����������� ������ 55 (2003), 15–19

UDK 517.986

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SPECTRAL STATES OF COMMUTATIVE L.M.C. ALGEBRAS

A. K. Gaur

Abstract. We characterize the commutative locally multiplicative convex (l.m.c.) algebras in terms of the spectral states. We also give a characterization of spectral states in terms of commutative semisimple l.m.c. algebras. Further, with the help of radicals of l.m.c. algebras we give a necessary and a sufficient condition for an algebra to be commutative modulo its radical.

1. Introduction

Let X be a locally m-convex (l.m.c.) algebra with unit e. We will follow the notations and terminologies of [4] and [6]. It is sufficient for our purpose to note that, for a given l.m.c. algebra X with unit e there exists a separating family of submultiplicative seminorms {P_{α}} on X which generates the topology and is such that P_{α}(e) = 1 for all α in the index set I. Given such an aglebra, we denote by P (X) the class of all such families of seminorms on X, and by (X, {P_{α}}) the algebra X with a particular family of seminorms {P_{α}} ∈ P (X).

For every α ∈ I, let X_{α} denote the unital Banach algebra. Using Bonsall and Duncan’s notation [2], the spectral state of X_{α} is denoted by Ω(X_{α}) and Ω(X_{α}) = {f ∈ X_{α}^{∗} : f(e) = 1, |f(x)| ≤ ρ_{α}(x), x ∈ X_{α}}, where ρ_{α}(·) is the spectral radius of x_{α} and ∥x_{α}∥_{α} = P_{α}(x). (See Michael [6]). Ω(X_{α}) is a weak^{∗}-compact convex subset of the complex plane. The set of all spectral states of X is denoted by Ω(X). If q_{α}^{∗}: X → X_{α} is the quotient map and q_{α}^{∗} is the adjoint of q_{α}, then we define Ω(X) = ⋃ q_{α}^{∗}(Ω(X_{α})).

Given (X, {P_{α}}), we define the set D_{α}(X, P_{α}; e) = {f ∈ X′ : f(e) = 1 and |f(x)| ≤ P_{α}(X) for all x ∈ X} and we write

D(X, {P_{α}}; e) = ⋃{D_{α}(X, P_{α}; e)}.

Note that D_{α}(X, P_{α}; e) is isomorphic to D(X_{α}, ∥·∥_{α}; e_{α}) and D(X, {P_{α}}; e) depends upon the particular family of seminorms {P_{α}} ∈ P (X) chosen to associate with X.

AMS Subject Classification: 46 J 99, 46 J 15, 46 K 99

Keywords and phrases: Spectral states, probability measure, l.m.c. algebra, commutative modulo.

2. Spectral states and commutative l.m.c. algebras

Using the idea in Giles and Koehler [4], we can say that Ω(X) is the state space of (X, {P_{α}}) and Ω(X) = D(X, {P_{α}}; e). Note that Ω(X) does not depend on the particular family of seminorms {P_{α}} chosen to generate the topology.

Theorem 2.1. Let D_{X} = ⋂{ D(X, {P_{α}}; e) : {P_{α}} ∈ P (X) }. Then Ω(X) = D_{X}.

Proof. Let f ∈ Ω(x) and {P_{α}} ∈ P (X). If x ∈ X and α ∈ I with |f(x)| ≤ ρ_{α}(x_{α}) ≤ P_{α}(x), then there exists M > 0 and β ∈ I such that P_{α}(x) ≤ MP_{β}(x) and |f(x)| ≤ \sqrt[n]{P_{α}(x^{n})} ≤ \sqrt[n]{M}P_{β}(x) for every natural number n and every x ∈ X. This shows that f ∈ D_{X}. Conversely, suppose that f ∈ X′ is not a spectral state and f(e) = 1. Then for each α, there exists x_{α} ∈ X_{α} such that |f(x_{α})| > ρ_{α}(q_{α}(x_{α})). By Lemma 2.8, [2], there exist seminorms v_{α} on X_{α} equivalent to the usual norm ∥ · ∥_{α} such that |f(x_{α})| > v_{α}(q_{α}(x_{α})). Let P_{α} = v_{α}, α ∈ I. Then P_{α} ∈ P (X), but f ∉ D(X, {P_{α}}; e). This implies that f ∉ D_{X} and hence Ω(X) = D_{X}.

Let X be a commutative l.m.c. algebra. Let Φ_{X} be the set of all multiplicative linear functionals on X and let Φ_{α} be the set of all multiplicative linear functionals on X_{α}. Also, suppose that ψ_{α} = q_{α}^{∗}(Φ_{α}). This means that for each α, Φ_{α} is homeomorphic to ψ_{α}. Let X̂ and X̂_{α} be the Gelfand transformations on X and X_{α}, respectively. Denote a compact Hausdorff space by E and suppose μ(E) denotes the set of all probability measures on E. (For more on these measures, see [1]).

Proposition 2.1. For a commutative l.m.c. algebra X with unit and for f ∈ X′, the following are equivalent.

