A characterization of linear-multiplicative functionals in topological algebras

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Seria I: PRACE MATEMATYCZNE XLVI (1) (2006), 79-91

Maciej Łuczak

A characterization of linear-multiplicative functionals in topological algebras

Abstract. The classical Gleason–Kahane–Żelazko theorem for complex Banach algebras was generalized for not necessary linear functionals by Kowalski and Słodkowski.

We prove a version of the Kowalski–Słodkowski theorem for real Banach algebras and also for real and complex A-pseudoconvex algebras.

2000 Mathematics Subject Classification: 46H05, 46H10.

Key words and phrases: A-pseudoconvex algebra, real algebra, Gleason–Kahane–

Żelazko theorem, Kowalski–Słodkowski theorem.

1. Introduction. The following characterization of linear-multiplicative functionals in complex Banach algebras, given by Gleason and Kahane–Żelazko ([2], [3], [8]), is known as the Gleason–Kahane–Żelazko theorem:

Theorem 1.1 (G–K–Ż thm. for complex Banach algebras) Let A be a complex Banach algebra. Let T : A → C be a linear functional such that

T a ∈ σ(a) (a ∈ A).

Then T is multiplicative.

This theorem for real Banach algebras was proved by Kulkarni ([5]):

Theorem 1.2 (G–K–Ż thm. for real Banach algebras) Let A be a real Ba- nach algebra with unit. Let T : A → C be an R-linear functional such that

T 11 = 1,

(T a)²+ (T b)²∈ σ(a²+ b²) (ab = ba, a, b ∈ A).

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Versions of the above two theorems for A-pseudoconvex algebras were given in [7]:

Theorem 1.3 (G–K–Ż thm. for complex A-pseudoconvex algebras) Let A be a complex complete A-pseudoconvex algebra with unit. Let T : A → C be a continuous linear functional such that

T a ∈ σ(a) (a ∈ A).

Theorem 1.4 (G–K–Ż thm. for real A-pseudoconvex algebras) Let A be a real complete A-pseudoconvex algebra with unit. Let T : A → C be a continuous R-linear functional such that

T 11 = 1,

(T a)²+ (T b)²∈ σ(a²+ b²) (ab = ba, a, b ∈ A).

A characterization for not linear functionals in complex Banach algebras was given by Kowalski–Słodkowski ([4]):

Theorem 1.5 Let A be a complex Banach algebra. Let T : A → C be a functional (not necessary linear) such that

T (0) = 0, (i)

T (a) − T (b) ∈ σ(a − b) (a, b ∈ A).

(ii)

Then T is linear and multiplicative.

From the proof given by Kowalski and Słodkowski in [4] it follows that The- orem 1.5 can be reformulated in the following way:

Theorem 1.6 Let A be a complex Banach algebra. Let T : A → C be a functional such that

T (a) ∈ σ(a) (a ∈ A).

Then the following conditions are equivalent:

(1) T is additive.

(2) Real differentials of T are constant.

(3) T is linear and multiplicative.

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In the paper we will prove a version of Theorem 1.6 for real Banach algebras (Thm. 3.6) and for real (Thm. 3.8) and complex (Thm. 4.3) A-pseudoconvex algebras.

2. Definitions and notation. In this paper, all algebras are real or complex algebras with the unit, denoted by 11.

Let A be a real algebra (resp. vector space). Its complexification is the (complex) algebra (resp. vector space) A_C= A + iA = {a + ib : a, b ∈ A} (with the algebraic operations defined in an obvious way).

For every complex algebra (resp. vector space) A we denote by A_R the algebra (resp. vector space) A with the scalar multiplication restricted to the field of the reals.

The symbol inv(A) will denote the set of all invertible elements of the algebra A.

In a complex algebra A the spectrum of an element a ∈ A is defined to be the set σ(a) = {λ ∈ C : λ11 − a 6∈ inv(A)}. In a real algebra A the spectrum of an element a ∈ A is the set σ(a) = {α + iβ ∈ C : (α11 − a)²+ β²11 6∈ inv(A)}. We have (see [1]

or [6]) σ_A(a) = σ_A

C(a) = {λ ∈ C : λ11 − a 6∈ inv(AC)}.

A topological (resp. semitopological) algebra is a topological vector space equipped with an associative jointly (resp. separately) continuous multiplication.

Let A be a real semitopological algebra. If Φ(A) is a basis of its neighbourhoods of zero, then Φ(A_C) = {U + iU : U ∈ Φ(A)} is a basis of neighbourhoods of zero in the algebra A_C. The algebra A_C with this topology is the complex semitopological algebra. If A is a topological algebra, then A_C is also a topological algebra. If A is complete, then A_Cis complete as well.

