Functional Analysis

A product system E over a semigroup P is a family of Hilbert spaces {E_s:s\in P} together with multiplications E_s \times E_t\to E_{st}. We view E as a unitary- valued cocycle on P, and consider twisted crossed products A \times_{\beta,E} P involving E and an action \beta of P by endomorphisms of a C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations of E, and consider a twisted crossed product B_P \times_{\tau,E} P which is universal for covariant representations of E when E has finite-dimensional fibres, and in general is slightly larger. In particular, when P=N and \dim E_1=\infty, our algebra B_\NN \times_{\tau,E} N is a new infinite analogue of the Toeplitz-Cuntz algebras TO_n. Our main theorem is a characterisation of the faithful representations of B_P \times_{\tau,E} P.

Asymptotic expansions of Green functions and spectral densities associated with partial differential operators are widely applied in quantum field theory and elsewhere. The mathematical properties of these expansions can be clarified and more precisely determined by means of tools from distribution theory and summability theory. (These are the same, insofar as recently the classic Cesaro-Riesz theory of summability of series and integrals has been given a distributional interpretation.) When applied to the spectral analysis of Green functions (which are then to be expanded as series in a parameter, usually the time), these methods show: (1) The "local" or "global" dependence of the expansion coefficients on the background geometry, etc., is determined by the regularity of the asymptotic expansion of the integrand at the origin (in "frequency space"); this marks the difference between a heat kernel and a Wightman two-point function, for instance. (2) The behavior of the integrand at infinity determines whether the expansion of the Green function is genuinely asymptotic in the literal, pointwise sense, or is merely valid in a distributional (cesaro-averaged) sense; this is the difference between the heat kernel and the Schrodinger kernel. (3) The high-frequency expansion of the spectral density itself is local in a distributional sense (but not pointwise). These observations make rigorous sense out of calculations in the physics literature that are sometimes dismissed as merely formal.

We show that, as conjectured by Adrien Douady back in 1972, every complete metric space is homeomorphic (moreover, isometric) to the locus of zeros of an analytic map between two Banach spaces. As a corollary, a paracompact topological space admits the structure of a Banach analytic space if and only if it is metrizable with a complete metric.

We introduce and study dual n 1 -Appell-like systems which are the simple generalization of generalized dual Appell systems in Infinite-Dimensional Analysis (IDA). We study connected with these systems objects of IDA: the analogues of Kondratiev spaces, S -transform, characterization theorems etc. The results we obtained are useful to application in the theory of probability.

Consider a coaction δ of a locally compact group G on a \cstar algebra A , and a closed normal subgroup N of G . We prove, following results of Echterhoff for abelian G , that Mansfield's imprimitivity between A × δ| G/N and $A\times_\delta G\times_{\deltahat,r}N$ implements equivalences between Mansfield induction of representations from A×G/N to A×G and restriction of representations from A×G × r N to A×G , and between restriction of representations from A×G to A×G/N and Green induction of representations from A×G to A×G × r N . This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.

Dynamical inverse problem of representation theory, which has its origin in a classical paper of E.P.Wigner on a determination of commutation relations of quantum mechanical quantities by the quantum dynamical equations, is illustrated on the simplest examples.

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra O 2 . We further prove that if A is a simple separable unital nuclear C*-algebra, then O 2 ⊗A≅ O 2 , and if, in addition, A is purely infinite, then O ∞ ⊗A≅A. The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If A is a continuous field of C*-algebras over a compact manifold or finite CW complex X with fiber A(x) over x∈X, such that the algebra of continuous sections of A is separable and exact, then there is a family of injective homomorphisms ϕ x :A(x)→ O 2 such that for every continuous section a of A the function x↦ ϕ x (a(x)) is continuous. Moreover, one can say something about the modulus of continuity of the functions x↦ ϕ x (a(x)) in terms of the structure of the continuous field. In particular, we show that the continuous field θ↦ A θ of rotation algebras posesses unital embeddings ϕ θ in O 2 such that the standard generators u(θ) and v(θ) are mapped to Lip 1/2 functions.

Borchers and Wiesbrock have demonstrated certain results concerning the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables. These results are abstracted and analyzed as essentially operator-theoretic. Criteria are then demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms. All this is combined to establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results. This analysis is then applied to questions of isotony and covariance for local algebras.

The unital endomorphisms of B(H) of (Powers) index n are classified by certain U(n)-orbits in the set of non-degenerate representations of the Cuntz algebra O_n on H. Using this, the corre- sponding conjugacy classes are identified, and a set of labels is given. This set is given as P modulo a certain non-smooth equivalence, where P is a set of pure state on the UHF algebra of Glimm type n^infinity. Several subsets of P, giving concrete examples of non- conjugate shifts, are worked out in detail, including sets of product states, and a set of nearest neighbor states.

Let F_n be the fixed-point algebra of the gauge action of the circle on the Cuntz algebra O_n. For every pure state \rho of F_n and every representation \theta of C(T) we construct a representation of O_n, and we use the resulting class of representations to parameterize the space of all states of O_n which extend \rho. We show that the gauge group acts transitively on the pure extensions of \rho and that the action is p-to-1 with p the period of \rho under the usual shift. We then use the above representations of O_n to construct endomorphisms of B(H) which we classify up to conjugacy in terms of the parameters \rho and \theta. In particular our construction yields every ergodic endomorphism \alpha whose tail algebra ⋂ k α k (B(H)) has a minimal projection, and our results classify these ergodic endomorphisms by an equivalence relation on the pure states of F_n. As examples we analyze the ergodic endomorphisms arising from periodic pure product states of F_n, for which we are able to give a geometric complete conjugacy invariant, generalizing results of Stacey, Laca, and Bratteli-Jorgensen-Price on the shifts of Powers.