is a difference of two orthogonal
∗-homomorphisms. Thus, there are no nontrivial ∗-linear n-homomorphisms
between C⋆-algebras.
1. Introduction
Let A and B be algebras and n ≥ 2 an integer. A linear map φ : A → B is an
n-homomorphism if for all a1, a2, . . . , an ∈ A,
φ(a1a2 · · · an) = φ(a1)φ(a2) · · ·φ(an).
A 2-homomorphism is then just a homomorphism, in the usual sense, between
algebras. Furthermore, every homomorphism is clearly also an n-homomorphism
for all n ≥ 2, but the converse is false, in general. The concept of n-homomorphism
was studied for complex algebras by Hejazian, Mirzavaziri, and Moslehian [7]. This
concept also makes sense for rings and (semi)groups. For example, an AEn-ring is a
ring R such that every additive endomorphism φ : R → R is an n-homomorphism;
Feigelstock [4, 5] classified all unital AEn-rings.
In [7], Hejazian et al. ask: Is every ∗-preserving n-homomorphism between C⋆-
algebras continuous? We answer in the affirmative by proving that every involutive
n-homomorphism φ : A → B between C⋆-algebras is in fact norm contractive:
‖φ‖ ≤ 1. Surprisingly, the arguments for the even and odd n cases are disjoint
and, thus, are discussed in different sections. When n = 3, automatic continuity is
reported by Bračič and Moslehian [2], but note that the proof of their Theorem 2.1
does not extend to the nonunital case since the unitization of a 3-homomorphism
is not a 3-homomorphism, in general.
Using these automatic continuity results, we prove the following stronger results:
If n > 2 is even, every ∗-linear n-homomorphism φ : A → B between C⋆-algbras
is in fact a ∗-homomorphism. If n ≥ 3 is odd, every ∗-linear n-homomorphism
φ : A→ B is a difference φ(a) = ψ1(a)−ψ2(a) of two orthogonal ∗-homomorphisms
ψ1 ⊥ ψ2. Regardless, for all integers n ≥ 3, every positive linear n-homomorphism
MSC 2000 Classification: Primary 46L05; Secondary 47B99, 47L30.
c©1997 American Mathematical Society
http://arxiv.org/abs/0704.0910v3
2 EFTON PARK AND JODY TROUT
is a ∗-homomorphism. Note that if ψ is a ∗-homomorphism, then −ψ = 0− ψ is a
norm contractive ∗-preserving 3-homomorphism that is not positive linear.
There is also a dichotomy between the unital and nonunital cases. When the
domain algebra A is unital, there is a simple representation of an n-homomorphism
as a certain n-potent multiple of a homomorphism (discussed in the Appendix.)
The nonunital case is more subtle. For example, if A and B are nonunital (Banach)
algebras such that An = Bn = {0}, then every linear map L : A→ B (bounded or
unbounded) is, trivially, an n-homomorphism (see Examples 2.5 and 4.3 of [7]).
The outline of the paper is as follows: In Section 2, we prove automatic continuity
for the even case and in Section 3 for the odd case. In Section 4, we prove our
nonexistence results. A key fact in many of our proofs is the Cohen Factorization
Theorem [3] of C⋆-algebras. (See Proposition 2.33 [8] for an elementary proof of this
important result.) Finally, in Appendix A, we collect some facts about n-potents
that we need.
The authors would like to thank Dana Williams and Tom Shemanske for their
helpful comments and suggestions.
2. Automatic Continuity: The Even Case
In this section, we prove that when n > 2 is even, every involutive (i.e., ∗-linear)
n-homomorphism between C⋆-algebras is completely positive and norm contractive,
which generalizes the well-known result for ∗-homomorphisms (n = 2). Recall that
a linear map θ : A→ B between C⋆-algebras is positive if a ≥ 0 implies θ(a) ≥ 0 or,
equivalently, for every a ∈ A there is a b ∈ B such that θ(a∗a) = b∗b. We say that
θ is completely positive if, for all k ≥ 1, the induced map θk : Mk(A) → Mk(B),
θk((aij)) = (θ(aij)), on k × k matrices is positive.
Theorem 2.1. Let H be a Hilbert space. If n ≥ 2 is even, then every involutive
n-homomorphism from a C*-algebra A into B(H) is completely positive.
Proof. Let φ : A → B(H) be an involutive n-homomorphism. We may assume
n = 2k > 2. Let 〈·, ·〉 denote the inner product on H. By Stinespring’s Theorem
[9] (see Prop. II.6.6 [1]), φ is completely positive if and only for any m > 1 and
elements a1, . . . , am ∈ A and vectors v1, . . . , vm ∈ H we have
i,j=1
〈φ(a∗i aj)vj , vi〉 ≥ 0.