A semitopological algebra A is called A-pseudoconvex if its topology is given by means of a family {p_α} of k_α-homogenous seminorms (k_α ∈ (0, 1] and can be different for every α) such that, for every x and α, there exist positive numbers A(x, α) and B(x, α) (depending on x and α) such that p_α(xy) ≤ A(x, α)p_α(y) and p_α(yx) ≤ B(x, α)p_α(y) for all y ∈ A.

In a real or complex algebra A a function T : A → C is called a complex functional. If T is also linear (R-linear if A is real), it is called a complex linear functional.

If, in addition, T is nonzero and multiplicative, then it is called a complex linear- multiplicative functional. In a real or complex algebra A the set of all complex continuous linear-multiplicative functionals is denoted by M(A).

We will sometimes assume that in an algebra B:

M(B) is not empty, (S1)

\

f ∈M(B)

ker f = {0}.

(S2)

For example, if B is a commutative semisimple Banach algebra, then conditions (S1) and (S2) hold true.

Let A and B be complex topological vector spaces. If T : A → B is a map such that for every a, x ∈ A there exists

lim

C3z→0

T (a + zx) − T (a)

z ,

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then we say that T is entire (holomorphic on A).

Let A and B be complex or real topological vector spaces. If T : A → B is a map such that for some a, x ∈ A there exists

lim

R3r→0

T (a + rx) − T (a)

r ,

then we denote it by T_x⁰(a) and call a real differential at the point a and the direction x.

3. Real algebras.

Lemma 3.1 Let X be a real or complex topological vector space. If a functional T : X → C is additive and continuous, then it is R-linear.

Proof Straightforward.

Lemma 3.2 Let A and B be real algebras. Let T : A → BC be a map such that T 11 = 11. Let eT : A_C→ B_C be a map defined by

T (a + ib) := T (a) + iT (b)e (a + ib ∈ A_C).

Then:

(1) T is R-linear if and only if eT is C-linear.

(2) T is R-linear and multiplicative if and only if eT is C-linear and multiplicative.

Proof Straightforward.

Lemma 3.3 Let A be a complex topological vector space and let B be a real topological vector space. Let T : A → B_C be an entire map and let U, V : A → B be such that T = U + iV . Then for every x, a ∈ A there exist real differentials U_x⁰(a), V_x⁰(a) and

U_x⁰(a) = V_ix⁰(a), V_x⁰(a) = −U_ix⁰ (a).

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Proof T is entire, and so for every x, a ∈ A we have lim

C3z→0

T (a + zx) − T (a)

z = lim

R3α→0

T (a + αx) − T (a)

α =

= lim

R3α→0

U (a + αx) + iV (a + αx) − U (a) − iV (a)

α =

= lim

R3α→0

U (a + αx) − U (a)

α + iV (a + αx) − V (a) α

=

= lim

R3α→0

U (a + αx) − U (a)

α + i lim

R3α→0

V (a + αx) − V (a)

α =

= U_x⁰(a) + iV_x⁰(a).

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Similarly lim

C3z→0

T (a + zx) − T (a)

z = lim

R3α→0

T (a + αix) − T (a)

αi = V_ix⁰(a) − iU_ix⁰ (a).

Hence for every x, a ∈ A we have

U_x⁰(a) + iV_x⁰(a) = V_ix⁰(a) − iU_ix⁰ (a),

and therefore (1) holds true.

Lemma 3.4 Let A and B be real topological vector spaces. Let T : A → BC be a map and let eT : A_C→ B_C be a map defined by

T (a + ib) := T (a) + iT (b).e If eT is an entire map, then for every direction x ∈ A

T_x⁰(a) = c ∈ B_C (a ∈ A) (real differentials of T are constant).

Proof Let U, V : A → B and eU , eV : A_C → B be such that T = U + iV and T = ee U + i eV . For every a + ib ∈ A_Cwe have

U (a + ib) + i ee V (a + ib) = eT (a + ib) = T (a) + iT (b) =

= U (a) + iV (a) + i(U (b) + iV (b)) =

= U (a) − V (b) + i(V (a) + U (b)), and so

U (a + ib) = U (a) − V (b),e V (a + ib) = V (a) + U (b).e Hence for every x + iy, a + ib ∈ A_C we have

Ue_x+iy⁰ (a + ib) = lim

R3r→0

U (a + ib + r(x + iy)) − ee U (a + ib)

r =

= lim

r→0

U (a + rx + i(b + ry)) − ee U (a + ib)

r =

= lim

r→0

U (a + rx) − V (b + ry) − U (a) + V (b)

r =

= lim

r→0

U (a + rx) − U (a)

r −V (b + ry) − V (b) r

=

= U_x⁰(a) − V_y⁰(b).

