On operator-commutative subalgebras of Jordan algebras

Gerald Hessenberger

Institut für Mathematik, Universität Innsbruck Technikerstraße 25/7, A- 6020 Innsbruck, Austria

0. INTRODUCTION

In this paper we define the notion of operator-commutative subsets of a Jordan algebra. For Jordan algebras without 2-torsion this agrees with the concept of strong associativity. The advantage of our definition is the independence of characteristic and of the existence of a unit (cf. [4]). In addition to that we do not only consider inverse-closed but also quasi-inverse-closed subalgebras:

A subalgebra of a Jordan algebra is called operator-commutative if all its Uoperators commute and all formal U-operators of Lie brackets [5] vanish. A subalgebra is called (quasi-)inverse-closed if its set of (quasi-)invertible elements is equal to the intersection of the subalgebra with the set of (quasi-)invertible elements of the whole algebra.

Our main result is the construction of the smallest operator-commutative, (quasi-)inverse-closed subalgebra of a Jordan algebra containing a fixed element of this algebra.

In the case of a (complex) Banach Jordan algebra we can even find a closed subalgebra with the above properties. As was shown in [1], this subalgebra contains the image of the spectral functional calculus.

1. OPERATOR-COMMUTATIVITY

1.1. PROPOSITION. Let A be any associative algebra over an arbitrary commutative unital ring. For the usual Jordan product U_{x}y:= xyx, x, y ∈ A, the following formulas hold:

U_{x∘y} − 2 (U_{x}U_{y} + U_{y}U_{x}) = U_{[x,y]},
x ∘ U_{y}x − U_{x}y^{2} − U_{y}x^{2} = [x, y]^{2}.

Proof. For any z ∈ A we have

U_{x∘y}z − 2 (U_{x}U_{y}z + U_{y}U_{x}z) = (xy + yx)z(xy + yx) − 2 (xyzyx + yxzxy)
= xyzxy − xyzyx + yxzyx − yxzxy
= xyz[x, y] − yxz[x, y] = [x, y]z[x, y]
= U_{[x,y]}z,
x ∘ U_{y}x − U_{x}y^{2} − U_{y}x^{2} = x(yxy) + (yxy)x − xy^{2}x − yx^{2}y
= xy(xy − yx) − yx(xy − yx) = (xy − yx)^{2}
= [x, y]^{2}.

1.2. DEFINITION. Now let (A, U, ^{2}) be a Jordan algebra and x, y ∈ A. The formal U-operator of [x, y] is defined by

U_{[x,y]} := U_{x∘y} − 2 (U_{x}U_{y} + U_{y}U_{x}),

the formal square of [x, y] is

[x, y]^{2} := x ∘ U_{y}x − U_{x}y^{2} − U_{y}x^{2}.

These definitions, due to McCrimmon [5], make sense even if [x, y] is not existing in A. By proposition 1.1 there is consistency in the case of special Jordan algebras.

Suppose that A has a unit 1_{A} and B is any subalgebra containing 1_{A}. We call B an operator-commutative subalgebra of A if

[U_{B}, U_{B}] = 0,  (OC 1)
U_{[B,B]} = 0.  (OC 2)

For a non-unital Jordan algebra A over R with unital hull Â := R ⊕ A, a subalgebra B is called operator-commutative if B̂ := R ⊕ B is an operatorcommutative subalgebra of Â.

1.3. LEMMA. (a) Let A be an arbitrary Jordan algebra. In all scalar extensions of A the following formulas hold

x ∘ U_{y}x = y ∘ U_{x}y,  (JA 21)
U_{x∘y,x} = U_{x}V_{y} + V_{y}U_{x},  (JA 22)
U_{x∘y} + U_{U_{x}y,y} = U_{x}U_{y} + U_{y}U_{x} + V_{x}U_{y}V_{x},  (JA 23)
U_{x∘y} + V_{U_{x}y^{2}} = U_{x}U_{y} + U_{y}U_{x} + V_{x,y}V_{y,x},  (JA 24)
U_{x∘y} + U_{x^{2},y^{2}} = U_{x}U_{y} + U_{y}U_{x} + U_{x,y}^{2},  (JA 25)
[x, y]^{2} + U_{x}y^{2} + U_{y}x^{2} = x ∘ U_{y}x = y ∘ U_{x}y,  (JA 26)
= V_{x,y}(x ∘ y) − 2 U_{x}y^{2},  (JA 27)
= U_{x,y}(x ∘ y) − x^{2} ∘ y^{2}.  (JA 28)

(b) If A has a unit 1_{A} then

U_{[x,y]}1_{A} = [x, y]^{2}.

