Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras
In this paper, we proved that a bijective mapping Φ : A → B satisfies Φ(a ◦ b + baΦ) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)* (where ◦ is the special Jordan product on A and B, respectively), for all elements a, b ∈ A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are...
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irk-123456789-1886772023-03-12T01:28:59Z Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras Ferreira, J.C.M. Marietto, M.G.B. In this paper, we proved that a bijective mapping Φ : A → B satisfies Φ(a ◦ b + baΦ) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)* (where ◦ is the special Jordan product on A and B, respectively), for all elements a, b ∈ A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism. 2021 Article Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras / J.C.M. Ferreira, M.G.B. Marietto // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 61–70. — Бібліогр.: 7 назв. — англ. 1726-3255 DOI:10.12958/adm1482 2020 MSC: 47B48, 46L10 http://dspace.nbuv.gov.ua/handle/123456789/188677 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, we proved that a bijective mapping Φ : A → B satisfies Φ(a ◦ b + baΦ) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)* (where ◦ is the special Jordan product on A and B, respectively), for all elements a, b ∈ A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism. |
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Ferreira, J.C.M. Marietto, M.G.B. |
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Ferreira, J.C.M. Marietto, M.G.B. Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras Algebra and Discrete Mathematics |
| author_facet |
Ferreira, J.C.M. Marietto, M.G.B. |
| author_sort |
Ferreira, J.C.M. |
| title |
Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras |
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Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras |
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Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras |
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Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras |
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Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras |
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mappings preserving sum of products a ◦ b + ba* on factor von neumann algebras |
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Інститут прикладної математики і механіки НАН України |
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2021 |
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| citation_txt |
Mappings preserving sum of products a ◦ b + ba* on factor von Neumann algebras / J.C.M. Ferreira, M.G.B. Marietto // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 61–70. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT ferreirajcm mappingspreservingsumofproductsabbaonfactorvonneumannalgebras AT mariettomgb mappingspreservingsumofproductsabbaonfactorvonneumannalgebras |
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2025-07-16T10:50:34Z |
| last_indexed |
2025-07-16T10:50:34Z |
| _version_ |
1837800404438482944 |
| fulltext |
“adm-n1” — 2021/4/10 — 20:38 — page 61 — #65
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 1, pp. 61–70
DOI:10.12958/adm1482
Mappings preserving sum of products a ◦ b + ba
∗
on factor von Neumann algebras
J. C. M. Ferreira and M. G. B. Marietto
Communicated by V. M. Futorny
Abstract. Let A and B be two factor von Neumann algebras.
In this paper, we proved that a bijective mapping Φ : A → B satisfies
Φ(a ◦ b + ba∗) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)∗ (where ◦ is the special
Jordan product on A and B, respectively), for all elements a, b ∈ A,
if and only if Φ is a ∗-ring isomorphism. In particular, if the von
Neumann algebras A and B are type I factors, then Φ is a unitary
isomorphism or a conjugate unitary isomorphism.
1. Introduction
Let R and C be the real and complex number fields, respectively,
and A and B be two algebras over C. If a, b ∈ A (resp., a, b ∈ B), let
a ◦ b = 1
2
(ab + ba). We call ◦ the special Jordan product on A (resp.,
special Jordan product on B). We say that a mapping Φ : A → B preserves
product if Φ(ab) = Φ(a)Φ(b), for all elements a, b ∈ A, that preserves
special Jordan product if Φ(a ◦ b) = Φ(a) ◦ Φ(b), for all elements a, b ∈ A,
that preserves Jordan product if Φ(ab+ ba) = Φ(a)Φ(b) + Φ(b)Φ(a), for
all elements a, b ∈ A, and that preserves Lie product if Φ(ab − ba) =
Φ(a)Φ(b)− Φ(b)Φ(a), for all elements a, b ∈ A. We say that a mapping
Φ : A → B is additive if Φ(a+ b) = Φ(a) + Φ(b), for all elements a, b ∈ A
and that is a ring isomorphism if Φ is an additive bijection that preserves
products.