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Analogously

Ve_x+iy⁰ (a + ib) = V_x⁰(a) + U_y⁰(b), Ue_i(x+iy)⁰ (a + ib) = U_−y⁰ (a) − V_x⁰(b), Ve_i(x+iy)⁰ (a + ib) = V_−y⁰ (a) + U_x⁰(b).

By Lemma 3.3 we know that for every x + iy, a + ib ∈ A_C Ue_x+iy⁰ (a + ib) = eV_i(x+iy)⁰ (a + ib), Ve_x+iy⁰ (a + ib) = − eU_i(x+iy)⁰ (a + ib).

Hence

U_x⁰(a) − V_y⁰(b) = V_−y⁰ (a) + U_x⁰(b), V_x⁰(a) + U_y⁰(b) = − U_−y⁰ (a) − V_x⁰(b) . Since V_−y⁰ (a) = −V_y⁰(a),

U_x⁰(a) − V_y⁰(b) = −V_y⁰(a) + U_x⁰(b), U_y⁰(a) + V_x⁰(b) = V_x⁰(a) + U_y⁰(b).

If we substitute y := x above, and add and substract these formulas, we get U_x⁰(a) = U_x⁰(b) (a, b ∈ A),

V_x⁰(b) = V_x⁰(a) (a, b ∈ A).

Hence for every x ∈ A

U_x⁰(a) = c₁∈ B (a ∈ A), V_x⁰(a) = c₂∈ B (a ∈ A).

Finally, since T_x⁰(a) = U_x⁰(a) + iV_x⁰(a),

T_x⁰(a) = c ∈ B_C (a ∈ A).

Lemma 3.5 Let A be a real topological vector space. Let T : A → C be a functional with

T (0) = 0 (1)

and all real differentials T_x⁰ constant, i.e. for every x ∈ A T_x⁰(a) = c (a ∈ A).

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Then T is R-linear.

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Proof We follow an argument used in the proof of Lemma 2.1 in [4].

For every a, b ∈ A we define a function f_a,b: R → C by f_a,b(x) := T (xa + b) (x ∈ R).

Then

f_a,b⁰ (x) = lim

r→0

f_a,b(x + r) − f_a,b(x)

r =

= lim

r→0

T ((x + r)a + b) − T (xa + b)

r =

= lim

r→0

T (xa + b + ra) − T (xa + b)

r =

= T_a⁰(xa + b).

Hence

f_a,b⁰ (x) = c ∈ C (x ∈ R).

Since f_a,b⁰ is constant, f_a,b is affine, i.e.

f_a,b(x) = f_a,b(1) − f_a,b(0)x + f_a,b(0).

Hence

T (xa + b) = T (a + b) − T (b)x + T (b).

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Substituting b = 0 in (3) and using (1) we obtain T (xa) = xT (a), so T is R-homogeneous.

Substituting in (3) a =¹₂(c − d), b = d, x = 2 we obtain T (c) = T 2 −¹₂(c − d) + d =

= T (xa + b) =

= T (¹₂(c − d) + d) − T (d)2 + T (d) =

= T (c + d) − T (d),

so T is additive.

Theorem 3.6 Let A be a real Banach algebra. Let T : A → C be a functional such that

T 11 = 1, (i)

(T a)²+ (T b)²∈ σ(a²+ b²) (ab = ba, a, b ∈ A).

(ii)

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(1) T is additive.

(3) eT is entire.

(4) T is R-linear and multiplicative.

Proof (4) ⇒ (3) T is linear, and then by Lemma 3.2, eT is C-linear, and so entire.

(3) ⇒ (2) Follows by Lemma 3.4.

(2) ⇒ (1) From (ii) we have

(T (0))²+ (T (0))²∈ σ(0) = {0}, so T (0) = 0. To conclude we use Lemma 3.5.

(1) ⇒ (4) From (ii), for b = 0 we have (T a)²∈ σ(a²). Then

|T a|²≤ |σ(a²)| ≤ ka²k ≤ kak².

Hence T is continuous. The result follows by Lemma 3.1 and the G–K–Ż theorem.

Corollary 3.7 Let A and B be real Banach algebras, where B is commutative and semisimple. Let T : A → B be a map such that

T 11 = 11, (i)

σ (T a)²+ (T b)² ⊂ σ(a²+ b²) (ab = ba, a, b ∈ A).