We proceed as follows: for each 1 ≤ i ≤ m use the Cohen Factorization Theorem
[3] to factor ai = ai1 · · · aik into a product of k elements. Thus, their adjoints factor
as a∗i = a
ik · · · a
i1. Since n = 2k, we compute
i,j=1
〈φ(a∗i aj)vj , vi〉 =
i,j=1
〈φ(a∗ik · · ·a
i1aj1 · · · ajk)vj , vi〉
i,j=1
〈φ(aik)
∗ · · ·φ(ai1)
∗φ(aj1) · · ·φ(ajk)vj , vi〉
φ(aj1) · · ·φ(ajk)vj ,
φ(ai1) · · ·φ(aik)vj〉
= 〈x, x〉 ≥ 0,
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 3
where x =
i=1 φ(ai1) · · ·φ(aik)vi ∈ H. The result now follows. �
Even though the previous result is a corollary of the more general theorem below,
we have included it because the proof technique is different.
Lemma 2.2. Let φ : A → B be an n-homomorphism. Then, for all k ≥ 1, the
induced maps φk : Mk(A) → Mk(B) on k × k matrices are n-homomorphisms.
Moreover, if φ is involutive (φ(a∗) = φ(a)∗), then each φk is also involutive.
Proof. Given n matrices a1 = (a1ij), . . . , a
n = (anij) in Mk(A), we can express
their product a1a2 · · · an = (aij), where the (i, j)-th entry aij is given by the formula
aij =
m1,··· ,mn−1=1
a1im1a
· · · anmn−1j .
Since φk(a
1a2 · · · an) = (φ(aij)) by definition and
φ(aij) =
m1,··· ,mn−1=1
φ(a1im1a
· · ·anmn−1j)
m1,··· ,mn−1=1
φ(a1im1)φ(a
) · · ·φ(anmn−1j)
= [φk(a
1)φk(a
2) · · ·φk(a
n)]ij ,
it follows that φk : Mk(A) → Mk(B) is an n-homomorphism. Now suppose that φ
is involutive. We compute for all a = (aij) ∈Mk(A):
∗) = φk((a
ji)) = (φ(a
ji)) = (φ(aji)
∗) = φk(a)
and hence each φk :Mk(A) →Mk(B) is involutive. �
Theorem 2.3. Let φ : A → B be an involutive n-homomorphism between C*-
algebras. If n ≥ 2 is even, then φ is completely positive. Thus, φ is bounded.
Proof. We may assume n = 2k > 2. Since φ is linear, we want to show that for
every a ∈ A we have φ(a∗a) ≥ 0. By the Cohen Factorization Theorem, for any
a ∈ A we can find a1, ..., ak ∈ A such that the factorization a = a1 · · · ak holds.
Thus, the adjoint factors as a∗ = a∗k · · ·a
1. Since n = 2k and φ is n-multiplicative
and ∗-preserving,
φ(a∗a) = φ(a∗k · · · a
1a1 · · · ak)
= φ(ak)
∗ · · ·φ(a1)
∗φ(a1) · · ·φ(ak)
= (φ(a1) · · ·φ(ak))
∗(φ(a1) · · ·φ(ak))
= b∗b ≥ 0,
where b = φ(a1) · · ·φ(ak) ∈ B. Thus, φ is a positive linear map. By the previous
lemma, all of the induced maps φk : Mk(A) → Mk(B) on k × k matrices are
involutive n-homomorphisms and are positive. Hence, φ is completely positive and
therefore bounded [1]. �
We now wish to show that if n ≥ 2 is even, then an involutive n-homomorphism
is actually norm-contractive. First, we will need generalizations of the familiar
C⋆-identity appropriate for n-homomorphisms.
4 EFTON PARK AND JODY TROUT
Lemma 2.4. Let A be a C⋆-algebra. For all k ≥ 1, we have that
‖x‖2k = ‖(x∗x)k‖
‖x‖2k+1 = ‖x(x∗x)k‖
for all x ∈ A.
Proof. In the even case, we have easily that
‖x‖2k = (‖x‖2)k = ‖x∗x‖k = ‖(x∗x)k‖
by the functional calculus since x∗x ≥ 0. In the odd case, we compute again using
the C⋆-identity and functional calculus:
‖x(x∗x)k‖2 = ‖(x(x∗x)k)∗(x(x∗x)k)‖
= ‖(x∗x)kx∗x(x∗x)k‖
= ‖(x∗x)2k+1‖ = ‖(x∗x)‖2k+1
= (‖x‖2)2k+1 = (‖x‖2k+1)2;
the result follows by taking square roots. �
Theorem 2.5. Let φ : A → B be an involutive n-homomorphism of C⋆-algebras.
