Ann. Soc. Math. Polon. Ser. I Comment. Math. Prace Mat. 20 (1978), 279–281

ON BANACH ALGEBRAS WHICH ARE HILBERT SPACES

WOJCIECH CHOJNACKI (WARSZAWA)

Let F denote a field that is either the real field R or the complex field C. Let H denote the real quaternion algebra.

An algebra A over F with identity e is called a Hilbert algebra with identity if A is a Hilbert space such that the norm derived from an inner product has the following properties:

1^{o} ∥e∥ = 1,

2^{o} ∥a · b∥ ≤ ∥a∥ · ∥b∥ for a, b ∈ A. If F = R, A is called a real Hilbert algebra, and if F = C, a complex Hilbert

algebra.

L. Ingelstam proved in [2] the following

Theorem. A real Hilbert algebra with identity is isomorphic to R, C or H.

In the present note we shall give a new simpler proof of this theorem.

We shall begin our considerations with the following

Lemma. Let A be a real Hilbert algebra with identity e. Let Φ be a multiplicative linear functional on A. Then Φ is injective.

Proof. In virtue of the Riesz theorem, Φ represents in the form

(1)  Φ(x) = (x, a),

where x, a ∈ A. It is well known that ∥Φ∥ = 1 for an arbitrary multiplicative linear functional Φ on a Banach algebra with identity. Hence and by (1) we obtain

(2)  ∥a∥ = 1.

By the Cauchy-Schwarz inequality and by 1^{o}, (2) we have

(3)  1 = Φ(e) = (e, a) ≤ ∥e∥ · ∥a∥ = 1.

In particular (e, a) = ∥e∥ · ∥a∥. But the equality in the Cauchy-Schwarz occurs exactly in the case when both a and e are linearly dependent. Therefore we can write a = λe, λ ∈ R. Using 1^{o}, (3) we obtain

λ = λ(e, a) = (a, e) = 1,

so actually we have a = e and

(4)  Φ(x) = (x, e).

Now let us consider the following linear operator

T_{a}(x) = a · x, a, x ∈ A.

Using 2^{o} and the fact that T_{a}(e) = a we obtain

(5)  ∥T_{a}∥ = ∥a∥.

Let T_{a}^{∗} denote the adjoint operator. By the elementary fact that ∥T_{a}^{∗}∥ = ∥T_{a}∥ and by (5) we obtain

(6)  ∥T_{a}^{∗}∥ = ∥a∥.

Now by the Cauchy-Schwarz inequality and by 1^{o}, (6) we have

(7) ∥a∥^{2} = (T_{a}(e), a) = (e, T_{a}^{∗}(a)) ≤ ∥e∥ · ∥T_{a}^{∗}(a)∥ ≤ ∥T_{a}^{∗}∥ · ∥a∥ = ∥a∥^{2}.

Hence, as before, T_{a}^{∗}(a) = λe for some λ ∈ R, and by 1^{o}, (7) we have

λ = λ(e, e) = (e, T_{a}^{∗}(a)) = ∥a∥^{2}.

Consequently

(8)  T_{a}^{∗}(a) = ∥a∥^{2} · e.

Now by (8) we obtain for a, b ∈ A

(9)  (a · b, a) = (T_{a}(b), a) = (b, T_{a}^{∗}(a)) = ∥a∥^{2} · (b, e).

By (9), 1^{o} we have for a ∈ A

\begin{aligned}((a + e) · a, a + e) − (a^{2}, a) & = ∥a + e∥^{2} · (a, e) − ∥a∥^{2} · (a, e) \\ & = (1 + 2(a, e)) · (a, e). \end{aligned} ; (10)

But

(11)  ((a + e) · a, a + e) − (a^{2}, a) = ∥a∥^{2} + (a^{2}, e) + (a, e).

By (4) we have

(12)  (a^{2}, e) = Φ(a^{2}) = (Φ(a))^{2} = (a, e)^{2}.

Taking account of (10), (11), (12) we obtain ∥a∥^{2} = |(a, e)| = |Φ(a)|. Hence ker Φ = {0}. The proof of the lemma is completed.

Now we shall prove the main theorem. Let A be a real Hilbert algebra with identity. In order to prove that A is isomorphic to R, C or H it is sufficient to show that A is a real division algebra (cf. Theorem 7, Chapter I, §14 of F. F. Bonsall, J. Duncan [1]). Let x be an arbitrary non-zero element of A. Let us consider the smallest closed subalgebra A_{0} with identity containing x. A_{0} is a commutative real Hilbert algebra with identity. Suppose that x has no inverse in A_{0}. Then x belongs to a certain maximal ideal M. Obviously, M is closed. Let ϕ be the canonical homomorphism of A_{0} onto A_{0}/M. A_{0}/M is a commutative real division algebra and therefore A_{0}/M is isomorphic to R or C. In both cases A_{0}/M contains a closed subalgebra B isomorphic to R. Let η be an isomorphism of B onto R. Since ψ is continuous, it follows that ψ^{−1}(B) is a real Hilbert algebra with identity. Φ = η ∘ ψ is a multiplicative linear functional on ψ^{−1}(B). In virtue of the lemma, Φ is injective. In particular, M = {0} contrary to our assumption of the non-triviality of M. Consequently x has inverse in A_{0}, and so in A. A is then a real division algebra. This completes the proof.

An immediate consequence of our theorem is the following

Corollary. A complex Hilbert algebra with identity is isomorphic to C.

Proof. Consider a complex Hilbert algebra A with identity as a real Hilbert algebra with the inner product ⟨x, y⟩ = Re(x, y) for x, y ∈ A. In virtue of our theorem, A is isomorphic to R, C or H. Among the last three algebras C is the only complex Banach algebra. This remark completes the proof. □

References

1. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin–Heidelberg–New York, 1973.

2. L. Ingelstam, Hilbert algebras with identity, Bull. Amer. Math. Soc. 69 (1963), 794–796.