Algebraic Aspects of the Paving and Feichtinger Conjectures

The Paving Conjecture in operator theory and the Feichtinger Conjecture in frame theory are both problems that are equivalent to the Kadison-Singer problem concerning extensions of pure states. In all three problems, one of the difficulties is that the natural multiplicative structure appears to be incompatible - the unique extension problem of Kadison-Singer is compatible with a linear subspace, but not a subalgebra; likewise, the payable operators is known to be a linear subspace but not a subalgebra; the Feichtinger Conjecture does not even have a linear structure. The Paving Conjecture and the Feichtinger Conjecture both have special cases in terms of exponentials in L(2)[0, 1]. We introduce convolution as a multiplication to demonstrate a possible attack for these special cases.