Lobachevskii Journal of Mathematics

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Vol. 18, 2005, 127–130 \begin{matrix}c \\ ◯ \end{matrix} M.I Karahanyan

M.I Karahanyan

ON THE ABSTRACT THEOREM OF PICARD

(submitted by D. Kh. Mushtari)

Abstract. Let A be a complex Banach algebra with unit. It was shown by Williams [1] that elements a, b ∈ A commute if and only if sup_{λ∈C} ∥ exp(λb)a exp(−λb)∥ < ∞. This result allows us to obtain an analog of the von Neumann-Fuglede-Putnam theorem in case of normal elements in a complex Banach algebra. In the present paper the results by Williams [1] and Khasbardar et Thakare [2] are refined by using [3, 4, 5]. An abstract version of Picard theorem is obtained in this context.

The set P (A) of all normalized states φ : A → C [6, 7] on a complex Banach algebra A with unit is a weak-star compact convex subset in the space A^{⋆} (conjugate space). The set V (a) = {φ(a) : φ ∈ P (A)} is called (algebraic) numerical range of a ∈ A. In particular, if A is an algebra B(H) of all bounded linear operators defined in the complex Hilbert space H, and T ∈ B(H), then its (algebraic) numerical range V (T) coincides with the closure of the general numerical range of the operator T [8]. An element h ∈ A is called Hermitian if V (h) ⊂ R, were R is the field of reals. This condition is equivalent to ∥ exp(ith)∥ = 1 for all t ∈ R. We also note that an element h ∈ A is called quasi-Hermitian[5] if ∥ exp(ith)∥ = o(| t |^{\frac{1}{2}} ) when | t |→ ∞, t ∈ R. This condition provides sp(h) ⊂ R, where sp(h) is the spectrum ofh, but the numerical range V (h) may be not contained in the real axis. It is easy to see that for any a ∈ A there holds sp(a) ⊂ V (a). If an element a ∈ A admits representation a = h + ik, where h, k are Hermitian elements, then a is called Hermitian decomposable, and a^{+} = h − ik is called the Hermitian conjugate of a. When [h, k] = hk − kh = 0, then the element a = h + ik is called normal element, if h, k are quasi-Hermitian elements and [h, k] = 0, then the element a is called quasi-normal.

Let J be a closed bi-ideal in A. Then the factor-algebra A/J is a Banach algebra with respect to the factor-norm ∥| • ∥|, and ã = a + J = π_{J}(a) ∈ A/J, where π_{J} : A → A/J is a canonical homomorphism generated by J -ideal.

Let {a_{n}}_{0}^{∞}, {b_{n}}_{0}^{∞} be sequences of elements in Banach algebra A such

that lim_{n→∞} ∥a_{n}∥^{\frac{1}{n}} = ρ_{a} < ∞, lim_{n→∞} ∥b_{n}∥^{\frac{1}{n}} = ρ_{b} < ∞, and let c ∈ A.

Then the function f(λ) = A(λ)cB(λ) = ∑_{0}^{∞} \frac{c_{n}λ^{n}}{n!} , where A(λ) = ∑_{0}^{∞} \frac{a_{n}λ^{n}}{n!} ,

B(λ) = ∑_{0}^{∞} \frac{b_{n}λ^{n}}{n!} , is an A-valued entire function of exponential type σ_{f} ≤

ρ_{a} + ρ_{b}, and c_{n} = ∑_{p=0}^{n} < a_{p}cb_{n−p} > is a generalized commutator of the

corresponding elements. If the elements a, b, c ∈ A, then one writes ca = bc(modJ) if ca − bc ∈ J.

Theorem 1. Let A be a complex Banach algebra with unit, J be a closed bi-ideal in A, and \{a_{n}\}_{0}^{∞}, \{b_{n}\}_{0}^{∞} ∈ A be such that lim_{n→∞} ∥a_{n}∥^{\frac{1}{n}} < ∞,

lim_{n→∞} ∥b_{n}∥^{\frac{1}{n}} < ∞. Then c_{n} = ∑_{p=0}^{n} < a_{p}cb_{n−p} >∈ J for all n ≥ 1 if and

only if ∥|f̃(λ)∥| = o(|λ|) for |λ| → ∞, λ ∈ C. Proof: Let π_{J} : A → A/J be a canonical homomorphism generated by J. Since, c_{n} ∈ J, c̃_{n} = ˜0. Hence f̃(λ) = ∑_{0}^{∞} \frac{c̃_{n}λ^{n}}{n!} = c̃_{0}, i.e., ∥|f̃(λ)∥| = ∥|c̃_{0}∥|,

and so \frac{∥|f̃(λ)∥|}{|λ|} → 0, |λ| → ∞.

Conversely assume that ∥|f̃(λ)∥| = o(|λ|) for |λ| → ∞, λ ∈ C. Then for a A/J-valued entire function f̃(λ) = ∑_{0}^{∞} \frac{c̃_{n}λ^{n}}{n!} , there holds f̃(λ) ≡ c_{0}

by the Liouville theorem. Hence, for all n ≥ 1 one has c̃_{n} = 0, and therefore c_{n} ∈ J. This completes the proof.

In particular, when a_{n} = a^{n} and b_{n} = b^{n}, where a, b ∈ A the following result holds:

Theorem 2. Let A be a complex Banach algebra with unit, J be a closed bi-ideal in A, and a, b, c ∈ A. Then ca = bc(modJ) if and only if ∥| exp(λb̃)c̃ exp(−λã)∥| = o(|λ|), |λ| → ∞, λ ∈ C. The proof is similar to that of Theorem 1.