If φ is bounded, then φ is norm contractive (‖φ‖ ≤ 1).
Proof. Suppose n = 2k is even. Then for all x ∈ A we have
(x∗x)k
= φ(x∗x · · ·x∗x) =
φ(x∗)φ(x)
φ(x)∗φ(x)
Thus by the previous lemma,
‖φ(x)‖n = ‖φ(x)‖2k
= ‖(φ(x)∗φ(x))k‖ = ‖φ((x∗x)k)‖
≤ ‖φ‖‖(x∗x)k‖ = ‖φ‖‖x‖2k = ‖φ‖‖x‖n,
which implies that ‖φ‖ ≤ 1 by taking n-th roots.
The proof for the odd case n = 2k + 1 is similar. �
3. Automatic Continuity: The Odd Case
The positivity methods above do not work when n is odd, since the negation
of a ∗-homomorphism defines an involutive 3-homomorphism that is (completely)
bounded, but not positive. We need the following slight generalization of Lemma
3.5 of Harris [6].
Lemma 3.1. Let A be a C⋆-algebra and let λ 6= 0 and k ≥ 1. If a ∈ A then
λ ∈ σ((a∗a)k) if and only if there does not exist an element c ∈ A with
(1) c (λ− (a∗a)k) = a.
Proof. If λ 6∈ σ((a∗a)k), then c = a(λ− (a∗a)k)−1 ∈ A satisfies
c (λ− (a∗a)k) = a(λ− (a∗a)k)−1(λ− (a∗a)k) = a.
and so (1) holds.
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 5
On the other hand, if λ ∈ σ((a∗a)k) then, by the commutative functional
calculus, there is a sequence {bm}
1 in the unitization A
+ with bm 6→ 0 but
dm =def (λ − (a
∗a)k)bm → 0. Since λ 6= 0 we must have
a∗(aa∗)k−1(abm) = (a
∗a)kbm = λbm − dm 6→ 0,
which implies abm 6→ 0. Hence, there does not exist an element c ∈ A that can
satisfy equation (1), since this would imply that
abm = c (λ− (a
∗a)k)bm → 0,
which is a contradiction. This proves the lemma. �
We now prove automatic continuity for involutive n-homomorphisms of C⋆-
algebras for all odd values of n. Note that we do not assume that A is unital,
nor do we appeal to the unitization φ+ : A+ → B+ of φ, which is not an n-
homomorphism, in general.
Theorem 3.2. Let φ : A → B be an involutive n-homomorphism between C⋆-
algebras. If n ≥ 3 is odd, then ‖φ‖ ≤ 1, i.e., φ is norm contractive.
Proof. Let n = 2k + 1 where k ≥ 1. Given any a ∈ A and λ > 0 such that that
λ 6∈ σ((a∗a)k), there is, by the previous lemma, an element c ∈ A such that
a = c (λ− (a∗a)k) = (λc− c(a∗a)k).
Noting that c(a∗a)k is a product of 2k + 1 = n elements in A, and φ is a ∗-linear
n-homomorphism, we compute:
φ(a) = φ(λc − c(a∗a)k) = λφ(c) − φ(c(a∗a)k)
= λφ(c) − φ(c)(φ(a)∗φ(a))k = φ(c)(λ − (φ(a)∗φ(a))k)
which yields that there is an element φ(c) ∈ B with:
φ(c)(λ − (φ(a)∗φ(a))k) = φ(a).
By the previous lemma, we conclude that λ 6∈ σ((φ(a)∗φ(a))k). Thus, we have
shown the following inclusion of spectra:
σ((φ(a)∗φ(a))k) ⊆ σ((a∗a)k) ∪ {0}.
Therefore, by the spectral radius formula [1, II.1.6.3] and the generalization of the
C⋆-identity in Lemma 2.4, we must deduce that:
‖φ(a)‖2k = ‖(φ(a)∗φ(a))k‖
= r((φ(a)∗φ(a))k) ≤ r((a∗a)k)
= ‖(a∗a)k‖ = ‖a‖2k,
which implies that ‖φ(a)‖ ≤ ‖a‖ for all a ∈ A, as desired. �
Note that the argument in the previous proof does not work for n = 2k even,
since we would need to employ (a∗a)k−1a which is a product of 2k − 1 = n − 1
elements as needed, but not self-adjoint, in general. Thus, we could not appeal
to the spectral radius formula for self-adjoint elements and Lemma 3.1 would not
apply. Hence, the even and odd n arguments are essentially disjoint.