2020 MSC: 47B48, 46L10.
Key words and phrases: ∗-ring isomorphisms, fator von Neumann algebras.
https://doi.org/10.12958/adm1482
“adm-n1” — 2021/4/10 — 20:38 — page 62 — #66
62 ∗-ring isomorphisms on factor von Neumann algebras
Let A and B be two ∗-algebras over C. We say that a mapping
Φ : A → B preserves involution if Φ(a∗) = Φ(a)∗, for all elements a ∈ A,
and that Φ is a ∗-ring isomorphism if Φ is a ring isomorphism that
preserves involution.
A von Neumann algebra A is a weakly closed, self-adjoint algebra of
operators on a Hilbert space H, which are denote in this paper by lower-
case Latin letters, containing the identity operator 1A, called elements of
A. A von Neumann algebra A is called factor if its center contains the
multiples of the identity operator only. It is well known that the factor
von Neumann algebra A is prime, that is, if a and b are elements of A
such that aAb = 0, then either a = 0 or b = 0.
The study of additivity or ring isomorphism of mappings preserving
products, special Jordan products, Jordan products and Lie products on
operator algebras have been received fair amount of attentions. Some
examples of these studies may be found in [1, 4–6]. On the other hand,
many mathematicians devoted themselves to study ∗-ring isomorphisms
of mappings satisfying new types of products on von Neumann algebras,
for example [2, 3, 7].
Let A and B be two ∗-algebras. We say that a mapping Φ : A → B
preserves sum of products a ◦ b+ ba∗ if
Φ(a ◦ b+ ba∗) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)∗, (1)
for all elements a, b ∈ A.
Inspired by the research described in [2], the aim of this paper is to
prove that a bijective mapping preserving sum of products a ◦ b+ ba∗ on
factor von Neumann algebras is a ∗-ring isomorphism. Our main result
reads as follows.
Main Theorem. Let A and B be two factor von Neumann algebras with
1A and 1B the identities of them, respectively. Then every bijective mapping
Φ : A → B satisfies Φ(a ◦ b + ba∗) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)∗, for all
elements a, b ∈ A, if and only if Φ is a ∗-ring isomorphism.
2. The proof of main theorem
In order to prove the Main Theorem we need to prove several lemmas.
We begin with the following lemma.
Lemma 1. Let A and B be two factor von Neumann algebras with 1A
the identity of A. Then every bijective mapping Φ : A → B satisfying
“adm-n1” — 2021/4/10 — 20:38 — page 63 — #67
J. C. M. Ferreira, M. G. B. Marietto 63
Φ(a ◦ b + ba∗) = Φ(a) ◦ Φ(b) + Φ(b)Φ(a)∗, for all elements a, b ∈ A, is
additive.
We shall organize the proof of Lemma 1 in a series of claims, based on
the techniques used by Li et al. [2]. Finally, we would like to mention the
following well-known result that will be used throughout this paper.
Let p1 be an arbitrary non-trivial projection of A and write p2 = 1−p1.
Then A has a Peirce decomposition A = A11 ⊕A12 ⊕A21 ⊕A22, where
Aij = piApj (i, j = 1, 2), satisfying the following multiplicative relations:
AijAkl ⊆ δjkAil, where δjk is the Kronecker delta function.
We begin with the following claim.
Claim 1. Let a, b, c be elements of A such that Φ(c) = Φ(a)+Φ(b). Then
hold the following identities: (i) Φ(t◦c+ct∗) = Φ(t◦a+at∗)+Φ(t◦b+bt∗)
and (ii) Φ(c◦ t+ tc∗) = Φ(a◦ t+ ta∗)+Φ(b◦ t+ tb∗), for all element t ∈ A.
