Norm controlled inversion in Quasi-Banach algebras
by
Anders Dahlner
Lund University, Sweden
Coauthors: Alexandru Aleman (Lund University)

Let A be a semi-simple commutative p-normed quasi-Banach algebra, 0 < p <= 1, with unit element e and Gelfand transform x --> [^x]. Consider for the extremal problems:

K\nu = sup { ||(e-x)-1||p : x in A,  ||x||p <= 1,   max | ^ x   | p <= \nu},     0 < \nu < 1,

C\delta = sup { ||x-1||p : x in A,  ||x||p <= 1,   min | ^ x   |p >= \delta} ,     0 < \delta < 1.

We ask when these these suprema are bounded in terms of \nu and \delta, respectively.

(i) Finiteness of the quantity K_\nu is described in terms of a ``uniform spe ctral radius'' of the algebra, r_N(A), N = 1, 2, 3, ... .

Let \delta₁(A) = inf{\delta: 0 < \delta < 1,  C_\delta < \infty}, and pu t r(A) = lim_N r_N(A).^M

(ii) We prove that

r(A) 1 + r(A) <= \delta1(A).

(iii) For almost any commutative quasi-Banach algebra A with compact Gelfand  transform we prove that

r(A)=0,     if and only if,     \delta1(A)=0.

(iv) For a large class of weights \omega we determine upper bounds of r_N(A) , where A=l^p(\omega), 0 < p < 1, is a Beurling type algebra.

In the case of Banach algebras such problems has earlier been considered by J-E . Björk, O. El-Fallah, A. Ezzaaraoui, N.K Nikolski, H. S. Shapiro, A. Olofsson and Zarrabi M.

Date received: June 14, 2002