6 EFTON PARK AND JODY TROUT
4. Nonexistence of Nontrival Involutive n-homomorphisms of
C⋆-algebras
Our first main result is the nonexistence of nontrivial n-homomorphisms on
unital C⋆-algebras for all n ≥ 3. We do the unital case first since it is much simpler
to prove and helps to frame the argument for the nonunital case.
Theorem 4.1. Let φ : A→ B be an involutive n-homomorphism between the C⋆-
algebras A and B, where A is unital. If n ≥ 2 is even, then φ is a ∗-homomorphism.
If n ≥ 3 is odd, then φ is the difference φ(a) = ψ1(a) − ψ2(a) of two orthogonal
∗-homomorphisms ψ1 ⊥ ψ2 : A→ B.
Proof. In either case, by Proposition A.1, the element e = φ(1) ∈ B is an
n-potent (en = e) and is self-adjoint, because
e = φ(1) = φ(1∗) = φ(1)∗ = e∗.
Also, there is an associated algebra homomorphism ψ : A→ B defined for all a ∈ A
by the formula
ψ(a) = en−2φ(a) = φ(a)en−2
such that φ(a) = eψ(a) = ψ(a)e. In either case, ψ is ∗-linear since φ is ∗-linear and
e is self-adjoint and commutes with the range of φ:
ψ(a∗) = en−2φ(a∗) = en−2φ(a)∗ =
en−2φ(a)
= ψ(a)∗.
Now, if n = 2k is even, e = en = (ek)∗ek ≥ 0 and so e = p is a projection. Thus,
φ(a) = pψ(a) = ψ(a)p = pψ(a)p is a ∗-homomorphism. If n ≥ 3 is odd, then by
Lemma A.8, e is the difference of two orthogonal projections e = p1−p2 which must
commute with both ψ and φ by the functional calculus. Define ψ1, ψ2 : A → B
by ψi(a) = piψ(a)pi for all a ∈ A and i = 1, 2. Then ψ2 ⊥ ψ2 are orthogonal
∗-homomorphisms, and
ψ1(a)− ψ2(a) = p1ψ(a)− p2ψ(a) = eψ(a) = φ(a)
for all a ∈ A, from which the desired result follows. �
Corollary 4.2. Let φ : A→ B be a linear map between C⋆-algebras. If A is unital,
the following are equivalent for all integers n ≥ 2:
a.) φ is a ∗-homomorphism.
b.) φ is a positive n-homomorphism.
c.) φ is an involutive n-homomorphism and φ(1) ≥ 0.
Proof. Clearly (a) =⇒ (b) =⇒ (c). If n ≥ 2 is even, then (c) =⇒ (a) by the
previous result. If n ≥ 3 is odd, then by the previous result, we only need to show
that φ is positive. Let n = 2k + 1. Given any a ∈ A, by the Cohen Factorization
Theorem, we can write a = a1 · · · ak. Since φ(1) ≥ 0, by hypothesis, and n = 2k+1,
we compute:
φ(a∗a) = φ(a∗1a) = φ(a∗k · · · a
11a1 · · ·ak)
= φ(ak)
∗ · · ·φ(a1)
∗φ(1)φ(a1) · · ·φ(ak)
φ(a1) · · ·φ(ak)
φ(a1) · · ·φ(ak)
= b∗φ(1)b ≥ 0,
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 7
where b = φ(a1) · · ·φ(ak) ∈ B. Thus, φ is positive linear and therefore a ∗-
homomorphism. �
Next, we extend our nonexistence results to the nonunital case, by appealing to
approximate unit arguments (which require continuity!) and the following impor-
tant factorization property of ∗-preserving n-homomorphisms.
Lemma 4.3 (Coherent Factorization Lemma). Let φ : A → B be an involutive
n-homomorphism of C⋆-algebras. For any 1 ≤ k ≤ n and any a ∈ A, if a =
a1 · · ·ak = b1 · · · bk in A, then
φ(a1) · · ·φ(ak) = φ(b1) · · ·φ(bk) ∈ B.
Note that, in general, φ(a) 6= φ(a1) · · ·φ(ak) when 1 < k < n.
