No abstract

No abstract

Softcover reprint ofthe hardcover 2nd edition 1987 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speeific statement, that such names are exempt from the relevant probreak tective laws and regulations and therefore free for general use.

No abstract

We study a class of representations of the Cuntz algebras O N , N = 2, 3, . . . , acting on L 2 (T) where T = R 2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O N -irreducibles decompose when restricted to the subalgebra UHF N ⊂ O N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of O N on Hilbert space H such that the generators S i as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L 2 (T) to L 2 T d , d > 1; even to L 2 (T) where T is some fractal version of the torus which carries more of the algebraic information encoded in our representations.1991 Mathematics Subject Classification. Primary 46L55, 47C15; Secondary 42C05, 22D25, 11B85.