In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in p for some p implies boundedness below in p for all p. We use the established result to enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.

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The definition and study of causal operators are based on the representation theory of group algebras. We study the structure of the spectra of causal operators, obtain conditions for causal invertibility and state criteria for a causal operator to belong to the radical.

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We develop a theory of almost periodic elements in Banach algebras and present an abstract version of a noncommutative Wiener's Lemma. The theory can be used, for example, to derive some of the recently obtained results in time-frequency analysis such as the spectral properties of the finite linear combinations of time-frequency shifts.