Approximation of infinite matrices by matricial Haar polynomials

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong toL ^∞ (T) may be one-to-one identified with Toeplitz matrices fromB(l ₂) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:ms—a unital commutative subalgebra ofl ^∞,C(l ₂) the matricial analogue of the space of all continuous periodic functionsC(T), the matricial Haar polynomials, etc. In Section 1 we present some results concerning the spacems—a concept important for this generalization, the proof of the main theorem being given in the second section. Partially supported by EUROMMAT ICA1-CT-2000-70022. Partially supported by V-Stabi-RUM/1022123. Partially supported by EUROMMAT ICA1-CT-2000-70022 and V-Stabi-RUM/1022123.