Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 18, 2005, 127 – 130

© M.I Karahanyan

ABSTRACT. Let $A$ be a complex Banach algebra with unit. It was shown by Williams [1] that elements $a,b\in A$ commute if and only if $\underset{\lambda \in C}{sup}\parallel exp\left(\lambda b\right)aexp\left(-\lambda b\right)\parallel <\infty$. 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 $P\left(A\right)$ of all normalized states $\varphi :A\to C$ [6, 7] on a complex Banach algebra $A$ with unit is a weak-star compact convex subset in the space $A^{\star }$ (conjugate space). The set $V\left(a\right)=\left\{\varphi \left(a\right):\varphi \in P\left(A\right)\right\}$ is called (algebraic) numerical range of $a\in A$. In particular, if $A$ is an algebra $B\left(H\right)$ of all bounded linear operators defined in the complex Hilbert space $H$, and $T\in B\left(H\right)$, then its (algebraic) numerical range $V\left(T\right)$ coincides with the closure of the general numerical range of the operator $T$ [8]. An element $h\in A$ is called Hermitian if $V\left(h\right)\subset \mathbf{\text{R}}$, were $\mathbf{\text{R}}$ is the field of reals. This condition is equivalent to $\parallel exp\left(ith\right)\parallel =1$ for all $t\in \mathbf{\text{R}}$. We also note that an element $h\in A$ is called quasi-Hermitian[5] if $\parallel exp\left(ith\right)\parallel =o\left(\mid t{\mid }^{\frac{1}{2}}\right)$ when $\mid t\mid \to \infty$, $t\in \mathbf{\text{R}}$. This condition provides $sp\left(h\right)\subset \mathbf{\text{R}}$, where $sp\left(h\right)$ is the spectrum of$h$, but the numerical range $V\left(h\right)$ may be not contained in the real axis. It is easy to see that for any $a\in A$ there holds $sp\left(a\right)\subset V\left(a\right)$. If an element $a\in A$ 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 $\left[h,k\right]=hk-kh=0$, then the element $a=h+ik$ is called normal element, if $h,k$ are quasi-Hermitian elements and $\left[h,k\right]=0$, 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 $\parallel \mid \bullet \parallel \mid$, and $ã=a+J={\pi }_J\left(a\right)\in A∕J$, where ${\pi }_J:A\to A∕J$ is a canonical homomorphism generated by $J$ -ideal.

Let ${\left\{a_n\right\}}_0^{\infty }$, ${\left\{b_n\right\}}_0^{\infty }$ be sequences of elements in Banach algebra $A$ such that $\underset{n\to \infty }{\overset{}{lim}}\parallel a_n{\parallel }^{\frac{1}{n}}={\rho }_a<\infty$, $\underset{n\to \infty }{\overset{}{lim}}\parallel b_n{\parallel }^{\frac{1}{n}}={\rho }_b<\infty$, and let $c\in A$. Then the function $f\left(\lambda \right)=A\left(\lambda \right)cB\left(\lambda \right)=\sum _0^{\infty }\frac{c_n{\lambda }^n}{n!}$, where $A\left(\lambda \right)=\sum _0^{\infty }\frac{a_n{\lambda }^n}{n!}$, $B\left(\lambda \right)=\sum _0^{\infty }\frac{b_n{\lambda }^n}{n!}$, is an $A$-valued entire function of exponential type ${\sigma }_f\le {\rho }_a+{\rho }_b$, and $c_n=\sum _{p=0}^n<a_p{cb}_{n-p}>$ is a generalized commutator of the corresponding elements. If the elements $a,b,c\in A$, then one writes $ca=bc\left(modJ\right)$ if $ca-bc\in J$.