Proof. Clearly, we may assume 1 < k < n. Since φ is ∗-linear, the range
φ(A) ⊂ B is a self-adjoint linear subspace of B (but not necessarily a subalgebra,
in general). Given any d = φ(c) ∈ φ(A), using the Cohen Factorization Theorem,
write d = d1 · · · dn = φ(c1) · · ·φ(cn) where di = φ(ci) for 1 ≤ i ≤ n. Consider the
following computation:
φ(a1) · · ·φ(ak)d = φ(a1) · · ·φ(ak)φ(c1) · · ·φ(cn)
= φ(a1 · · · akc1 · · · cn−k)φ(cn−k+1) · · ·φ(cn)
= φ(b1 · · · bkc1 · · · cn−k)φ(cn−k+1) · · ·φ(cn)
= φ(b1) · · ·φ(bk)φ(c1) · · ·φ(cn)
= φ(b1) · · ·φ(bk)d.
Let f = φ(a1) · · ·φ(ak) − φ(b1) · · ·φ(bk). Then fd = 0 for all d ∈ φ(A) ⊂ B, and
thus fd = 0 for all d in the ∗-subalgebra Aφ of B generated by φ(A). In particular,
for the element
da = φ(a
k) · · ·φ(a
1)− φ(b
k) · · ·φ(b
1) = f
∗ ∈ Aφ.
Hence, ff∗ = fda = 0 and so ‖f‖
2 = ‖ff∗‖ = 0 by the C⋆-identity. Therefore,
φ(a1) · · ·φ(ak)− φ(b1) · · ·φ(bk) = f = 0,
and the result is proven. �
Definition 4.4. An approximate unit for a (nonunital) C⋆-algebra A is a net
{eλ}λ∈Λof elements in A indexed by a directed set Λ such that
a.) 0 ≤ eλ and ‖eλ‖ ≤ 1 for all λ ∈ Λ;
b.) eλ ≤ eµ if λ ≤ µ in Λ;
c.) For all a ∈ A,
‖aeλ − a‖ = lim
‖eλa− a‖ = 0.
Every C⋆-algebra has an approximate unit, which is countable (Λ = N) if A is
separable (see Section II.4 of Blackadar [1].)
Theorem 4.5. Suppose φ : A → B is an involutive n-homomorphism of C⋆-
algebras, where A is nonunital. Then, for all a ∈ A, the limit
ψ(a) = lim
φ(eλ)
n−2φ(a) = lim
φ(a)φ(eλ)
8 EFTON PARK AND JODY TROUT
exists, independently of the choice of the approximate unit {eλ} of A, and defines
a ∗-homomorphism ψ : A→ B such that
φ(a) = lim
φ(eλ)ψ(a)
for all a ∈ A.
Proof. We may assume n ≥ 3. Given a ∈ A, use the Cohen Factorization
Theorem to factor a = a1a2 · · · an. Define a map ψ : A→ B by
ψ(a) = φ(a1a2)φ(a3) · · ·φ(an) = φ(a1) · · ·φ(an−2)φ(an−1an),
which is well-defined by the Coherent Factorization Lemma. The continuity of φ
implies that
φ(eλ)
n−2φ(a) = lim
φ(eλ)
n−2φ(a1) · · ·φ(an)
= lim
φ(en−2λ a1a2)φ(a3) · · ·φ(an)
= φ(a1a2)φ(a3) · · ·φ(an) = ψ(a) ∈ B.
It follows that we can write:
ψ(a) = lim
φ(eλ)
n−2φ(a) = lim
φ(a)φ(eλ)
and so ψ : A→ B is linear since φ is linear. Moreover, since φ is ∗-linear, it follows
that ψ is also ∗-linear:
ψ(a)∗ =
φ(a1a2)φ(a3) · · ·φ(an)
= φ(an)
∗ · · ·φ(a3)
φ(a1a2)
= φ(a∗n) · · ·φ(a
3)φ(a
= φ(a∗n1a
n2)φ(a
n−1) · · ·φ(a
= ψ((a∗n1a
n2)(a
n−1) · · · (a
= ψ(a∗n · · ·a
1) = ψ(a
In the computation above, we factored an = an2an1 and set a12 = a1a2 to obtain
the factorization a∗ = a∗n · · ·a
1 = (a
n−1 · · · a
12 into n elements. Given
a, b ∈ A with factorizations a = a1 · · · an and b = b1 · · · bn, the fact that φ is an
n-homomorphism implies:
ψ(a)ψ(b) =
φ(a1a2)φ(a3) · · ·φ(an)
φ(b1b2)φ(b3) · · ·φ(bn)
= φ((a1a2)a3 · · · an(b1b2))φ(b3) · · ·φ(bn)
= φ((ab1)b2)φ(b3) · · ·φ(bn)
= ψ(ab);
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 9
note that ab = (ab1)b2b3 · · · bn is a factorization of ab into n elements. A second
proof of multiplicativity goes as follows:
ψ(ab) = lim
φ(eλ)
φ(ab) = lim
φ(eλ)
φ( lim
= lim
φ(eλ)
n−2 lim
φ(aen−2µ b)
= lim
φ(eλ)
n−2 lim
φ(a)φ(eµ)
n−2φ(b)
= lim
φ(eλ)
n−2φ(a) lim
φ(eµ)
n−2φ(b)
= ψ(a)ψ(b).