Combination of Theorem 2 with the generalized theorem of von Neumann-Fuglede-Putnam [3, 4] gives:

Corollary 3. Let a, b ∈ A be quasinormal elements, c ∈ A, and J ∈ A be a closed bi-ideal. If ∥| exp(λb̃)c̃ exp(−λã)∥| = o(|λ|), |λ| → ∞, λ ∈ C, then a^{+}c = cb^{+}(modJ)[9].

Corollary 4. Let A be a complex Banach algebra with unit, a, b ∈ A be quasinormal elements and c ∈ A. If ∥ exp(λa)c exp(−λb)∥ = o(|λ|), |λ| → ∞, λ ∈ C, then a^{+}c = c b^{+}.

Let D : A → A be a continuous A-derivation (i.e., D ∈ BL(A) and D(ab) = (Da)b + a(Db), where a, b ∈ A). Then in the factor-algebra A/J the A/J-derivation D_{J} : A/J → A/J is defined by the formula D_{J}(ã) = π_{J}(Da) = D̃a. It is obvious that D_{J} ∈ BL(A/J). We denote aut(A) the group of all automorphisms of the algebra A. Then, exp(λD) ∈ aut(A) for each λ ∈ C, and exp(λD_{J}) ∈ aut(A/J).

Theorem 5. Let A be a complex Banach algebra with unit, J be a closed bi-ideal in A, and D : A → A be a continuous A-derivation. Then Da ∈ J, where a ∈ A, if and only if ∥|(exp λD_{J})(ã)∥| = o(|λ|), |λ| → ∞, λ ∈ C.

The proof of this theorem is similar to those of Theorems 1 and 2.

Corollary 6. Let A be a complex Banach algebra with unit and D : A → A be a continuous A-derivation. Then a ∈ ker(D) if and only if ∥(exp λD)(a)∥ = o(|λ|), |λ| → ∞, λ ∈ C.

The following result is a generalization of the Picard’s theorem for entire functions. Theorem 7. Let A be a complex Banach algebra with unit, J be a closed bi-ideal in A and D : A → A be a continuous A-derivation. If for the element a ∈ A one has Da ∉ J, then ⋃_{λ∈C} V ((exp λD_{J})(ã)) = C.

Proof. Let φ ∈ P (A/J). We consider an entire function f_{φ}(λ) = φ(exp(λD_{J})(ã)). Then the range of the f_{φ} is contained in the set U_{J} = ⋃_{λ∈C} V ((exp λD_{J})(ã)). Let us assume that C \ U_{J} contains at least two

points. Then according to the Picard’s theorem for the entire functions one has f_{φ}(λ) ≡ const Hence, for all natural n ≥ 1 there holds φ(D_{J}^{n}(ã)) = 0, and in particular, D_{J}(ã) = 0, i.e., Da ∈ J. However, this contradicts the condition of the theorem. So, C \ U_{J} contains at most one point. It remains to show that U_{J} = C. Since for any fixed λ ∈ C the operator exp(λD_{J}) belongs to aut(A/J), for any element ã ∈ A/J one has sp(ã) = sp((exp λD_{J})(ã)).

130 M.I KARAHANYAN

Let ζ_{0} ∈ sp(ã) and ζ ∈ C be a complex number. Then on the line passing through the points ζ_{0} and ζ there exists a point ζ_{1} such that ζ_{1} ∈ V ((exp λD_{J})(ã)) for some λ ∈ C. However, [ζ_{0}, ζ_{1}] ⊂ V ((exp λD_{J})(ã)), and since V ((exp λD_{J})(ã)) is a convex set, we have ζ ∈ V ((exp λD_{J})(ã)), and hence, U_{J} = C. This completes the proof.

In the case when the ideal J equals {0} one has:

Corollary 8. Let A be a complex Banach algebra with unit and D : A → A is a continuous A-derivation. If the element a ∉ ker(D), then

⋃_{λ∈C} V ((exp λD)(a)) = C.

Corollary 9. Let A be a complex Banach algebra with unit and a, b, c ∈ A are such elements that ac ≠ cb (modJ), where J is a closed bi-ideal in A. Then ⋃_{λ∈C} V (exp(λã)c̃exp(−λb̃)) = C.

References

[1] J.P. Williams, ”On commutativity and numerical range in Banach algebras.”, J. Functional Analysis, 10, pp.326-329, 1972.

[2] S.K. Khasbardar, N.K. Thakare, ”Commutativity in a Banach Algebra,” Bolletino U.M.I. (5), 15-A, pp. 581-584, 1978

[3] E.A. Gorin, M.I. Karahanyan, ”Asymptotic variant of Fuglede-Putnam theorem on commutators for Banach algebra elements” (in Russian), Math. Zametki 22, N22, pp. 179-188, 1977.

[4] M.I. Karahanyan, ”Asymptotic properties of commutators of Banach algebras elements.”, Izv. Akad. Nauk Armenii, Matematika, 19, N6, pp. 405-421, 1978.

[5] M.I. Karahanyan, ”Asymptotic properties of commutators.”, Izv. Akad. Nauk Armenii, Matematika, 29, N1, pp. 43-49, 1994.

[6] J.F. Bonsall and J. Dunkan, ”Complete Normed Algebras.”, Springer-Verlag, 1973.

[7] E.A. Gorin, ”Estimation of a partial involution in a Banach algebra.”, Russian Journal of Mathematical Physics, 5, N1, pp. 117-118, 1997.

[8] P.R. Halmosh, ”A Hilbert Space Problem Book.”, Princeton, New-Jersey, 1967.

[9] I.M. Karahanyan, M.I. Karahanyan, ”On the commutativity in Banach algebra.”, Izv. Akad. Nauk Armenii, Matematika, 34, N4, pp. 76-81, 1999.

Yerevan State University, Department of Mathematics

Received January 31, 2005; revised version July 19, 2005