On the left and right joint spectra in Banach algebras
by
Andrzej Sołtysiak
Poznan University
Coauthors: Che-Kao Fong (Carleton University, Ottawa)

Let A be a complex Banach algebra with the unit 1_A. The symbol rad A denotes the Jacobson radical of the algebra A. If a₁, ..., a_n are elements of A, then their left joint spectrum , denoted by \sigma_l(a₁, ... , a_n), is defined as follows:

\sigmal(a1, ... , an)={(\lambda1, ... , \lambdan) in Cn\colon  n å j=1  A(aj-\lambdaj) =/= A}.

(Here a_j-\lambda_j stands for a_j-\lambda_j1_A.) The right joint spectrum \sigma_r(a₁, ... , a_n) is defined similarly.

It is clear from the definition that if the algebra A/rad A is commutative, then the left and right spectra of an arbitrary n-tuple of elements in A are equal. We show that the converse is also true.

Theorem. If \sigma_l(a₁, ... , a_n) subset \sigma_r(a₁, ... , a_n) for an arbitrary n-tuple (a₁, ... , a_n) of elements of the Banach algebra A, then A/rad A is commutative.

We also recall some questions connected with the above theorem which remain open.

This is a joint work with Che-Kao Fong from Carleton University in Ottawa, published in Studia Math. 97 (1990), 151-156.

Date received: May 30, 2000