Proof. For an arbitrary element t ∈ A, we have
Φ(t ◦ c+ ct∗) = Φ(t) ◦ Φ(c) + Φ(c)Φ(t)∗
= Φ(t) ◦ (Φ(a) + Φ(b)) + (Φ(a) + Φ(b))Φ(t)∗
= Φ(t) ◦ Φ(a) + Φ(a)Φ(t)∗ +Φ(t) ◦ Φ(b) + Φ(b)Φ(t)∗
= Φ(t ◦ a+ at∗) + Φ(t ◦ b+ bt∗).
Similarly, we obtain (ii).
Claim 2. Φ(0) = 0.
Proof. From the surjectivity of Φ there exists an element b ∈ A such that
Φ(b) = 0. It follows that
Φ(0) = Φ(0◦b+b0∗) = Φ(0)◦Φ(b)+Φ(b)Φ(0)∗ = Φ(0)◦0+0Φ(0)∗ = 0.
Claim 3. For arbitrary elements a11 ∈ A11, b12 ∈ A12, b21 ∈ A21 and
c22 ∈ A22 hold: (i) Φ(a11 + b12) = Φ(a11) + Φ(b12), (ii) Φ(a11 + b21) =
Φ(a11)+Φ(b21), (iii) Φ(b12+c22) = Φ(b12)+Φ(c22) and (iv) Φ(b21+c22) =
Φ(b21) + Φ(c22).
Proof. From the surjectivity of Φ, we can choose an element c = c11 +
c12 + c21 + c22 ∈ A such that Φ(c) = Φ(a11) + Φ(b12). By Claim 1(i), we
have
Φ(p2 ◦ c+ cp∗2) = Φ(p2 ◦ a11 + a11p
∗
2) + Φ(p2 ◦ b12 + b12p
∗
2) = Φ(
3
2
b12).
“adm-n1” — 2021/4/10 — 20:38 — page 64 — #68
64 ∗-ring isomorphisms on factor von Neumann algebras
This results that p2 ◦ c + cp∗2 = 3
2
b12 which implies that c12 = b12 and
c21 = c22 = 0. Thus, Φ(c11 + b12) = Φ(a11) + Φ(b12). It follows that, for
an arbitrary element d12 ∈ A12
Φ(d12 ◦ (c11 + b12) + (c11 + b12)d
∗
12)
= Φ(d12 ◦ a11 + a11d
∗
12) + Φ(d12 ◦ b12 + b12d
∗
12)
which implies that
Φ(
1
2
c11d12 + b12d
∗
12) = Φ(
1
2
a11d12) + Φ(b12d
∗
12).
Hence,
Φ(p2 ◦ (
1
2
c11d12 + b12d
∗
12) + (
1
2
c11d12 + b12d
∗
12)p
∗
2)
= Φ(p2 ◦ (
1
2
a11d12) + (
1
2
a11d12)p
∗
2) + Φ(p2 ◦ (b12d
∗
12) + (b12d
∗
12)p
∗
2)
which yields Φ(3
4
c11d12) = Φ(3
4
a11d12). This shows that c11d12 = a11d12
which leads to c11 = a11.
Similarly, we prove the cases (ii), (iii) and (iv).
Claim 4. For arbitrary elements a12 ∈ A12 and b21 ∈ A21 holds Φ(a12 +
b21) = Φ(a12) + Φ(b21).
Proof. Choose an element c = c11 + c12 + c21 + c22 ∈ A such that Φ(c) =
Φ(a12) + Φ(b21). Hence, for an arbitrary element d12 ∈ A12, we have
Φ(d12 ◦ c+ cd∗12) = Φ(d12 ◦ a12 + a12d
∗
12) + Φ(d12 ◦ b21 + b21d
∗
12)
= Φ(a12d
∗
12) + Φ(d12 ◦ b21)
which yields
Φ(p2 ◦ (d12 ◦ c+ cd∗12) + (d12 ◦ c+ cd∗12)p
∗
2)
= Φ(p2 ◦ (a12d
∗
12) + (a12d
∗
12)p
∗
2) + Φ(p2 ◦ (d12 ◦ b21) + (d12 ◦ b21)p
∗
2)
= Φ(b21d12).
This results that p2 ◦ (d12 ◦ c+ cd∗12) + (d12 ◦ c+ cd∗12)p
∗
2 = b21d12 which
implies that c11 = c22 = 0 and c21 = b21. Thus,
Φ(c12 + b21) = Φ(a12) + Φ(b21).
“adm-n1” — 2021/4/10 — 20:38 — page 65 — #69
J. C. M. Ferreira, M. G. B. Marietto 65
It follows that, for an arbitrary element d21 ∈ A21, we have
Φ(d21 ◦ (c12 + b21) + (c12 + b21)d
∗
21)
= Φ(d21 ◦ a12 + a12d
∗
21) + Φ(d21 ◦ b21 + b21d
∗
21)
= Φ(d21 ◦ a12) + Φ(b21d
∗
21).
This implies that
Φ(p1 ◦ (d21 ◦ c12 + b21d
∗
21) + (d21 ◦ c12 + b21d
∗
21)p
∗
1)
= Φ(p1 ◦ (d21 ◦ a12) + (d21 ◦ a12)p
∗
1) + Φ(p1 ◦ (b21d
∗
21) + (b21d
∗
21)p
∗
1)
= Φ(a12d21).
It follows that p1◦(d21◦c12+b21d
∗
21)+(d21◦c12+b21d
∗
21)p
∗
1 = a12d21 which
yields c12d21 = a12d21. This allows us to conclude that c12 = a12.
Claim 5. For arbitrary elements a11 ∈ A11, b12 ∈ A12, c21 ∈ A21 and
d22 ∈ A22 hold: (i) Φ(a11 + b12 + c21) = Φ(a11) + Φ(b12) + Φ(c21) and (ii)
Φ(b12 + c21 + d22) = Φ(b12) + Φ(c21) + Φ(d22).
Proof. Choose an element f = f11 + f12 + f21 + f22 ∈ A such that
Φ(f) = Φ(a11) + Φ(b12) + Φ(c21). By Claim 4 we have
Φ(p2 ◦ f + fp∗2)
= Φ(p2 ◦ a11 + a11p
∗
2) + Φ(p2 ◦ b12 + b12p
∗
2) + Φ(p2 ◦ c21 + c21p
∗
2)
= Φ(
3
2
b12) + Φ(
1
2
c21) = Φ(
3
2
b12 +
1
2
c21).
This results that p2 ◦ f + fp∗2 = 3
2
b12 +
1
2
c21 which leads to f12 = b12,
f21 = c21 and f22 = 0. It follows that,
Φ(f11 + b12 + c21) = Φ(a11) + Φ(b12) + Φ(c21).
Next, for an arbitrary element d12 ∈ A12, we have
Φ(d12 ◦ (f11 + b12 + c21) + (f11 + b12 + c21)d
∗
12)
= Φ(d12 ◦ a11+a11d
∗
12)+Φ(d12 ◦ b12+b12d
∗
12)+Φ(d12 ◦ c21+c21d
∗
12)
which implies that
Φ(d12 ◦ (f11 + c21) + b12d
∗
12) = Φ(
1
2
a11d12) + Φ(b12d
∗
12) + Φ(d12 ◦ c21).
“adm-n1” — 2021/4/10 — 20:38 — page 66 — #70
66 ∗-ring isomorphisms on factor von Neumann algebras
As consequence, we get
Φ(p2 ◦ (d12 ◦ (f11 + c21) + b12d
∗
12) + (d12 ◦ (f11 + c21) + b12d
∗
12)p
∗
2)
= Φ(p2 ◦ (
1
2
a11d12) + (
1
2
a11d12)p
∗
2) + Φ(p2 ◦ (b12d
∗
12) + (b12d
∗
12)p
∗
2)
+ Φ(p2 ◦ (d12 ◦ c21) + (d12 ◦ c21)p
∗
2)
which results
Φ(
3
4
f11d12 + c21d12) = Φ(
3
4
a11d12) + Φ(c21d12) = Φ(
3
4
a11d12 + c21d12).
This shows that 3
4
f11d12+c21d12 = 3
4
a11d12+c21d12 which yields f11 = a11.
Similarly, we prove the case (ii).
Claim 6. For arbitrary elements a11 ∈ A11, b12 ∈ A12, c21 ∈ A21 and
d22 ∈ A22 holds Φ(a11+b12+c21+d22) = Φ(a11)+Φ(b12)+Φ(c21)+Φ(d22).
Proof. Choose an element f = f11 + f12 + f21 + f22 ∈ A such that
Φ(f) = Φ(a11) + Φ(b12) + Φ(c21) + Φ(d22). By Claim 5(i) we have
Φ(p1 ◦ f + fp∗1) = Φ(p1 ◦ a11 + a11p
∗
1) + Φ(p1 ◦ b12 + b12p
∗
1)
+ Φ(p1 ◦ c21 + c21p
∗
1) + Φ(p1 ◦ d22 + d22p
∗
1)
= Φ(2a11)+Φ(
1
2
b12)+Φ(
3
2
c21) = Φ(2a11+
1
2
b12+
3
2
c21).
It follows that p1 ◦ f + fp∗1 = 2a11 + 1
2
b12 + 3
2
c21 which implies that
f11 = a11, f12 = b12 and f21 = c21. Next, by Claim 5(ii) we have
Φ(p2 ◦ f + fp∗2) = Φ(p2 ◦ a11 + a11p
∗
2) + Φ(p2 ◦ b12 + b12p
∗
2)
+ Φ(p2 ◦ c21 + c21p
∗
2) + Φ(p2 ◦ d22 + d22p
∗
2)
= Φ(
3
2
b12) + Φ(
1
2
c21) + Φ(2d22) = Φ(
3
2
b12 +
1
2
c21 + 2d22).
This results that p2◦f+fp∗2 = 3
2
b12+
1
2
c21+2d22 which yields f22 = d22.
Claim 7. For arbitrary elements a12, b12 ∈ A12 and a21, b21 ∈ A21 hold:
(i) Φ(a12+ b12) = Φ(a12)+Φ(b12) and (ii) Φ(a21+ b21) = Φ(a21)+Φ(b21).
Proof. First, we note that the following identity holds
(p1 + a12) ◦ (p2 + b12) + (p2 + b12)(p1 + a12)
∗
=
1
2
a12 +
1
2
b12 + a∗12 + b12a
∗
12.
“adm-n1” — 2021/4/10 — 20:38 — page 67 — #71
J. C. M. Ferreira, M. G. B. Marietto 67
Hence, by Claim 6 we have
Φ(
1
2
a12 +
1
2
b12) + Φ(a∗12) + Φ(b12a
∗
12) = Φ(
1
2
a12 +
1
2
b12 + a∗12 + b12a
∗
12)
= Φ((p1 + a12) ◦ (p2 + b12) + (p2 + b12)(p1 + a12)
∗)
= Φ(p1 + a12) ◦ Φ(p2 + b12) + Φ(p2 + b12)Φ(p1 + a12)
∗
= (Φ(p1) + Φ(a12)) ◦ (Φ(p2) + Φ(b12))
+ (Φ(p2) + Φ(b12))(Φ(p1)
∗ +Φ(a12)
∗)
= Φ(p1) ◦ Φ(p2) + Φ(p2)Φ(p1)
∗ +Φ(p1) ◦ Φ(b12) + Φ(b12)Φ(p1)
∗
+Φ(a12) ◦ Φ(p2)+Φ(p2)Φ(a12)
∗+Φ(a12) ◦ Φ(b12)+Φ(b12)Φ(a12)
∗
= Φ(p1 ◦ p2 + p2p
∗
1) + Φ(p1 ◦ b12 + b12p
∗
1) + Φ(a12 ◦ p2 + p2a
∗
12)
+ Φ(a12 ◦ b12 + b12a
∗
12)
= Φ(
1
2
b12) + Φ(
1
2
a12) + Φ(a∗12) + Φ(b12a
∗
12).
This allows us to conclude that Φ(a12 + b12) = Φ(a12) + Φ(b12).
Similarly, we prove the case (ii) using the identity
(p2 + a21) ◦ (p1 + b21) + (p1 + b21)(p2 + a21)
∗
=
1
2
a21 +
1
2
b21 + a∗21 + b21a
∗
21.
Claim 8. For arbitrary elements a11, b11 ∈ A11 and a22, b22 ∈ A22 hold:
(i) Φ(a11+ b11) = Φ(a11)+Φ(b11) and (ii) Φ(a22+ b22) = Φ(a22)+Φ(b22).
Proof. Choose an element c = c11 + c12 + c21 + c22 ∈ A such that Φ(c) =
Φ(a11) + Φ(b11). By Claim 7(i), for an arbitrary element d12 ∈ A12 we
have
Φ(d12 ◦ c+ cd∗12) = Φ(d12 ◦ a11 + a11d
∗
12) + Φ(d12 ◦ b11 + b11d
∗
12)
= Φ(
1
2
a11d12) + Φ(
1
2
b11d12) = Φ(
1
2
(a11 + b11)d12).
This shows that d12◦c+cd∗12 = 1
2
(a11+b11)d12 which implies c11 = a11+b11
and c12 = c21 = c22 = 0.
Similarly, we prove the case (ii).
Claim 9. Φ is an additive mapping.
Proof. The result is an immediate consequence of Claims 6, 7 and 8.
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68 ∗-ring isomorphisms on factor von Neumann algebras
Lemma 2. Let B be a factor von Neumann algebra with identity 1B and
an element a ∈ B. If B satisfies the condition a ◦ b + ba∗ = 0, for all
element b ∈ B, then a ∈ iR1B (i is the imaginary number unit).
Proof. When b = 1B, we get that a+ a∗ = 0. Replacing this last result in
given condition, we obtain ab− ba = 0, for all element b ∈ B. This implies
that a belongs to the center of B which results that a ∈ iR1B.
In the remainder of this paper, all lemmas satisfy the conditions of
the main theorem.
Lemma 3. Φ(iR1A) = iR1B and Φ(C1A) = C1B.
Proof. For arbitrary elements λ ∈ R and a ∈ A we have
0 = Φ((iλ1A)a+ (iλ1A)
∗a) = Φ((iλ1A) ◦ a+ a(iλ1A)
∗)
= Φ(iλ1A) ◦ Φ(a) + Φ(a)Φ(iλ1A)
∗.
By Lemma 2 results that Φ(iλ1A) ∈ iR1B. Since λ is an arbitrary real
number, then we conclude that Φ(iR1A) ⊆ iR1B. Note that Φ−1 has the
same properties that Φ. Thus, applying a similar argument to the above
we obtain that iR1B ⊆ Φ(iR1A). Consequently, Φ(iR1A) = iR1B.
Let a ∈ A such that a∗ = −a (called an anti-self-adjoint element).