(a) For α ∈ I and μ ∈ μ(E)ψ_{α}, f(x) = ∫ X̂(x) dμ, x ∈ X.

(b) There exists a probability measure μ on Φ_{X} with compact (equicontinuous) support K, (see [7]), with f(x) = ∫ X̂(x) dμ, x ∈ X.

Proof. If K is a compact subset of Φ_{X}, then K is contained in some ψ_{α}. Hence,

(b) implies (a). The implication (a) ⟹ (b) follows by the definitions involved.

Proposition 2.2. There exists α ∈ I such that |e^{f(x)}| ≤ ∥e^{x_{α}}∥_{α}, x ∈ X if and only if

Re f(x) ≤ sup{ Re η_{α}(x_{α}) : η_{α} ∈ Φ_{α} }.  (∗)

Proof. For α ∈ I, |e^{f(x)}| ≤ ∥e^{x_{α}}∥_{α} ⇔ Re f(x) ≤ \frac{1}{n} ln ∥e^{nx_{α}}∥_{α}, where ln is the natural log function, n is a natural number, and x ∈ X. If sp(X_{α}, x_{α}) denotes the spectrum of x_{α}, see [4], then by Theorem 8, page 32 of [2], we have sup\{ Re λ : λ ∈ sp(X_{α}, x_{α}) \} = inf\{ \frac{1}{n} ln ∥e^{nx_{α}}∥_{α} : n is a natural number \} and hence ∥e^{f(x)}∥ ≤ ∥e^{x_{α}}∥_{α} is equivalent to condition (∗).

Remark 2.1. The above propositions provide us with a characterization for spectral states of l.m.c. algebras. Further, these characterizations also show that Ω(X) does not depend on the particular family of seminorms.

It is clear that Ω(X) contains all non-zero multiplicative linear functionals. Also, if X is a commutative l.m.c. algebra, then every probability measure on the Carrier space [8, p. 261] of X provides a spectral state on X.

For commutative X, Ω(X) is nonempty, but if H is an infinite dimensional complex Hilbert space and B(H) is the set of all bounded linear operators on H, Ω(B(H)) = ∅, see example 5, page 115, [2]. In fact for C^{∗}-algebra X, Ω(X) = ∅. On the other hand, if B(H) is the set of all compact operators on H, then Ω(B(H)) = {0}.

Example 2.1. Let E be a compact Hausdorff space and C(E) be the l.m.c. algebra of all complex-valued continuous functions on E. The topology on C(E) is of the uniform convergence. Then Φ_{C(E)} is isomorphic to E. The countable compact subsets of E are the compact subsets of Φ_{C(E)}. Let ϕ ∈ C(E) and for each natural number n, a_{n} ∈ E. Suppose λ_{n} ∈ [0, 1], then a linear functional f on C(E) is given by f(ϕ) = ∑_{n=1}^{∞} λ_{n}ϕ(a_{n}). These linear functionals define the spectral states of C(E). Let μ be a probability measure on E which vanishes at singletons. Then f is defined by integration with respect to μ such that f(e) = 1. Further, |f(ϕ)| ≤ ρ_{C(E)}(ϕ) and f(ϕ) ∈ co sp(C(E), ϕ) for each ϕ ∈ C(E), where co is the convex hull. Since f is defined by integration with respect to a probability measure μ with an uncountable support, f is not a spectral state.

Remark 2.2. If A is a finite dimensional complex Banach algebra with unit and Wedderburn decomposition A = A_{1} ⊕ A_{2} ⊕ ⋯ ⊕ A_{m} ⊕ R (where R is the radical of A and each A_{i} is a subalgebra of A that is isomorphic to a matrix algebra over the complex numbers), then Ω(A) is the convex hull of the normalized traces T_{i}(i = 1, 2,…, m), see Theorem 11, p. 119 [2]. Also, if R is the Jacobsen radical of A, then f(R) = {0} for each f in Ω(A).

3. Commutative semisimple algebra and spectral states

Definition 3.1. X_{α} is semisimple if the Gelfand transformation on X_{α} is one-to-one.

A commutative Banach algebra A is simple if Rad(A) = {0}.

So if we have a semisimple l.m.c. algebra, then a rich supply of spectral states is possible. We prove the following theorem which characterizes such algebras.

Theorem 3.1. Let X be an l.m.c. algebra with unit. Then X is commutative and semisimple if and only if Ω(X) separates the points of X.

Proof. Let X be a commutative semisimple l.m.c. algebra with unit. Then the complex homomorphisms of X separate the points of X, see Corollary 3.5.1, [5]. Hence, Ω(X) separates the points of X.

Conversely, suppose that Ω(X) separates the points of X. If f ∈ Ω(X), then f(ab) = f(ba), a, b ∈ X, by Theorem 4, p. 114 [2]. Hence, X is commutative. Since every f ∈ Ω(X) vanishes on the kernel of the Gelfand transformation X̂ on X, Proposition 2.1 proves that X̂ is one-to-one and X is semisimple by Definition 3.1 above.