Proof. (a) In the sequel all JA-numbers of formulae not listed above refer to [2]. We get JA 21 by applying JA 8 on the unit of the unital hull. JA 22 is a consequence of the left identity of JA 12 applied on z by exchanging y and z. JA 23 to JA 25 hold for special Jordan algebras whence by Macdonald’s Theorem they hold in general. JA 26 to JA 28 are consequences of JA 23 to JA 25 by applying these identities to the unit of the unital hull and using JA 21.

(b) The assertion holds by JA 21, JA 23, and the definitions of U_{[x,y]} and [x, y]^{2}.

We now give a characterization of operator-commutative subalgebras without recourse to the unital hull of the ambient algebra:

1.4. THEOREM. A subalgebra B of a non-unital Jordan algebra A over R is operator-commutative if and only if in addition to OC 1 and OC 2 the following identities hold:

[U_{B}, V_{B}] = 0,  (OC 3)
[V_{B}, V_{B}] = 0,  (OC 4)
U_{a}b^{2} − U_{b}a^{2} = 0 for all a, b ∈ B,  (OC 5)
[B, B]^{2} = 0.  (OC 6)

Proof. In the sequel we identify A with 0_{R} ⊕ A. Using the formulae Û_{u+v} = Û_{u} + Û_{u,v} + Û_{v}, Û_{u,1Â} = V̂_{u}, and U_{1Â} = IdÂ we obtain for r, s ∈ R resp. a, b ∈ B:

[Û_{r⊕a}, Û_{s⊕b}] = [Û_{a}, Û_{b}] + r [V̂_{a}, Û_{b}] + s [Û_{a}, V̂_{b}] + rs [V̂_{a}, V̂_{b}].

The first bracket of the right side vanishes if and only if OC 1 and OC 5 hold, whereas the fourth bracket is zero if and only if OC 4 holds. For the remaining two brackets we need OC 3 and the identity c^{2} ∘ d = 2U_{c}d, c, d ∈ B, a consequence of JA 9 and V_{x}^{2}|_{B} = 2 V_{x^{2}}|_{B} which is implied by OC 4. Thus [Û_{r⊕a}, Û_{s⊕b}] vanishes if and only if OC 1, OC 3, OC 4, and OC 5 hold for B. Now let r, s ∈ R and a, b ∈ B. Using JA 22 resp. JA 11 and JA 14 we obtain

Û_{[r⊕a,s⊕b]} = Û_{[x,y]} + 2r (Û_{x∘y,y} − V̂_{x}Û_{y} − Û_{y}V̂_{x})

+ 2s (Û_{x∘y,x} − Û_{x}V̂_{y} − V̂_{y}Û_{x}) + 2rs (2Û_{x,y} + V̂_{x∘y} − V̂_{x}V̂_{y} − V̂_{y}V̂_{x})

= Û_{[a,b]},

whence by 1.3(b) the vanishing of Û_{[r⊕a,s⊕b]} is equal to OC 2 and OC 6.

1.5. REMARK. In [4, I.3.2] the property OC 4 is called “strong associativity”.

1.6. DEFINITION. For elements u, v, x, y of a Jordan algebra A we put

U_{[x,y,v]} := U_{x∘y,x∘v} − 2 (U_{x}U_{y,v} + U_{y,v}U_{x}),
U_{[x,u,y,v]} := U_{x∘y,u∘v} + U_{x∘v,u∘y} − 2 (U_{x,u}U_{y,v} + U_{y,v}U_{x,u}).

Thus U_{[x,y,v]} is quadratic in x and linear in y and v, whereas U_{[x,u,y,v]} is multilinear. Moreover, U_{[x,y,y]} = 2 U_{[x,y]}, and U_{[x,x,y,v]} = 2 U_{[x,y,v]}.

