Functional Analysis

Some algebraic, geometric and geometroalgebraic characteristics of pairs of operators are discussed.

We introduce a method to study C*-algebras possessing an action of the circle group, from the point of view of its internal structure and its K-theory. Under relatively mild conditions our structure Theorem shows that any C*-algebra, where an action of the circle is given, arises as the result of a construction that generalizes crossed products by the group of integers. Such a generalized crossed product construction is carried out for any partial automorphism of a C*-algebra, where by a partial automorphism we mean an isomorphism between two ideals of the given algebra. Our second main result is an extension to crossed products by partial automorphisms, of the celebrated Pimsner-Voiculescu exact sequence for K-groups. The representation theory of the algebra arising from our construction is shown to parallel the representation theory for C*-dynamical systems. In particular, we generalize several of the main results relating to regular and covariant representations of crossed products.

The classical and quantum dynamics of noncanonically coupled os- cillators is investigated in its relation to Lie superalgebras. It is shown that the quantum dynamics admits a hidden (super)hamiltonian formulation and, hence, preserves the initial operator relations.

Using the continuous decomposition, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III λ ,0<λ<1 . Winsløw's fundamental homomorphism is a complete invariant. This removes the extra assumptions in the classification theorems of Loi and Winsløw and gives a complete classification up to cocycle conjugacy.

We prove a classification theorem for purely infinte simple C*-algebras that is strong enough to show that the tensor products of two different irrational rotation algebras with the same even Cuntz algebra are isomorphic. In more detail, let C be be the class of simple C*-algebras A which are direct limits A = lim A_k, in which each A_k is a finite direct sum of algebras of the form C(X) \otimes M_n \otimes O_m, where m is even, O_m is the Cuntz algebra, X is either a point, a compact interval, or the circle S^1, and each map A_k ---> A is approximately absorbing. ("Approximately absorbing" is defined in Section 1 of the paper.) We show that two unital C*-algebras A and B in the class C are isomorphic if and only if (K_0 (A), [1_A], K_1 (A)) is isomorphic to (K_0 (B), [1_B], K_1 (B)). This class is large enough to exhaust all possible K-groups: if G_0 and G_1 are countable odd torsion groups and g is in G_0, then there is a C*-algebra A in C with (K_0 (A), [1_A], K_1 (A)) isomorphic to (G_0, g, G_1). The class C contains the tensor products of irrational rotation algebras with even Cuntz algebras. It is also closed under several natural operations.

A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.

For the q -deformed canonical commutation relations a(f) a † (g)=(1−q)⟨f,g⟩1+q a † (g)a(f) for f,g in some Hilbert space H we consider representations generated from a vector Ω satisfying a(f)Ω=⟨f,ϕ⟩Ω , where ϕ∈H . We show that such a representation exists if and only if ∥ϕ∥≤1 . Moreover, for ∥ϕ∥<1 these representations are unitarily equivalent to the Fock representation (obtained for ϕ=0 ). On the other hand representations obtained for different unit vectors ϕ are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a natural q -analogue of the Cuntz algebra (obtained for q=0 ). We discuss the Conjecture that, for d<∞ , this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases q=±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.

Let A be a unital C ∗ -algebra, given together with a specified state ϕ:A→C . Consider two selfadjoint elements a,b of A, which are free with respect to ϕ (in the sense of the free probability theory of Voiculescu). Let us denote c:=i(ab−ba) , where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012.

We present explicit generators of an algebra of commuting difference operators with trigonometric coefficients. The operators are simultaneously diagonalized by recently discovered q-polynomials (viz. Koornwinder's multivariable generalization of the Askey-Wilson polynomials). From the viewpoint of physics the algebra can be interpreted as consisting of the quantum integrals of a novel difference-type integrable sytem. This system generalizes the Calogero-Moser systems associated with non-exceptional root systems.

The paper gives a negative answer to the question whether one can perturb a Fredholm pair of index 0 by compact operators to a pair with exact Koszul complex.