Thus, ψ is a well-defined ∗-homomorphism. Finally, we compute:
φ(eλ)ψ(a) = lim
φ(eλ)φ(a1a2)φ(a3) · · ·φ(an)
= lim
φ(eλ(a1a2)a3 · · · an) = lim
φ(eλa)
= φ(a).
Using similar factorizations, the fact that {enλ} is also an approximate unit for
A, and the fact that the strict completion of the C⋆-algebra C⋆(φ(A)) generated
by the range φ(A) is the multiplier algebra M(C⋆(ψ(A))), we obtain the nonunital
version of Proposition A.1.
Corollary 4.6. Suppose that A and B are C⋆-algebras with A nonunital, and let
φ : A → B be an involutive n-homomorphism with associated ∗-homomorphism
ψ : A→ B. Then there is a self-adjoint n-potent e = e∗ = en ∈M(C∗(φ(A))) such
that φ(eλ) → e strictly for any approximate unit {eλ} of A, and with the property
φ(a) = eψ(a) = ψ(a)e
ψ(a) = en−2φ(a)
for all a ∈ A.
Proof. By the previous proof, we can define e ∈ M(C⋆(φ(A))) on generators
φ(a) by
eφ(a) = lim
φ(eλ)φ(a) = φ(a1a2 · · · an−1)φ(an) ∈ C
⋆(φ(A))
for any a = a1 · · · an ∈ A. It follows that:
enφ(a) = lim
φ(eλ)
nφ(a)
= lim
φ(enλ)φ(a1)φ(a2) · · ·φ(an)
= lim
φ((enλ)a1a2 · · · an−1)φ(an)
= φ(a1 · · · an−1)φ(an) = eφ(a),
which implies e ∈ M(C⋆(φ(A))) is n-potent. The fact that e = e∗ follows from
φ(eλ)
∗ = φ(e∗λ) = φ(eλ). The other statements follow from the previous proof. �
The dichotomy between the unital and nonunital cases is now clear. If A is unital,
then C⋆(φ(A)) ⊂ B is a unital C⋆-subalgebra of B with unit ψ(1) = φ(1)n−1 ∈ B
(which is a projection!) and so
M(C⋆(ψ(A))) = C⋆(φ(A)) ⊂ B.
10 EFTON PARK AND JODY TROUT
However, for A nonunital, we cannot identify the multiplier algebra M(C⋆(φ(A)))
as a subalgebra of B, or evenM(B), unless φ is surjective. In general, we only have
inclusions ψ(A) ⊂ C⋆(φ(A)) ⊂ B.
Now that we know, as in the unital case, every involutive n-homomorphism is an
n-potent multiple of a ∗-homomorphism, we can prove the following general version
of Theorem 4.1 and its corollary in a similar manner using Lemma A.8.
Theorem 4.7. Let φ : A→ B be an involutive n-homomorphism of C⋆-algebras. If
n ≥ 2 is even, then φ is a ∗-homomorphism. If n ≥ 3 is odd, then φ is the difference
φ(a) = ψ1(a)− ψ2(a) of two orthogonal ∗-homomorphisms ψ1 ⊥ ψ2 : A→ B.
Corollary 4.8. For all n ≥ 2 and C⋆-algebras A and B, φ : A → B is a positive
n-homomorphism if and only if φ is a ∗-homomorphism.
Appendix A. On n-homomorphisms and n-potents
An element x ∈ A is called an n-potent if xn = x. Note that if φ : A→ B is an n-
homomorphism, then φ(x) = φ(xn) = φ(x)n ∈ B is also an n-potent. The following
important result is Proposition 2.2 [7], whose proof is included for completeness.
Proposition A.1. If A is a unital algebra (or ring) and φ : A → B is an n-
homomorphism, then there is a homomorphism ψ : A → B and an n-potent e =
en ∈ B such that φ(a) = eψ(a) = ψ(a)e for all a ∈ A. Also, e commutes with the
range1 of φ, i.e., eφ(a) = φ(a)e for all a ∈ A.