1.7. LEMMA. (a) The following symmetry relations hold for a Jordan algebra A:

U_{[x,y]} = U_{[y,x]},
U_{[x,y,v]} = U_{[x,v,y]},
U_{[x,u,y,v]} = U_{[y,v,x,u]},
U_{[x,u,y,v]} = U_{[x,u,v,y]}.

(b) If A is a special Jordan algebra we have

U_{[x,y,v]} = U_{[x,y],[x,v]},
U_{[x,u,y,v]} = U_{[x,y],[u,v]} + U_{[x,v],[u,y]}.

Proof. Part (a) is clear by 1.6. The assertion of (b) is obtained by linearization of the first identity of 1.1.

1.8. LEMMA. Let A be a Jordan algebra and B a subalgebra of A.

(a) If OC 1 holds for B, then

[U_{B}, U_{B,B}] = 0,  (OC 1a)
[U_{B,B}, U_{B,B}] = 0.  (OC 1b)

(b) If OC 2 holds for B, then

U_{[B,B,B]} = 0,  (OC 2a)
U_{[B,B,B,B]} = 0.  (OC 2b)

Proof. The assertions are obtained by linearization of OC 1 resp. OC 2.

1.9. THEOREM. Let A be a unital Jordan algebra and B ⊆ A a subalgebra containing 1_{A}. If B is operator-commutative, the identities OC 3, OC 4, OC 5, and OC 6 hold for B.

Proof. By OC 1a resp. OC 1b we get

[U_{B}, V_{B}] = [U_{B}, U_{B,1_{A}}] = 0 = [U_{B,1_{A}}, U_{B,1_{A}}] = [V_{B}, V_{B}].

Moreover, OC 5 and OC 6 are obtained by applying OC 1 resp. OC 2 to 1_{A} and using 1.3(b).

1.10. COROLLARY. A subalgebra B of a unital Jordan algebra A with 1_{A} ∈ B is operator-commutative if and only if B̂ is an operator-commutative subalgebra of Â.

Proof. The assertion holds by 1.4 and 1.9.

1.11. THEOREM. In any Jordan algebra A without 2-torsion a subalgebra B ⊆ A is operator-commutative if and only if OC 4 holds.

Proof. If OC 4 holds for B it does so for B̂, whence we can assume that A is unital and 1_{A} ∈ B. Thus it suffices to show OC 1 and OC 2. But OC 1 holds by JA 9 and OC 4, since these identities yield 4 [U_{B}, U_{B}] = 0. The same identities also imply

2 [x, y]^{2} = x ∘ (V_{y}^{2}x − V_{y^{2}}x) − (V_{x}^{2} − V_{x^{2}})y^{2} − (V_{y}^{2} − V_{y^{2}})x^{2}
= V_{x}V_{y}V_{x}y − 2 V_{x}V_{x}y^{2} + 2 V_{x^{2}}y^{2} − V_{y}V_{x^{2}}y
= V_{x}V_{x}V_{y}y − V_{x}V_{x}V_{y}y + V_{x^{2}}V_{y}y − V_{x^{2}}V_{y}y = 0,

whence [x, y]^{2} = 0. Using [5, (0.28)] and OC 4 we therefore get for a, b ∈ B:

2 U_{[a,b]} = V_{[a,b]^{2}} − [V_{a}, V_{b}]^{2} = V_{0} − 0^{2} = 0.

1.12. THEOREM. A subalgebra B of an associative algebra A is operatorcommutative if and only if the following conditions hold:

[B, B] ⊆ C(A),  (AOC 1)
[a, b]a ∈ C(A) for all a, b ∈ B,  (AOC 2)

where C(A) denotes the center of A.

Proof. For a, b ∈ B and c ∈ A we have

[V_{a}, V_{b}]c = abc − bac − cab + cba = [[a, b], c],

whence AOC 1 and OC 4 are equivalent. If we assume that AOC 1 = OC 4 holds for B, we obtain

[U_{a}, V_{b}]c = abca − baca − acab + acba
= [a, b][c, a] + [[a, b], a]c + a[[a, b], c]
= [c, [a, b]a] + 0c + a0 = −[[a, b]a, c].

Thus (AOC 1, AOC 2) is equivalent to (OC 3, OC 4). Now we show that AOC 1 and AOC 2 imply OC 1. For this purpose assume that AOC 1 and AOC 2 hold for B, whence

[U_{a}, U_{b}]c = abcba − bacab = [a, b][c, ab] + [[a, b], ab]c + ab[[a, b], c]
= [a, b]cab − [a, b]abc + 0c + a0 = c([a, b]a)b − ([a, b]a)bc
= [a, b]a[c, b] = ac([a, b]b) − a([a, b]b)c
= [[b, a]b, ac] = 0.

For the implication of OC 5 by AOC 1 consider the identity

U_{a}b^{2} − U_{b}a^{2} = ab^{2}a − ba^{2}b = [[a, b], ab] = 0.

Analogously, OC 6 is a consequence of AOC 1 and AOC 2 since

[a, b]^{2} = [a, b]ab − b[a, b]a = [[a, b]a, b] = 0.

Finally, AOC 1 and OC 6 imply OC 2:

U_{[a,b]}c = [a, b]^{2}c = 0c = 0.

1.13. COROLLARY. Let A be an associative algebra.

(a) If A is semiprime, the operator-commutative subalgebras are just the commutative algebras.

(b) If A has no 2-torsion and B is a subalgebra of A, then B is operatorcommutative if and only if [B, B] ⊆ C(A). In particular, the latter condition implies [B, B]^{2} = 0.

Proof. (a) For commutative subalgebras AOC 1 and AOC 2 trivially hold. Conversely, if B is an operator-commutative subalgebra, OC 2 yields U_{[B,B]} = 0. Thus by semiprimeness B is commutative.

(b) The assertion is clear by 1.11 and the proof of 1.12.

2. (QUASI-)INVERSE-CLOSED SUBALGEBRAS AND ADJOINED INVERSES

2.1. DEFINITION. Let A be a unital Jordan algebra and B ⊆ A a subalgebra containing 1_{A}. We call B inverse-closed in A, if for all invertible elements of A lying in B, the inverse is an element of B, i.e.

B^{×} = A^{×} ∩ B,  (IC 1)

where A^{×}, B^{×} denote the sets of invertible elements of A resp. B. For a non-unital Jordan algebra A an arbitrary subalgebra B is called quasiinverse-closed in A if

Q(B) = Q(A) ∩ B,  (QIC 1)

where Q(A), Q(B) denote the sets of quasi-invertible elements of A resp. B.

2.2. REMARK. For a unital Jordan algebra A over a field R, an inverse-closed subalgebra B, and a ∈ B, the spectrum of a in A and in B is the same:

Sp(a, A): = {λ ∈ R: (λ · 1_{A} − a) ∉ A^{×}}
= {λ ∈ R: (λ · 1_{A} − a) ∉ B^{×}} =: Sp(a, B).  (IC 2)

For a non-unital Jordan algebra A over a field R, a quasi-inverse-closed subalgebra B, and a ∈ B, analogously

Sp′(a, A): = {0} ∪ {λ ∈ R: λ^{−1} · a ∉ Q(A)}
= {0} ∪ {λ ∈ R: λ^{−1} · a ∉ Q(B)} =: Sp′(a, B). (QIC 2)

2.3. LEMMA. For any Jordan algebra the following identity holds:

U_{U(x)y,U(x)z} = U_{x}U_{y,z}U_{x}.  (JA 6′)

Proof. JA 6′ is obtained by linearization of JA 6.

2.4. DEFINITION. For a commutative ring R, an R-module M, and X ⊂ M let _{R}⟨X⟩ denote the R-linear hull of X in M. Analogously, for an associative algebra C and Y ⊆ C let _{C}⟨Y ⟩ denote the subalgebra of C generated by the elements of Y. Now consider a Jordan algebra A. The multiplication algebra of A is defined as

M(A):=_{End(A)} ⟨U_{A} ∪ V_{A}⟩.

If A has a unit 1_{A}, then

M(A) =_{End(A)} ⟨U_{A}⟩,

since V_{x} = U_{x,1_{A}} = U_{x+1_{A}} − U_{x} − U_{1_{A}}.

2.5. LEMMA. Let A be a unital Jordan algebra over R, and B ∋ 1_{A} a subalgebra of A such that OC 1 holds in B. Consider a ∈ A^{×} ∩ B and B[a^{−1}], the subalgebra of A generated by B and a^{−1}. Then B[a^{−1}] =_{R} ⟨{U_{a^{−n}}B : n ∈ IN_{0}}⟩.

Proof. For G:= {U_{a^{−n}}B : n ∈ IN_{0}} and M :=_{R} ⟨G⟩ we get B ⊆ M ⊆ B[a^{−1}]. By the inclusion 1_{A} ∈ B it suffices to show U_{M}M ⊆ M. We therefore consider x:= U_{a^{−k}}b, y:= U_{a^{−l}}c, and z:= U_{a^{−m}}d, where k, l, m ∈ IN_{0}, b, c, d ∈ B. By OC 1 the operator U_{a} commutes with all U-operators of B, whence the same is true for U_{a}^{−1}. Using JA 6 and its linearization JA 6′ we obtain

U_{x}y = U_{a^{−k}}U_{b}U_{a^{−k}}U_{a^{−l}}c
= U_{a}^{−k}U_{b}U_{a}^{−k−l}c
= U_{a}^{−2k−l}(U_{b}c) ∈ G,
U_{x,z}y = U_{a^{−k−m}}U_{U(a^{m})b,U(a^{k})d}U_{a^{−k−m}}U_{a^{−l}}c
= U_{a}^{−k−m}U_{U(a^{m})b,U(a^{k})d}U_{a}^{−k−l−m}c
= U_{a}^{−2k−l−2m}(U_{U(a^{m})b,U(a^{k})d}c) ∈ G.

Thus U_{G}G ⊆ G and U_{G,G}G ⊆ G which implies U_{M}M ⊆ M. We conclude that M is a subalgebra of A and therefore equal to B[a^{−1}].

2.6. COROLLARY. In the situation of 2.5 the identities OC 1 and OC 3 hold for B[a^{−1}], and we have

M(B[a^{−1}]) = M(B)[U_{a}^{−1}].

Proof. Define G and x, y ∈ G as in the proof of 2.5. By OC 1 the operator U_{a}^{−1} commutes with all U-operators of B. Using JA 6 resp. JA 6′ we obtain the identities

U_{x} = U_{a}^{−k}U_{b}U_{a}^{−k} = U_{a}^{−2k}U_{b},
U_{x,y} = U_{a}^{−k−l}U_{U(a^{l})b,U(a^{k})c}U_{a}^{−k−l} = U_{a}^{−2k−2l}U_{U(a^{l})b,U(a^{k})c}.

Thus OC 1, OC 1a, and OC 1b hold for G whence OC 1 holds in B[a^{−1}], the unital hull of G. Now OC 3 is a consequence of OC 1 by 1.9. Moreover, we get the relations U_{G} ⊆ M(B)[U_{a}^{−1}], and U_{G,G} ⊆ M(B)[U_{a}^{−1}] which imply U_{B[a^{−1}]} ⊆ M(B)[U_{a}^{−1}] and therefore M(B[a^{−1}]) ⊆ M(B)[U_{a}^{−1}]. The converse inclusion being trivially true, the two sets are equal.

2.7. LEMMA. In the situation of 2.5 let k, l, m, n ∈ IN_{0}, and b, c, d, e ∈ B. We have

U_{[U(a^{−k})b,U(a^{−l})c]} = U_{a}^{−k−l}U_{[b,c]}U_{a}^{−k−l},  (∗ 1)
U_{[U(a^{−k})b,U(a^{−l})c,U(a^{−n})e]} = U_{a}^{−k−l−n}U_{[b,U(a^{n})c,U(a^{l})e]}U_{a}^{−k−l−n}, (∗ 2)
U_{[U(a^{−k})b,U(a^{−m})d,U(a^{−l})c,U(a^{−n})e]}
= U_{a}^{−k−l−m−n}U_{[U(a^{m})b,U(a^{k})d,U(a^{n})c,U(a^{l})e]}U_{a}^{−k−l−m−n}.  (∗ 3)

