A SIMPLE CHARACTERIZATION OF COMMUTATIVE H∗-ALGEBRAS

Commutative H∗-algebras are characterized without postulating the existence of Hilbert space structure.


Introduction.
Let M be the space of all maximal regular ideals in a commutative H * -algebra A and let x(M), M ∈ M, denote the Gelfand transform of x, Loomis [3] (in the sequel we use notation of Naimark [5]).Then it is easy to show (see Theorem 1 below) that the series x(M) ȳ(M) converges absolutely for all x, y ∈ A. Also, if we assume that each minimal self-adjoint idempotent in A has norm one, then it is true that for each bounded linear function f on A(f ∈ A * ) there exists a ∈ A such that f (x) = x(M) ā(M) for all x ∈ A.
In this note we show that these properties could be used to characterize commutative proper H * -algebras of this kind.More specifically we show that each semi-single completely symmetric, Naimark [5], Banach algebra with the above properties is a proper H * -algebra with respect to some Hilbertian norm which is equivalent to its original norm.Also, there is a characterization of all proper commutative H * -algebras.

Characterizations.
Let A be a complex commutative Banach algebra.We do not assume that A has an identity and so, because of this, we have to deal with regular maximal ideals.An ideal I in A is regular if the algebra A/I has an identity.If M is maximal regular ideal then it is closed and the algebra A/M is isomorphic to the complex field (Gelfand-Mazur theorem, complex case, Loomis [3, 22F]).It follows that there exists a continuous linear functional F M , Loomis [3, 23B], such that M = {x ∈ A : F M (x) = 0}, i.e., M is the kernel (null space) of F M .
The Gelfand transform x() (we use the Naimark's notion, Naimark [5], here) of x is defined by setting x(M) = F M (x) (Loomis uses the notion x ∧ in Loomis [3, 23B]), where M is a regular maximal ideal in A.
The algebra A is said to be semi-simple if ∩ M∈ M M = (0) (as it is stated above, M denotes the space of all maximal regular ideals as A).Equivalent condition: mapping x → x() is one to one.The algebra A is said to be completely symmetric, Naimark [5], if it has an involution x → x * such that x * (M) = x(M) for all M ∈ M.
A proper H * -algebra is a Banach algebra A with an involution x → x * and a scalar product ( , ) such that (x, x) = x 2 and (xy, z) = (y, x * z) = (x, zy * ) for all x, y, z ∈ A. Note that A is semi-simple.For simplicity, a nonzero self-adjoint idempotent will be called projection (e.g., Saworotnow [6]).A projection e is minimal if it is not a sum of two projections whose product is zero.
A completely symmetric commutative Banach algebra is a Banach algebra with invo- Theorem 1.Each proper commutative H * -algebra A is completely symmetric in the sense of Naimark [5].Also, the series M∈M |x(M)| 2 converges for each x ∈ A and if we assume that each minimal projection in A has norm one, then each bounded linear functional Proof.First and second parts of the theorem follow from Loomis [3, 27G].For each M ∈ M there exists a minimal projection e M such that . Theorem 2. Let A be a semi-simple commutative completely symmetric Banach algebra.Assume further that the series M∈M |x(M)| 2 converges for each x ∈ A and that for each bounded linear functional f on A there exists a ∈ A such that f (x) = M∈M x(M) ā(M) for all x ∈ A. Then there exists a Hilbertian norm 2 on A, equivalent to the original norm such that A is an H * -algebra with respect to the scalar product ( , ) associated with 2 and the original involution.Also, each minimal projection in A has norm 1.