In [4], it is shown that if X is a complex l.m.c. algebra with unit, then for each x ∈ X,

co sp(X, x) ⊆ ⋂\{ V (X, \{P_{α}\}; x) : \{P_{α}\} ∈ P (X) \} ⊆ \overline{co} sp(X, x)

where V (X, {P_{α}}; x) is the numerical range of x in X.

If X is commutative modulo its radical, then co sp(X, x) = {f(x) : f ∈ Ω(X)}. This follows from the fact that the following condition in [4]

⋂{ f(x) : f ∈ D(X, {P_{α}}; e) }

can be replaced by

{ f(x) : f ∈ ⋂ D(X, {P_{α}}; e) }

Inspired by this observation, we have the following theorem.

Theorem 3.2. Let X be a complete l.m.c. algebra with unit. Then X is commutative modulo Rad(X) if and only if co sp(X, x) = {f(x) : f ∈ Ω(X)} for every x ∈ X.

Proof. Let X be commutative modulo Rad(X). By Proposition 24.16 in [3], it follows that for a, x, y ∈ X, a(xy − yx) is quasi-regular or quasi-invertible, see [5, p.13]. This implies that ρ_{X}(xy −yx) = 0. Thus, for each α ∈ I, ρ_{α}(x_{α}y_{α} −y_{α}x_{α}) = 0, which proves that X_{α} is commutative modulo Rad(X_{α}).

For each x ∈ X, { f_{α}(x_{α}) : f_{α} ∈ Ω(X_{α}) } ⊂ co sp(X_{α}, x_{α}) and since X_{α} is commutative modulo Rad(X_{α}), we have sp(X_{α}, x_{α}) = { ϕ_{α}(x_{α}) : ϕ_{α} ∈ Φ_{α} }. Further, since Φ_{α} ⊂ Ω(X_{α}), and Ω(X_{α}) is convex, we have co sp(X_{α}, x_{α}) ⊂ { f_{α}(x_{α}) : f_{α} ∈ Ω(X_{α}) }. Hence we have established that if X is commutative modulo Rad(X), then co sp(X_{α}, x_{α}) = { f_{α}(x_{α}) : f_{α} ∈ Ω(X_{α}) }.

Since the family of spectra is a well directed family, we have co sp(X, x) = ⋃ co sp(X_{α}, x_{α}), see Theorem 1 [4]. By the definition of Ω(X), we prove that co sp(X_{α}, x_{α}) = { f_{α}(x_{α}) : f_{α} ∈ Ω(X_{α}) } implies that co sp(X, x) = { f(x) : f ∈ Ω(X) } for every x ∈ X.

Conversely, suppose that for each x ∈ X, co sp(X, x) = { f(x) : f ∈ Ω(X) }. Since sp(X, xy − yx) = {0}, we have the commutativity of each X_{α} modulo its radical. Hence, a(xy − yx) is quasi-regular in X. This shows that xy − yx belongs to Rad(X) and hence X is commutative modulo Rad(X).

Corollary 3.1. If X is a complete l.m.c. algebra with unit and X is commutative modulo Rad(X), then

Γ_{X} = { x ∈ X : {sup |f(x)| : f ∈ Ω(X)} < ∞ }
= { x ∈ X : V (X, {P_{α}}; x) is bounded } = U_{X}
= { x ∈ X : ρ_{X}(x) < ∞ } = R_{X}.

Proof. Γ_{X} = U_{X} follows from Theorem 4, [4]. By Theorem 3.2, {f(X) : f ∈ Ω(X)} = co sp(X, x), hence U_{X} = R_{X}.

Corollary 3.2. Ω(X) is weak^{∗}-bounded.

Proof. The result follows from Corollary 3.1 and the fact that Ω(X) is weak^{∗}- bounded if and only if sp(X, x) is bounded for each x ∈ X.

Acknowledgements. The author thanks the referee for his or her constructive comments and suggestions which clearly have improved the clarity of the paper.

REFERENCES

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[2] Bonsall, F.F., Duncan J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Vol. I, London Math. Soc, Lect. Notes 2, Cambridge Univ. Press, Cambridge, 1971.

[3] Bonsall, F.F., Duncan, J., Complete Normed Algebras, Ergebnisse der Mathematik 80, Springer Verlag, Berlin, 1973.

[4] Giles, J.R., Koehler, D.O., On Numerical Ranges of Elements of Locally m-convex Algebras, Pac. J. Math. 49 (1973), 79–91.

[5] Larsen, R., Banach Algebras. An Introduction, Marcel Dekker, Inc., N.Y., 1973.

[6] Michael, E.A., Locally Multiplicatively Convex Topological Algebras, Amer. Math. Soc. Mem., 11 (1952).

[7] Royden, H.L., Real Analysis, Macmillian Publishing Co., 1968.

[8] Wilansky, A., Functional Analysis, Blaisdell Publishing Co., 1964.

(received 15.03.2001, in revised form 13.06.2003)

Department of Mathematics and Computer Science, Duquesne University, Pittsburgh, PA 15282, U.S.A. E-mail: gaur@mathcs.duq.edu