Proof. By 2.6 the identities OC 1 and OC 3 hold for B[a^{−1}]. Hence U(a^{−k})b ∘ U(a^{−l})c = U(a^{−k−l})(b ∘ c) as a consequence of OC 3. Using JA 6 and OC 1 we therefore obtain

U_{[U(a^{−k})b,U(a^{−l})c]}
= U_{U(a^{−k})b∘U(a^{−l})c} − 2 (U_{U(a^{−k})b}U_{U(a^{−l})c} + U_{U(a^{−l})c}U_{U(a^{−k})b})
= U_{U(a^{−k−l})(b∘c)} − 2 (U_{a}^{−k}U_{b}U_{a}^{−k}U_{a}^{−l}U_{c}U_{a}^{−l} + U_{a}^{−l}U_{c}U_{a}^{−l}U_{a}^{−k}U_{b}U_{a}^{−k})
= U_{a}^{−k−l}U_{b∘c}U_{a}^{−k−l} − 2 U_{a}^{−k−l}(U_{b}U_{c} + U_{c}U_{b})U_{a}^{−k−l}
= U_{a}^{−k−l}U_{[b,c]}U_{a}^{−k−l}.

Similarly, application of the identities U(a^{−k})b ∘ U(a^{−l})c = U(a^{−k−l−n})(b ∘ U(a^{n})c), U(a^{−k})b ∘ U(a^{−n})e = U(a^{−k−l−n})(b ∘ U(a^{l})e), JA 6, JA 6′, and OC 1 yields

U_{[U(a^{−k})b,U(a^{−l})c,U(a^{−n})e]} = U_{U(a^{−k})b∘U(a^{−l})c,U(a^{−k})b∘U(a^{−n})e} − 2 (U_{U(a^{−k})b}U_{U(a^{−l})c,U(a^{−n})e} + U_{U(a^{−l})c,U(a^{−n})e}U_{U(a^{−k})b}) = U_{U(a^{−k−l−n})(b∘U(a^{n})c),U(a^{−k−l−n})(b∘U(a^{l})e)}

− 2 (U_{a}^{−k}U_{b}U_{a}^{−k−l−n}U_{U(a^{n})c,U(a^{l})e}U_{a}^{−l−n} + U_{a}^{−l−n}U_{U(a^{n})c,U(a^{l})e}U_{a}^{−k−l−n}U_{b}U_{a}^{−k})

= U_{a}^{−k−l−n}U_{b∘U(a^{n})c,b∘U(a^{l})e}U_{a}^{−k−l−n}

− 2 U_{a}^{−k−l−n}(U_{b}U_{U(a^{n})c,U(a^{l})e} + U_{U(a^{n})c,U(a^{l})e}U_{b})U_{a}^{−k−l−n}

= U_{a}^{−k−l−n}U_{[b,U(a^{n})c,U(a^{l})e]}U_{a}^{−k−l−n}.

Analogously, we get (∗ 3) using JA 6′.

2.8. THEOREM. (a) Let A be a unital Jordan algebra, B an operator-commuta-tive subalgebra containing 1_{A}, and a ∈ A^{×}∩B. Then the subalgebra C:= B[a^{−1}] is operator-commutative.

(b) Let A be a non-unital Jordan algebra over R, B an operator-commutative subalgebra, and a ∈ Q(A) ∩ B. Then the subalgebra C:= B[b], where b denotes the quasi-inverse of a, is operator-commutative.

Proof. (a) By 2.6 the identity OC 1 holds for B[a^{−1}]. Defining G as in the proof of 2.5 and using (∗ 1) to (∗ 3) we get OC 2, OC 2a and OC 2b in G. Thus OC 2 holds for B[a^{−1}], the linear hull of G.

(b) This follows by applying (a) to the unital hull of A and using the identity B̂[(1_{A} − a)^{−1}] = B̂[(1_{A} + b)] = R ⊕ B[b].

2.9. COROLLARY. (a) A unital maximal operator-commutative subalgebra of a unital Jordan algebra is inverse-closed.

(b) A maximal operator-commutative subalgebra of a Jordan algebra is quasiinverse-closed.

Proof. (a) By 2.8 the subalgebra B[a^{−1}], a ∈ A^{×} ∩ B is operator-commutative. Thus maximality of B implies B[a^{−1}] = B whence a^{−1} ∈ B resp. a ∈ B^{×}.

(b) This is shown similarly to (a).

3. THE (QUASI-)INVERSE-CLOSED OPERATOR-COMMUTATIVE HULL

3.1. LEMMA. Let A be a Jordan algebra over R.

(a) If K ⊆ Pot(A) is a chain of operator-commutative subalgebras of A, then the union C:= ⋃ K is operator-commutative.

(b) For any operator-commutative subalgebra B ⊆ A there exists a maximal operator-commutative subalgebra of A containing B.

Proof. (a) Let λ ∈ R and a, b ∈ C, i.e. a ∈ B_{1} resp. b ∈ B_{2} for B_{1}, B_{2} ∈ K. Since K is a chain with respect to inclusion, we have

B:= B_{1} ∪ B_{2} ∈ {B_{1}, B_{2}} ⊆ K.

Therefore a, b ∈ B, and B is operator-commutative, whence λa, a + b, U_{a}b, a^{2} ∈ B ⊆ C. Thus OC 1 to OC 6 hold for a and b. Now λ, a, and b being arbitrary, the claim is clear.

(b) By (a) this follows from Zorn’s Lemma.

3.2. COROLLARY. (a) Let A be a unital Jordan algebra over R, x ∈ A, and R_{1}[x]:=_{A} ⟨{1_{A}, x}⟩. Then there exists an inverse-closed, operator-commutative subalgebra of A containing R_{1}[x].

(b) Let A be an arbitrary Jordan algebra over R, x ∈ A, and R[x] :=_{A} ⟨{x}⟩. Then there exists a quasi-inverse-closed, operator-commutative subalgebra of A containing R[x].

Proof. By 2.9 and 3.1.(b) it suffices to show that R_{1}[x] and R[x] are operatorcommutative:

(a) For a special Jordan algebra J, u ∈ J, and m, n ∈ IN_{0} we have the identities [U_{u^{m}}, U_{u^{n}}] = 0 resp. U_{[u^{m},u^{n}]} = 0. By Shirshov’s Theorem, these identities hold for all Jordan algebras, whence OC 1 and OC 2 hold for the powers of x. Analogously, it is possible to show that OC 1a, OC 1b, OC 2a, and OC 2b hold. Thus OC 1 and OC 2 hold for R_{1}[x], the linear hull of the powers of x.

(b) By (a) the subalgebra R_{1}[x] of the unital hull of A is operator-commutative. Since R_{1}[x] = R ⊕ R[x], the same is true for R[x] by 1.2.

3.3. THEOREM. (a) Let A be a unital Jordan algebra over R, x ∈ A, and U the set of all inverse-closed, operator-commutative subalgebras of A containing R_{1}[x]. Then R_{1} ⟨[x]⟩ := ⋂ U is an inverse-closed, operator-commutative subalgebra of A. More precisely, it is the smallest such algebra containing x, called the inverse-closed operator-commutative hull of x.

(b) Let A be a unital Jordan algebra over R, x ∈ A, and U the set of all quasi-inverse-closed, operator-commutative subalgebras of A containing R[x].

Then R ⟨[x]⟩:= ⋂ U is a quasi-inverse-closed, operator-commutative subalgebra of A. More precisely, it is the smallest such algebra containing x, called the quasi-inverse-closed operator-commutative hull of x.

Proof. The sets R_{1} ⟨[x]⟩ resp. R ⟨[x]⟩ are well-defined since U ≠ ∅ by 3.2. Generally, any intersection of subalgebras is again a subalgebra. Moreover, the properties OC 1 to OC 6 resp. IC 1 and QIC 1 still hold for intersections.

4. BANACH JORDAN ALGEBRAS

4.1. LEMMA. Let A be a Banach Jordan algebra, i.e. a complex normed complete Jordan algebra such that all Jordan operators are continuous, and let B be a subalgebra of A.

(a) If B is operator-commutative, then the same is true for its closure \overline{B}.