Theorem 1. Let $A$ be a complex Banach algebra with unit, $J$ be a closed bi-ideal in $A$, and ${\left\{a_n\right\}}_0^{\infty },{\left\{b_n\right\}}_0^{\infty }\in A$ be such that $\underset{n\to \infty }{\overset{}{lim}}\parallel a_n{\parallel }^{\frac{1}{n}}<\infty$, $\underset{n\to \infty }{\overset{}{lim}}\parallel b_n{\parallel }^{\frac{1}{n}}<\infty$. Then $c_n=\sum _{p=0}^n<a_p{cb}_{n-p}>\in J$ for all $n\ge 1$ if and only if $\parallel \mid \tilde{f}\left(\lambda \right)\parallel \mid =o\left(\mid \lambda \mid \right)$ for $\mid \lambda \mid \to \infty$, $\lambda \in C$.

Proof: Let ${\pi }_J:A\to A∕J$ be a canonical homomorphism generated by $J$. Since, $c_n\in J$, ${\tilde{c}}_n=\tilde{0}$. Hence $\tilde{f}\left(\lambda \right)=\sum _0^{\infty }\frac{{\tilde{c}}_n{\lambda }^n}{n!}={\tilde{c}}_0$, i.e., $\parallel \mid \tilde{f}\left(\lambda \right)\parallel \mid =\parallel \mid {\tilde{c}}_0\parallel \mid$, and so $\frac{\parallel \mid \tilde{f}\left(\lambda \right)\parallel \mid }{\mid \lambda \mid }\to 0$, $\mid \lambda \mid \to \infty$.

Conversely assume that $\parallel \mid \tilde{f}\left(\lambda \right)\parallel \mid =o\left(\mid \lambda \mid \right)$ for $\mid \lambda \mid \to \infty$, $\lambda \in C$. Then for a $A∕J$-valued entire function $\tilde{f}\left(\lambda \right)=\sum _0^{\infty }\frac{{\tilde{c}}_n{\lambda }^n}{n!}$, there holds $\tilde{f}\left(\lambda \right)\equiv c_0$ by the Liouville theorem. Hence, for all $n\ge 1$ one has ${\tilde{c}}_n=0$, and therefore $c_n\in J$. This completes the proof.

In particular, when $a_n=a^n$ and $b_n=b^n$, where $a,b\in A$ the following result holds:

Theorem 2. Let $A$ be a complex Banach algebra with unit, $J$ be a closed bi-ideal in $A$, and $a,b,c\in A$. Then $ca=bc\left(modJ\right)$ if and only if $\parallel \mid exp\left(\lambda \tilde{b}\right)\tilde{c}exp\left(-\lambda \tilde{a}\right)\parallel \mid =o\left(\mid \lambda \mid \right)$, $\mid \lambda \mid \to \infty$, $\lambda \in C$.

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 $a,b\in A$ be quasinormal elements, $c\in A$, and $J\in A$ be a closed bi-ideal. If $\parallel \mid exp\left(\lambda \tilde{b}\right)\tilde{c}exp\left(-\lambda \tilde{a}\right)\parallel \mid =o\left(\mid \lambda \mid \right)$, $\mid \lambda \mid \to \infty$, $\lambda \in C$, then $a^{+}c=c{b}^{+}\left(modJ\right)$[9].

Corollary 4. Let $A$ be a complex Banach algebra with unit, $a,b\in A$ be quasinormal elements and $c\in A$. If $\parallel exp\left(\lambda a\right)cexp\left(-\lambda b\right)\parallel =o\left(\mid \lambda \mid \right)$, $\mid \lambda \mid \to \infty$, $\lambda \in C$, then $a^{+}c=c{b}^{+}$.

Let $D:A\to A$ be a continuous $A$-derivation (i.e., $D\in BL\left(A\right)$ and $D\left(ab\right)=\left(Da\right)b+a\left(Db\right)$, where $a,b\in A$). Then in the factor-algebra $A∕J$ the $A∕J$-derivation $D_J:A∕J\to A∕J$ is defined by the formula $D_J\left(\tilde{a}\right)={\pi }_J\left(Da\right)=\widetilde{Da}$. It is obvious that $D_J\in BL\left(A∕J\right)$. We denote $aut\left(A\right)$ the group of all automorphisms of the algebra $A$. Then, $exp\left(\lambda D\right)\in aut\left(A\right)$ for each $\lambda \in C$, and $exp\left(\lambda D_J\right)\in aut\left(A∕J\right)$.