Proof. Note that e = φ(1) = φ(1n) = φ(1)n = en ∈ B is an n-potent. Define a
linear map ψ : A→ B by ψ(a) = en−1φ(a) for all a ∈ A. For all a, b ∈ A,
ψ(ab) = en−2φ(ab) = en−2φ(a1n−2b)
en−2φ(a)
φ(1)n−2φ(b)
= ψ(a)ψ(b),
and so ψ is an algebra homomorphism. Furthermore,
eψ(a) = φ(1)(φ(1)n−2φ(a)) = φ(1)n−1φ(a) = φ(1n−1a) = φ(a).
Similarly, ψ(a)e = φ(a) for all a ∈ A. The final statement is a consequence of the
fact that for all a ∈ A,
eφ(a) = φ(1)φ(a1n−1) =
φ(1)φ(a)φ(1)n−2
φ(1) = φ(1a1n−2)e = φ(a)e.
The following computation will be more significant when we consider the nonuni-
tal case (see the proof of Theorem 4.5.)
Corollary A.2. Let φ and ψ be as in Proposition A.1 and n ≥ 3. Then for all
a ∈ A, if a = a1a2 · · ·an with a1, . . . , an ∈ A,
ψ(a) = φ(a1a2)φ(a3) · · ·φ(an).
1Note that the range φ(A) is not a subalgebra of B in general.
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 11
Proof. We compute as follows:
ψ(a) =def e
n−2φ(a) = φ(1)n−2φ(a1 · · · an)
= φ(1)n−2φ(a1) · · ·φ(an)
φ(1)n−2φ(a1)φ(a2)
φ(a3) · · ·φ(an)
= φ(1n−2a1a2)φ(a3) · · ·φ(an)
= φ(a1a2)φ(a3) · · ·φ(an). �
Definition A.3. Let A be a unital algebra. An n-partition of unity is an ordered
n-tuple (e0, e1, . . . , en−1) of idempotents (e
k = ek) that sum to the identity e0 +
e1 + · · · + en−1 = 1 and are pairwise mutually orthogonal, i.e., ejek = δjk1 for all
0 ≤ j, k ≤ n− 1, where δjk is the Kronecker delta.
Note that e0 = 1− (e1+ · · ·+ en−1) is completely determined by e1, e2, . . . , en−1
and is thus redundant in the notation for an n-partition of unity.
Definition A.4. Let ω0 = 0 and ωk = e
2πi(k−1)/(n−1) for 1 ≤ k ≤ n − 1.
Note that ω1 = 1 and ω1, . . . , ωn−1 are the (n − 1)-th roots of unity and Σn =
{ω0, ω1, . . . , ωn−1} are the n roots of the polynomial equation x
n−x = x(xn−1−1) =
If A is a complex algebra, we let Ã denote A, if A is unital, or the unitization
A+ = A⊕ C, if A is nonunital.
Theorem A.5. Let A be a complex algebra. If e ∈ A is an n-potent, there is a
unique n-partition of unity (e0, e1, . . . , en−1) in Ã such that
ωkek.
If A is nonunital, then e1, . . . , en−1 ∈ A.
Proof. Define the n polynomials p0, p1, . . . , pn−1 by
pk(x) =
j 6=k(x− ωj)
j 6=k(ωk − ωj)
In particular, p0(x) = 1− x
n−1. Each polynomial pk has degree n− 1 and satisfies
pk(ωk) = 1 and pk(ωj) = 0 for all j 6= k. It follows that pj(x)pk(x) = 0 for all
x ∈ Σn. We also claim that for all x ∈ C that
pk(x) = p0(x) + · · ·+ pn−1(x) = 1
(3) x =
ωkpk(x).
Indeed, these identities follow from the fact that these polynomial equations have
degree n− 1 but are satisfied by the n distinct points in Σn.
Now, given any xn = x in C it follows that pk(x)
2 = pk(x). Hence, for any n-
potent e ∈ A, if we define ek = pk(e) then (e0, e1, . . . , en−1) consists of idempotents
12 EFTON PARK AND JODY TROUT
e2k = pk(e)
2 = pk(e) = ek and satisfy, by (2),
pk(e) = 1Ã.
They are pairwise orthogonal, because ejek = pj(e)pk(e) = 0 for j 6= k. Moreover,
ωkpk(e) =
by Equation (3). For 1 ≤ k ≤ n− 1, note that pk(x) = xqk(x) for some polynomial
qk(x). Hence, if A is nonunital and 1 ≤ k ≤ n−1, we have ek = pk(e) = eqk(e) ∈ A,
since A is an ideal in Ã. �
The following result is the n-homomorphism version of the previous n-potent
result. Recall say that two linear maps ψi, ψj : A→ B are orthogonal (ψi ⊥ ψj) if
ψi(a)ψj(b) = ψj(b)ψi(a) = 0
for all a, b ∈ A.2
Proposition A.6. Let A and B be complex algebras. If A is unital then a linear
map φ : A → B is an n-homomorphism if and only if there exist n − 1 mutually
orthogonal homomorphisms ψ1, . . . , ψn−1 : A→ B such that for all a ∈ A,
φ(a) =
ωkψk(a).
Proof. (⇒) Let φ : A → B be an n-homomorphism. By Proposition A.1, there
is an n-potent e ∈ B and a homomorphism ψ : A → B such that φ(a) = eψ(a) =
ψ(a)e. Using the previous result, write e =
k=1 ωkek, where (e0, e1, . . . , en−1)
is the associated n-partition of unity in Ã defined by the polynomials pk. Since
ek = pk(e), we have that ekψ(a) = ψ(a)ek for 1 ≤ k ≤ n− 1. Define ψk : A → B
ψk(a) =def ekψ(a) = e
kψ(a) = ekψ(a)ek.
Then ψ1, . . . , ψn−1 are orthogonal homomorphisms and, for all a ∈ A,
φ(a) = eψ(a) =
ωkekψ(a) =
ωkψk(a).
(⇐) Follows from the fact that ωnk = ωk for all k = 1, . . . , n− 1. �
Remark A.7. If A is nonunital, the above result does not hold. One reason is
that the unitization φ+ : A+ → B+ of an n-homomorphism is not, in general, an
n-homomorphism. Also, if An = Bn = {0}, then every linear map L : A → B is
an n-homomorphism (See Examples 2.5 and 4.3 of Hejazian et al [7]).
Let Σn be the n roots of the polynomial equation x = x
n from Definition A.4.
If A is a C⋆-algebra, it follows that a normal n-potent e = en must have spectrum
σ(e) ⊆ Σn. Recall that a projection is an element p = p
∗ = p2 ∈ A. Two projections
p1 and p2 are orthogonal if p1p2 = 0. A tripotent is a 3-potent element e
3 = e ∈ A.
The following characterization of self-adjoint n-potents in C⋆-algebras is impor-
tant for our nonexistence results on n-homomorphisms.
2Note that the zero homomorphism is orthogonal to every homomorphism.
NONEXISTENCE OF NONTRIVIAL n-HOMOMORPHISMS 13
Lemma A.8. Let A be a C⋆-algebra.
a.) If n ≥ 2 is an even integer, the following are equivalent:
i.) e is a projection.
ii.) e is a positive n-potent.
iii.) e is a self-adjoint n-potent.
b.) If n ≥ 3 is an odd integer, the following are equivalent:
i.) e is a self-adjoint tripotent.
ii.) e = p1 − p2 is a difference of two orthogonal projections.
iii.) e is a self-adjoint n-potent.
Proof. In both the even and odd cases, (i) =⇒ (ii) =⇒ (iii) (See Theorem
A.5). Suppose (iii) holds. If n = 2k is even,
e = e∗ = en = e2k = (ek)∗(ek) ≥ 0,
and so the spectrum of e satisfies σ(e) ⊂ Σn∩[0,∞] = {0, 1}. Thus, e is a projection.
If n ≥ 3 is odd, then since e = e∗ we must have σ(e) ⊂ Σn ∩R = {−1, 0, 1}. Thus,
λ = λ3 for all λ ∈ σ(e), which implies e = e3 is tripotent. �
References
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[2] J. Bračič and S. Moslehian, On Automatic Continuity of 3-Homomorphisms on Banach
Algebras, to appear in Bull. Malays. Math. Sci. Soc. arXiv: math.FA/0611287.
[3] P. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959) 199–205.
[4] S. Feigelstock, Rings whose additive endomorphisms are N-multiplicative, Bull. Austral.
Math. Soc. 39 (1989), no. 1, 11–14.
[5] S. Feigelstock, Rings whose additive endomorphisms are n-multiplicative. II, Period. Math.
Hungar. 25 (1992), no. 1, 21–26.
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[7] M. Hejazian, M. Mirzavaziri, and M.S. Moslehian, n-homomorphisms, Bull. Iranian Math.
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Box 298900, Texas Christian University, Fort Worth, TX 76129
E-mail address: e.park@tcu.edu
6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755
E-mail address: jody.trout@dartmouth.edu
http://arxiv.org/abs/math/0611287
	1. Introduction
	2. Automatic Continuity: The Even Case
	3. Automatic Continuity: The Odd Case
	4. Nonexistence of Nontrival Involutive n-homomorphisms of C-algebras
	Appendix A. On n-homomorphisms and n-potents
	References
