Residual spectrum, surjunctive, group von Neumann algebra, directly finite, amenable group
Appeared as Int. Eq. Op. Th. 67 (2010), no. 1, 95--107.
Important note: an erratum appeared as Int. Eq. Op. Th. 69 (2011), no. 1, 149--150.
Preprint version available at arXiv 0906.2854
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Let X be a Banach space on which a discrete group Γ acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. We show, for instance, that this occurs if X is l2(Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair, and develop some of the basic properties of this construction.
The cases X=lp(Γ) for 1 ≤ p <2 or 2< p< ∞ are more difficult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary.
One of the questions posed in the paper asked, in effect, if every convolution operator on lp(H) which is bounded below must automatically be invertible; here, H denotes the integer Heisenberg group. After the paper was accepted, I learned that the answer is ``yes'', as a corollary of some existing results of R. Tessera, see
R. Tessera, Left inverses of matrices with polynomial decay. J. Funct. Anal. 259 (2010) no. 11, 2793--2813.
The original paper had a misuse of terminology: what was referred to throughout as the residual spectrum of an operator should actually have been called the set of points in the spectrum that are not approximate eigenvalues. That is:if $A\subseteq B(X)$ is a subalgebra, we should say that $(A,X)$ is a surjunctive pair when the spectrum of each $a\in A$ consits purely of approximate eigenvalues.
An erratum, with more details and line-by-line discussion of the necessary amendments, has been published in IEOT; see the link near the top of this page.