Then 0 = Φ((i1A)a+(i1A)a
∗) = Φ(a ◦ (i1A)+ (i1A)a
∗) = Φ(a) ◦Φ(i1A)+
Φ(i1A)Φ(a)
∗ which implies that Φ(a) = −Φ(a)∗. Since Φ−1 has the same
properties of Φ, then applying a similar reasoning we prove that Φ(a) =
−Φ(a)∗ implies a = −a∗. Consequently, a = −a∗ if and only if Φ(a) =
−Φ(a)∗ for any such element a ∈ A. From this fact, for arbitrary elements
λ ∈ C and a ∈ A such that a∗ = −a, we have 0 = Φ((λ1A)a+(λ1A)a
∗) =
Φ(a ◦ (λ1A) + (λ1A)a
∗) = Φ(a) ◦ Φ(λ1A) + Φ(λ1A)Φ(a)
∗ which implies
that Φ(a)Φ(λ1A) = Φ(λ1A)Φ(a). It follows that bΦ(λ1A) = Φ(λ1A)b,
for any element b ∈ B such that b∗ = −b. As consequence, we have
bΦ(λ1A) = Φ(λ1A)b, for an arbitrary element b ∈ B, since we can write
any element b ∈ B in the form b = b1+ ib2, where b1 = b−b∗
2
and b2 = b+b∗
2i
are anti-self-adjoint elements of B. This results that Φ(λ1A) ∈ C1B. It
follows that Φ(C1A) ⊆ C1B. Since Φ−1 has the same properties of Φ, then
applying a similar argument we obtain C1B ⊆ Φ(C1A). Consequently,
Φ(C1A) = C1B.
Lemma 4. Φ(1A) = 1B and there is a real number ν satisfying ν2 = 1
such that Φ(ia) = iνΦ(a), for all element a ∈ A.
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J. C. M. Ferreira, M. G. B. Marietto 69
Proof. By Lemma 3, there is a nonzero element λ ∈ C such that Φ(1A) =
λ1B. Fix a nontrivial projection p ∈ A. It follows that 2Φ(p) = Φ(2p) =
Φ(p ◦ 1A + 1Ap
∗) = Φ(p) ◦ Φ(1A) + Φ(1A)Φ(p)
∗ = λ(Φ(p) + Φ(p)∗). This
allows us to write Φ(p) = λq, where q = 1
2
(Φ(p) + Φ(p)∗). Also, 2Φ(p) =
Φ(2p) = Φ(1A ◦ p + p1∗
A
) = Φ(1A) ◦ Φ(p) + Φ(p)Φ(1A)
∗ = (λ + λ)Φ(p)
which results that λ + λ = 2. Yet, 2Φ(p) = Φ(2p) = Φ(p ◦ p + pp∗) =
Φ(p) ◦ Φ(p) + Φ(p)Φ(p)∗ = Φ(p)Φ(p) + Φ(p)Φ(p)∗ = (λ2 + λλ)q2 = 2λq2
which implies that q = q2. This show that q is a non-trivial projection
of B.
Let a be an element of A such that s = pap⊥ is a nonzero element. Then
1
2
Φ(s) = Φ(1
2
s) = Φ(p◦ s+ sp∗) = Φ(p)◦Φ(s)+Φ(s)Φ(p)∗ = (λq)◦Φ(s)+
λΦ(s)q which implies qΦ(s)q = 0, q⊥Φ(s)q⊥ = 0, (1 − λ)qΦ(s)q⊥ = 0
and (1 + λ)q⊥Φ(s)q = 0. If λ 6= ±1, then qΦ(s)q⊥ = 0 and q⊥Φ(s)q = 0
which implies that Φ(s) = 0 resulting in s = 0 which is a contradiction.
Thus we must have λ = 1 which results Φ(1A) = 1B, q
⊥Φ(s)q = 0 and
Φ(s) = qΦ(s)q⊥. Let ν be a nonzero real number such that Φ(i1A) = iν1B,
by Lemma 3. Then 2Φ(ip) = Φ(p ◦ (i1A) + (i1A)p
∗) = Φ(p) ◦ Φ(i1A) +
Φ(i1A)Φ(p)
∗ = 2iνq which implies that Φ(ip) = iνq. It follows that
1
2
Φ(is) = Φ(1
2
is) = Φ((ip) ◦ s + s(ip)∗) = Φ(ip) ◦ Φ(s) + Φ(s)Φ(ip)∗ =
(iνq)◦Φ(s)+Φ(s)(iνq)∗ = 1
2
iνΦ(s) which yields Φ(is) = iνΦ(s). Moreover,
as −1
2
Φ(s) = Φ(−1
2
s) = Φ((ip) ◦ (is) + (is)(ip)∗) = Φ(ip) ◦ Φ(is) +
Φ(is)Φ(ip)∗ = (iνq) ◦ (iνΦ(s)) + (iνΦ(s))(iνq)∗ = −1
2
ν2Φ(s) we still get
that ν2 = 1.