Theorem 5. Let $A$ be a complex Banach algebra with unit, $J$ be a closed bi-ideal in $A$, and $D:A\to A$ be a continuous $A$-derivation. Then $Da\in J$, where $a\in A$, if and only if $\parallel \mid \left(exp\lambda D_J\right)\left(\tilde{a}\right)\parallel \mid =o\left(\mid \lambda \mid \right)$, $\mid \lambda \mid \to \infty$, $\lambda \in C$.

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 $D:A\to A$ be a continuous $A$-derivation. Then $a\in ker\left(D\right)$ if and only if $\parallel \left(exp\lambda D\right)\left(a\right)\parallel =o\left(\mid \lambda \mid \right)$, $\mid \lambda \mid \to \infty$, $\lambda \in C$.

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 $D:A\to A$ be a continuous $A$-derivation. If for the element $a\in A$ one has $Da\notin J$, then $\bigcup _{\lambda \in C}V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)=C$.

Proof. Let $\varphi \in P\left(A∕J\right)$. We consider an entire function $f_{\varphi }\left(\lambda \right)=\varphi \left(exp\left(\lambda D_J\right)\left(\tilde{a}\right)\right)$. Then the range of the $f_{\varphi }$ is contained in the set $U_J=\bigcup _{\lambda \in C}V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$. Let us assume that $C\setminus U_J$ contains at least two points. Then according to the Picard’s theorem for the entire functions one has $f_{\varphi }\left(\lambda \right)\equiv const$ Hence, for all natural $n\ge 1$ there holds $\varphi \left(D_J^n\left(\tilde{a}\right)\right)=0$, and in particular, $D_J\left(\tilde{a}\right)=0$, i.e., $Da\in J$. However, this contradicts the condition of the theorem. So, $\mathbf{\text{C}}\setminus U_J$ contains at most one point. It remains to show that $U_J=\mathbf{\text{C}}$. Since for any fixed $\lambda \in C$ the operator $exp\left(\lambda D_J\right)$ belongs to $aut\left(A∕J\right)$, for any element $\tilde{a}\in A∕J$ one has $sp\left(\tilde{a}\right)=sp\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$.

Let ${\zeta }_0\in sp\left(\tilde{a}\right)$ and $\zeta \in C$ be a complex number. Then on the line passing through the points ${\zeta }_0$ and $\zeta$ there exists a point ${\zeta }_1$ such that ${\zeta }_1\in V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$ for some $\lambda \in C$. However, $\left[{\zeta }_0,{\zeta }_1\right]\subset V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$, and since $V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$ is a convex set, we have $\zeta \in V\left(\left(exp\lambda D_J\right)\left(\tilde{a}\right)\right)$, and hence, $U_J=C$. Thise proof completes the proof.

In the case when the ideal $J$ equals $\left\{0\right\}$ one has:

Corollary 8. Let $A$ be a complex Banach algebra with unit and $D:A\to A$ is a continuous $A$-derivation. If the element $a\notin ker\left(D\right)$, then $\bigcup _{\lambda \in C}V\left(\left(exp\lambda D\right)\left(a\right)\right)=C$.

Corollary 9. Let $A$ be a complex Banach algebra with unit and $a,b,c\in A$ are such elements that $ac\ne cb\left(modJ\right)$, where $J$ is a closed bi-ideal in $A$. Then $\bigcup _{\lambda \in C}V\left(exp\left(\lambda ã\right)\tilde{c}exp\left(-\lambda \tilde{b}\right)\right)=C$.

References

YEREVAN STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS