Isometric Dilations of Non-commuting Finite Rank N-tuples

A contractive n-tuple A = (A1, . . . , An) has a minimal joint isometric dilation S = (S1, . . . , Sn) where the Si’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra S generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra S is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an n-tuple B of d× d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B. In [15, 16], the first author and David Pitts studied a class of algebras coined free semigroup algebras. These are the wot–closed (nonself-adjoint) unital operator algebras generated by an n-tuple of isometries with pairwise orthogonal ranges. When these ranges span the whole space, the associated norm-closed self-adjoint algebra is a representation of the Cuntz algebra. This nonself-adjoint algebra can contain detailed information about fine unitary invariants of the corresponding C*-algebra representation. Indeed in [15] the set of atomic representations of the Cuntz algebra is completely classified. On the other hand, when the ranges span a proper subspace, the representation contains a multiple of the left regular representation of the free semigroup on n letters. The wot-closed algebra of the left regular representation is called the non-commutative analytic Toeplitz algebra. This nomenclature is justified by a good analogue of Beurling’s Theorem [29, 1, 15], hyper-reflexivity [15] and the relationship [16] between its automorphism group and the group of conformal automorphisms of the ball in C. 1991 Mathematics Subject Classification. 47L80. March 10, 1999. Revised May 1, 2000. The first author was partially supported by an NSERC grant. The second author was partially supported by NSERC and OGS scholarships. 1 2 K.R.DAVIDSON, D.W.KRIBS, AND M.E.SHPIGEL The connection with dilation theory derives from a theorem of Frahzo, Bunce and Popescu [19, 11, 26]. If A = (A1, . . . , An) is an n-tuple of operators such that AA∗ = ∑n i=1 AiA ∗ i ≤ I, then there is a unique minimal isometric dilation to isometries Si on a larger space with pairwise orthogonal ranges. Popescu [26] establishes the analogue of Wold’s decomposition which splits this into a direct sum of a multiple of the left regular representation and a representation of the Cuntz algebra. Moreover, Popescu [28] obtains the non-commutative analogue of von Neumann’s inequality in this context. We mention in passing that there has been recent interest in dilating commuting n-tuples as well [30, 3, 4]. On the other hand, representations of the Cuntz algebra correspond to endomorphisms of B(H) [31, 25, 8, 9]. This has created new interest in classifying these representations up to unitary equivalence. The well-known theorem of Glimm [22] shows that this classification is non-smooth because On is anti-liminal (or NGCR). Nevertheless, interesting classes of representations do lend themselves to a complete analysis. In [10], Bratteli and Jorgensen introduced a class of representations which turned out to be a special case of the atomic representations classified in [15] using nonself-adjoint techniques. In [9] they introduce a different class associated to finitely correlated states. The reader will see a lot of parallels between their results and ours, though the approach is quite different. In the end, they specialize to the subclass of diagonalizable shifts in order to obtain a classification theorem. In this paper, we obtain good unitary invariants for the class of all of these finitely correlated representations. The goal of this paper is two-fold. First we wish to understand the structure of the free semigroup algebra generated by the dilation of an n-tuple A in terms of information obtained from the n-tuple itself (and the algebra it generates). In particular, we seek unitary invariants for the associated C*-algebra representation. Secondly, we wish to determine whether these algebras are reflexive and even hyper-reflexive. In this paper, we focus on the case in which the n-tuple A acts on a finite dimensional space. Here we obtain a complete description of the algebra. This enables us to decompose the associated representation as a direct sum of irreducible representations and obtain complete unitary invariants. These algebras all turn out to be hyper-reflexive. In the last section, we discuss similarity invariants. One of the surprising consequences is a complete invariant for an irreducible n-tuple of d × d matrices up to similarity. An algorithm for determining if two n-tuples of matrices are similar is provided by Friedland [21]. But this method rapidly gets complicated. So it is not clear whether it is ISOMETRIC DILATIONS 3 superior to ours. We find that there is a finite set of no more than 1 + (n − 1)d polynomials pj so that another n-tuple B is similar to A if and only if pj(B) = 0 for all j. These polynomials are obtained from a computable set of generators of an ideal of the left regular free semigroup algebra as a right ideal, which amounts to computing an orthonormal basis for a certain subspace. In practice, one only needs generators as a two-sided ideal, and hence the actual number needed is normally smaller.