Let t be an element of A such that t = t∗. Then 2Φ(t) = Φ(2t) = Φ(t◦
1A+1At
∗) = Φ(t)◦Φ(1A)+Φ(1A)Φ(t)
∗ = Φ(t)+Φ(t)∗ which implies that
Φ(t) = Φ(t)∗. This results that 2Φ(it) = Φ(2it) = Φ(t ◦ (i1A) + (i1A)t
∗) =
Φ(t) ◦ Φ(i1A) + Φ(i1A)Φ(t)
∗ = Φ(t) ◦ (iν1B) + (iν1B)Φ(t)
∗ which allows
us to conclude that Φ(it) = iνΦ(t). Therefore, for an arbitrary element
a ∈ A, let us write a = a1 + ia2, where a1 = a+a∗
2
and a2 = a−a∗
2i
are
self-adjoint elements of A. Then we conclude that Φ(ia) = Φ(ia1 − a2) =
Φ(ia1)− Φ(a2) = iνΦ(a1) + iνΦ(ia2) = iνΦ(a1 + a2) = iνΦ(a).
Lemma 5. Φ : A → B is a ∗-ring isomorphism.
Proof. For arbitrary elements a, b ∈ A, we have
Φ(a ◦ b− ba∗) = Φ((ia) ◦ (−ib) + (−ib)(ia)∗)
= Φ(ia) ◦ Φ(−ib) + Φ(−ib)Φ(ia)∗
= (iνΦ(a)) ◦ (−iνΦ(b)) + (−iνΦ(b))(iνΦ(a))∗
= Φ(a) ◦ Φ(b)− Φ(b)Φ(a)∗. (2)
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70 ∗-ring isomorphisms on factor von Neumann algebras
Thus, from (1) and (2) we getΦ(a◦b) = Φ(a)◦Φ(b) andΦ(ba∗) = Φ(b)Φ(a)∗
and, as consequence, we deduce that Φ(a∗) = Φ(1Aa
∗) = Φ(1A)Φ(a)
∗ =
1BΦ(a)
∗ = Φ(a)∗ and Φ(ab) = Φ(a(b∗)∗) = Φ(a)Φ(b∗)∗ = Φ(a)(Φ(b)∗)∗ =
Φ(a)Φ(b). This allows us to conclude that Φ is a ∗-ring isomorphism.
The theorem is proved.
From the main theorem and the fact that every ring isomorphism
between type I factor von Neumann algebras is spatial, we have the
following corollary.
Corollary 1. Let A and B be two type I factor von Neumann algebras
acting on complex Hilbert spaces H and K, respectively. Then every bijec-
tive mapping Φ : A → B satisfies Φ(a◦ b+ ba∗) = Φ(a)◦Φ(b)+Φ(b)Φ(a)∗,
for all elements a, b ∈ A, if and only if there exists a unitary or conjugate
unitary operator U : H → K such that Φ(a) = UaU∗, for all element
a ∈ A.
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Contact information
João Carlos da
Motta Ferreira,
Maria das Graças
Bruno Marietto
Center for Mathematics, Computation and
Cognition, Federal University of ABC, Avenida
dos Estados, 5001, 09210-580, Santo André,
Brazil
E-Mail(s): joao.cmferreira@ufabc.edu.br,
graca.marietto@ufabc.edu.br
Web-page(s): http://www.ufabc.edu.br
Received by the editors: 21.10.2019.
mailto:joao.cmferreira@ufabc.edu.br
mailto:graca.marietto@ufabc.edu.br
http://www.ufabc.edu.br
J. C. M. Ferreira, M. G. B. Marietto
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