(ii)

(1) T is additive.

(3) eT is entire.

Proof (4) ⇒ (3) If T is R-linear, then by Lemma 3.2, eT is C-linear, and so entire.

(3) ⇒ (2) Follows by Lemma 3.4.

(2) ⇒ (1) Let f ∈ M(B). Then we have

f ◦ T (11) = f (T 11) = f (11) = 1, f ◦ T (a)2

+ f ◦ T (b)2

= f (T a)²+ (T b)² ⊂ σ (T a)²+ (T b)² ⊂ σ(a²+ b²).

Therefore f ◦ T satisfies (i) and (ii) in Theorem 3.6.

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Since

(f ◦ T )⁰_x(a) = lim

r→0

(f ◦ T )(a + rx) − (f ◦ T )(a)

r =

= f

lim

r→0

T (a + rx) − T a r

=

= f (T_x⁰(a)) = f (c) = const, condition (2) in Theorem 3.6 holds true.

Hence, we can use Theorem 3.6 and we obtain that f ◦ T is additive (for every f ∈ M(B)). Then f (T (x + y) − T (x) − T (y)) = 0 for every f ∈ M(B) and by (S1), (S2), T is additive.

(1) ⇒ (4) If T is additive and satisfies (i) and (ii), then f ◦ T is additive and satisfies conditions (i) and (ii) of Theorem 3.6. Hence f ◦T is linear and multiplicative

and by (S1), (S2) T is linear and multiplicative.

Since Lemmas 3.2, 3.3, 3.4, 3.5 are also true in A-pseudoconvex algebras, the following theorem is a counterpart of Theorem 3.6 in case algebra A is A-pseudoconvex and functional T is countinuous.

Theorem 3.8 Let A be a real complete A-pseudoconvex algebra. Let T : A → C be a continuous functional such that

T 11 = 1, (i)

(T a)²+ (T b)²∈ σ(a²+ b²) (ab = ba, a, b ∈ A).

(ii)

(1) T is additive.

(3) eT is entire.

Proof Similar to the proof of Theorem 3.6.

Corollary 3.9 Let A and B be real complete A-pseudoconvex algebras, where B satisfies (S1) and (S2). Let T : A → B be a map such that

T 11 = 11, (i)

σ[(T a)²+ (T b)²] ⊂ σ(a²+ b²) (ab = ba, a, b ∈ A).

(ii)

(1) T is additive.

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(3) eT is entire.

Proof Similar to the proof of Corollary 3.7.

4. Complex algebras. In [4] we can find a proof of the following lemma:

Lemma 4.1 Let A be a complex Banach algebra. Let T : A → C be a complex R- linear functional such that

T a ∈ σ(a) (a ∈ A).

Then T is C-linear.

Reasoning in the same way as in the proof of this lemma in [4], but with G–K–Ż theorem for A-pseudoconvex algebras instead of G–K–Ż theorem for Ba- nach algebras, we obtain a version of the above lemma for A-pseudoconvex algebras:

Lemma 4.2 Let A be a complex complete A-pseudoconvex algebra. Let T : A → C be a complex continuous R-linear functional such that

T a ∈ σ(a) (a ∈ A).

Then T is C-linear.

Now, we can prove versions of Theorem 3.8 and Corollary 3.9 in case of complex algebras.

Theorem 4.3 Let A be a complex complete A-pseudoconvex algebra. Let T : A → C be a continuous functional such that

T (a) ∈ σ(a) (a ∈ A).

(1) T is additive.

Proof (3) ⇒ (2) Straightforward.

(2) ⇒ (1) Follows by Lemm 3.5 for the algebra A_R(the algebra A as real algebra).

(1) ⇒ (3) T is additive and continuous, hence by Lemma 3.1 it is R-linear, and by Lemma 5.2, C-linear. The result follows by G–K–Ż theorem (for complex

A-pseudoconvex algebras).

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Corollary 4.4 Let A and B be complex complete A-pseudoconvex algebras, where B satisfies (S1) and (S2). Let T : A → B be a continuous map such that

σ(T a) ⊂ σ(a) (a ∈ A).

(1) T is additive.

Proof Similar to the proof of Corollary 3.7.

5. Final remarks. If T is a complex C-linear functional on a complex algebra A then the following two conditions are equivalent:

T 11 = 1, T a 6= 0 for a ∈ inv A, (A)

T a ∈ σ(a) for a ∈ A.