(b) If A is unital, 1_{A} ∈ B, and B is inverse-closed, then \overline{B} is inverse-closed, too.

(c) If B is quasi-inverse-closed, then the same is true for \overline{B}.

Proof. (a) Since the identities OC 1 to OC 6 hold for B, this follows from continuity of the mappings x ↦ U_{x}, x ↦ V_{x}, (x, y) ↦ U_{x}y, and (x, y) ↦ x ∘ y.

(b) Let a ∈ A^{×} ∩ \overline{B}. Then there exists a sequence (a_{n})_{n∈IN} in B such that a = lim_{n→∞} a_{n}. It is well-known that A^{×} is an open subset of A. Thus there is an index n_{0} such that all a_{n} are invertible for n ≥ n_{0}. This implies a_{n}^{−1} ∈ B for all n ≥ n_{0} since B is inverse-closed. By continuity of inversion we obtain a^{−1} = lim_{n→∞} a_{n}^{−1} ∈ \overline{B}, whence a ∈ \overline{B}^{×}.

(c) The assertion is clear if we apply (b) to B̂ = C ⊕ B.

4.2. COROLLARY. (a) Let A be a unital Banach Jordan algebra, and x ∈ A. Then there exists a closed inverse-closed, operator-commutative subalgebra of A containing C_{1}[x] and therefore \overline{C_{1}[x]}.

(b) Let A be an arbitrary Banach Jordan algebra, and x ∈ A. Then there exists a closed quasi-inverse-closed, operator-commutative subalgebra of A containing C[x] and therefore \overline{C[x]}.

Proof. By 3.1(b) there exists a maximal operator-commutative subalgebra of A containing C_{1}[x] resp. C[x]. This subalgebra is inverse-closed resp. quasiinverse-closed as a consequence of 2.9 and closed by maximality and 4.1(b)(c).

4.3. THEOREM. (a) Let A be a unital Jordan algebra over R, x ∈ A, and V the set of all closed inverse-closed, operator-commutative subalgebras of A containing C_{1}[x]. Then P_{1}(x):= ⋂ V is a closed inverse-closed, operator-commutative subalgebra of A. More precisely, it is the smallest such algebra containing x, and we have P_{1}(x) = \overline{C_{1} ⟨[x]⟩}.

(b) Let A be a unital Jordan algebra over R, x ∈ A, and V the set of all closed quasi-inverse-closed, operator-commutative subalgebras of A containing C[x]. Then P(x) := ⋂ V is a closed quasi-inverse-closed, operator-commutative subalgebra of A. More precisely, it is the smallest such algebra containing x, and we have P(x) = \overline{C ⟨[x]⟩}.

Proof. With the same arguments as in the proof of 3.3 we obtain that P_{1}(x) and P(x) are well-defined operator-commutative, (quasi-)inverse-closed subalgebras. If we define U as in 3.3 we have V ⊆ U. Therefore P_{1}(x) ⊇ C_{1} ⟨[x]⟩ and P_{1}(x) ⊇ \overline{C_{1} ⟨[x]⟩} resp. P(x) ⊇ C ⟨[x]⟩ and P(x) ⊇ \overline{C ⟨[x]⟩}. Since C_{1} ⟨[x]⟩ and C ⟨[x]⟩ are operator-commutative and (quasi-)inverse-closed, the same is true for their closures by 4.1. Thus \overline{C_{1} ⟨[x]⟩} ∈ V and P_{1}(x) ⊆ \overline{C_{1} ⟨[x]⟩} resp. \overline{C ⟨[x]⟩} ∈ V and P(x) ⊆ \overline{C ⟨[x]⟩}.

Acknowledgement. I am grateful to O. Loos and K. McCrimmon for valuable remarks.

REFERENCES

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3. O. Loos, Properly algebraic and spectrum-finite ideals in Jordan systems, Math. Proc. Camb. Phil. Soc. 114 (1993), 149–161.

4. J. Martínez Moreno, “Sobre algebras de Jordan normadas,” Tesis doctorales de la Universidad de Granada, Vol. 149, 1977.

5. K. McCrimmon, E. Zel’manov, The structure of strongly prime quadratic Jordan algebras, Adv. Math. 69 (1988), 133-222.