Proof. For each x, y ∈ A, define (x, y) = M∈M x(M) ȳ(M). This series converges absolutely for all x, y ∈
for each finite subset {M 1 ,...,M k } of M. Hence, the inner product ( , ) is defined everywhere on A. Let 2 be the corresponding norm, x 2 2 = (x, x) for all x ∈ A. Let us show that A is complete with respect to 2 .First, note that the completion A of A with respect to 2 is a proper H * -algebra (since x * 2 = x 2 for all x ∈ A).Hence, A is semi-simple.(It is a consequence of Loomis [3, 27A].)So we can apply [5, Sec. 12, Thm.1]: there exists C > 0 such that x 2 ≤ C x for all x ∈ A. Now, let {a n } be a sequence of numbers of A such that lim m,n a n −a m 2 = 0. Then there exists N > 0 such that a n 2 ≤ N for each n.For each fixed x ∈ A define (2.2) From |(x, a m )| < x 2 a m 2 ≤ NC x we conclude that f is a bounded linear functional on A. Hence, there exists a ∈ A so that f (x) = M∈M x(M) ā(M) for each x ∈ A.
Let us show that a−a n 2 → 0. Let > 0 be arbitrary, take n 0 so that a m −a n 2 < /2 if m,n > n 0 .Let n > n 0 and x ∈ A be fixed.Then a−a n Select m > n 0 so that and this implies that a − a n 2 < for each n > n 0 .So, A is complete with respect to 2 .It follows from [5, Sec. 12, Thm.1] that the norm 2 and the original norm on A are equivalent.
It is also easy to see that A is an H * -algebra with respect to the scalar product ( , ) (and the original involution).
Let us show that every minimal projection in A has norm one.First note that the product of any two distinct minimal projections e 1 and e 2 is zero, e 1 e 2 = 0.It follows from the fact that e = e 1 e 2 is also a projection and that ee i = e i , i = 1, 2. This means that if e ≠ 0, then both e = e 1 and e = e 2 , which is impossible, since e 1 ≠ e 2 .Thus e M 1 e M 2 = 0 if M 1 ≠ M 2 (as was remarked in a proof above).But this also means that every minimal projection e is of the form e = e M for some M ∈ M. It follows then that e(M ) = 1 and e For the general case we have Theorems 3 and 4 below, which constitute a characterization of any proper commutative H * -algebra.The characterization is stated in terms of multiplicative functionals (it could also be done in terms of ideals) (needless to say, Theorems 1 and 2 could be restated in terms of multiplicative functionals also).Theorem 3.For each proper commutative H * -algebra A there exists a real valued function k(q), defined on the set Q of all its continuous multiplicative linear functionals, with the following properties : (i) k(q) ≥ 1 for each q ∈ Q. (ii) The series q∈Q |q(x)| 2 k(q) converges for each x ∈ A. (iii) For each f ∈ A * there exists α ∈ A such that f (x) = q∈Q q(x)q(a)k(q) for each x ∈ A(A * denotes the dual of A).
Proof.It is easy consequence of Loomis [3, 27G] that for each nonzero member q of Q there exists a unique minimal projection e q such that q(x) = (x, e q ) e q −2 and x = q∈Q q(x)e q (2.5) for each x ∈ A (note that {e q } q≠0 is an orthogonal basis for A).We define the function k(q) by setting k(q) = e q 2 for each nonzero member q of Q and k(0) = 1.We leave it to the reader to verify that k(q) has desired properties.

Theorem 4.
Let A be a semi-simple commutative completely symmetric algebra and let Q be the set of all its continuous multiplicative linear functionals.Assume that there exists a real valued function k(q) on Q with properties (i), (ii), and (iii) in Theorem 3.
Then A is an H * -algebra with respect to some Hilbert space norm 2 equivalent to the original norm of A, and the original involution.
Proof.Define the scalar product ( , ) on A by setting (x, y) = q∈Q q(x)q y * k(q), (2.6) and take that corresponding norm 2 (with the property that (x, x) = x 2 2 ).Then we proceed as in the proof of Theorem 2.
The last part follows from Loomis[3, 10G]: If we assume that each minimal projection has norm one, thenx 2 = M∈M |x(M)| 2 and (x, a) = M∈M x(M) ā(M)for all x, a ∈ A (and there exists a ∈ A such that f (x) = (x, a) for all x ∈ A).Now we have a characterization of those commutative H * -algebra in which each minimal projection has norm one.