(B)

If we assume only R-linearity of T then we get:

Lemma 5.1 Let A be a complex Banach algebra and let T : A → C be an R-linear functional.

The following three conditions

T 11 = 1, (1)

T (i11) = i, (2)

T x 6= 0 for x ∈ inv A (3)

hold true if and only if

T x ∈ σ(x) for x ∈ A.

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Proof Let (1), (2) and (3) hold true. Then by (1) and (2) every x ∈ A can be written as

x = (T x)11 − y, where T y = 0.

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Using (5) and (3) we obtain

(T x)11 − x = y 6∈ inv A, hence T x ∈ σ(x).

Now, let (4) be satisfied. Then

(6) (T x)11 − x 6∈ inv(A).

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This implies, that if T x = 0, then x 6∈ inv A, and so we get (3). From (6) we also have, that

(T 11 − 1)11 = (T 11)11 − 11 6∈ inv(A), (T (i11) − i)11 = T (i11)11 − i11 6∈ inv(A).

Hence T 11 − 1 = 0, T (i11) − i = 0, so (1) and (2) are true. Thus we see that in case of R-linear functionals we have to use the conditions (1)–(3) instead of (A).

For example, we can formulate Lemma 4.2 as follows:

Lemma 5.2 Let A be a complex complete A-pseudoconvex algebra. Let T : A → C be a continuous R-linear functional such that

T 11 = 1, T (i11) = i T a 6= 0 for a ∈ inv(A).

Then T is C-linear.

The next examples show that in Lemma 5.2 all three conditions are necessary (even in case of Banach algebras).

Example 5.3 There exists a complex commutative Banach algebra A and a functional T : A → C which is continuous, R-linear and satisfies

T 11 = 1, T (i11) = i, but is not C-linear.

Take the real commutative Banach algebra R²with coordinate operations, and the unit 11 = (1, 1). We define the following real functionals T₁, T₂: R²→ R,

T₁(a, b) := a (a, b) ∈ R² , T₂(a, b) := b (a, b) ∈ R² . T₁and T₂are R-linear and continuous.

We define an algebra A as complexification R²_C of the algebra R². Then A is a complex commutative Banach algebra, where

11 = [(1, 1), (0, 0)], i11 = [(0, 0), (1, 1)].

We define a complex functional T : A → C by

T [a, b] := T₁a + iT₂b = T₁(a₁, a₂) + iT₂(b₁, b₂) = a₁+ ib₂, where a, b ∈ R²; a = (a₁, a₂), b = (b₁, b₂).

It is easily seen that T is continuous, additive, and T 11 = 1, T (i11) = i. Moreover T is R-homogeneous but it is not C-homogeneous.

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Example 5.4 There exists a complex commutative Banach algebra A and a complex functional T : A → C such that it is continuous, R-linear, and satisfies conditions

T 11 = 1, T a 6= 0 for a ∈ inv A, but it is not C-linear.

We define A := C and T : C → C by

T (a + ib) := a + i2b (a + ib ∈ C; a, b ∈ R).

T is continuous, additive, and T 1 = 1. Moreover, if T (a + ib) = 0 then a + ib 6∈ inv C.

The functional T is R-homogenous but it is not C-homogeneous.

If we put T (i11) = i instead of T 11 = 1 and define T as T (a + ib) := 2a + ib, then we also have an example of a functional which is not C-homogeneous.

References

[1] F. F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, New York 1973.

[2] A. M. Gleason, A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171–172.

[3] J.-P. Kahane, W. Żelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339–343.

[4] S. Kowalski, Z. Słodkowski, A characterization of multiplicative linear functionals in Banach algebras, Studia Math. 67 (1980), 215–223.

[5] S. H. Kulkarni, Gleason–Kahane–Żelazko theorem for real Banach algebras, Jour. Math. Phy.

Sci. 18 (S) (1984), 19–28.

[6] S. H. Kulkarni, B. V. Limaye, Real Function Algebras, Marcel Dekker, Inc., New York 1992.

[7] M. Łuczak, On the Gleason–Kahane–Żelazko theorem, Comment. Math. Prace Mat. 44(2) (2004), 245–253.

[8] W. Żelazko, A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83–85.

Maciej Łuczak

Koszalin University of Technology

Department of Civil and Environmental Engineering, Division of Mathematics and Faculty of Mathematics and Computer Science, Adam Mickiewicz University Śniadeckich 2, 75-453 Koszalin, Poland and

Umultowska 87, 61–614 Poznań, Poland E-mail: mluczak@amu.edu.pl

(Received: 06.05.2006)