Functional Analysis

A method for deforming C*-algebras is introduced, which applies to C*-algebras that can be described as the cross-sectional C*-algebra of a Fell bundle. Several well known examples of non-commutative algebras, usually obtained by deforming commutative ones by various methods, are shown to fit our unified perspective of deformation via Fell bundles. Examples are the non-commutative spheres of Matsumoto, the non-commutative lens spaces of Matsumoto and Tomiyama, and the quantum Heisenberg manifolds of Rieffel. In a special case, in which the deformation arises as a result of an action of R^{2d}, assumed to be periodic in the first d variables, we show that we get a strict deformation quantization.

We consider self-similar measures μ with support in the interval 0≤x≤1 which have the analytic functions { e i2πnx :n=0,1,2,...} span a dense subspace in L 2 (μ) . Depending on the fractal dimension of μ , we identify subsets P⊂ N 0 ={0,1,2,...} such that the functions { e n :n∈P} form an orthonormal basis for L 2 (μ) . We also give a higher-dimensional affine construction leading to self-similar measures μ with support in R ν . It is obtained from a given expansive ν -by- ν matrix and a finite set of translation vectors, and we show that the corresponding L 2 (μ) has an orthonormal basis of exponentials e i2πλ⋅x , indexed by vectors λ in R ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.

Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con- nected, which acts on both A and B by *-automorphisms, and let \s be a sub- multiplicative function from G to the nonnegative real numbers. If \s and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in \s. We give conditions on the action of G on A which imply the m-convexity of the dense subalgebra G\rtimes^{\s}A. A locally convex algebra is said to be m-con- vex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a manifold M, we develop a class of dense subalgebras for the crossed product L^{1}(G, C_{0}(M)), where C_{0}(M) denotes the continuous functions on M vanishing at infinity with the sup norm topology.We define Schwartz functions S(M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense m-convex Frechet *-subalgebra G\rtimes^ {\s} S(M) of rapidly vanishing, G-differentiable functions from G to S(M). If the reciprocal of \s is in L^{p}(G) for some p, we prove that our group algebras S^{\s}(G) are nuclear Frechet spaces, and that G\rtimes^{\s}A is the projective completion S^{\s}(G) \otimes A.

We define operator manifolds as manifolds on which a spectral measure on a Hilbert space is given as additional structure. The spectral measure mathematically describes space as a quantum mechanical observable. We show that the vectors of the Hilbert space can be represented as functions on the manifold. The arbitrariness of this representation is interpreted as local gauge freedom. In this way, the physical gauge principle is linked with quantum mechanical measurements of the position variable. We derive the restriction for the local gauge group to be U(m), where m is the number of components of the wave functions.

We study functional determinants for Dirac operators on manifolds with boundary and discuss the ellipticity of boundary problems by using the Calderón projector. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green functions. We finally apply this result to the case of a bidimensional disk under bag-like conditions.

It is known that the classical Hilbert--Schmidt theorem can be generalized to the case of compact operators in Hilbert A -modules H ∗ A over a W ∗ -algebra of finite type, i.e. compact operators in H ∗ A under slight restrictions can be diagonalized over A . We show that if B is a weakly dense C ∗ -subalgebra of real rank zero in A with some additional property then the natural extension of a compact operator from H B to H ∗ A ⊃ H B can be diagonalized with diagonal entries being from the C ∗ -algebra B .

It is known that a continuous family of compact operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators in Hilbert modules over a commutative W*-algebra. The aim of the present paper is to generalize this fact for a finite W*-algebra A not necessarily commutative. We prove that for a compact operator K acting in the right Hilbert A -module H ∗ A dual to H A under slight restrictions one can find a set of "eigenvectors" x i ∈ H ∗ A and a non-increasing sequence of "eigenvalues" λ i ∈A such that K x i = x i λ i and the autodual Hilbert A -module generated by these "eigenvectors" is the whole H ∗ A . As an application we consider the Schrödinger operator in magnetic field with irrational magnetic flow as an operator acting in a Hilbert module over the irrational rotation algebra A θ and discuss the possibility of its diagonalization.

For W*-algebras A and self-dual Hilbert A-modules M we show that every self-adjoint, ''compact'' module operator on M is diagonalizable. Some specific properties of the eigenvalues and of the eigenvectors are described.

It is well known that in the commutative case, i.e. for A=C(X) being a commutative C*-algebra, compact selfadjoint operators acting on the Hilbert C*-module H A (= continuous families of such operators K(x) , x∈X ) can be diagonalized if we pass to a bigger W*-algebra L ∞ (X)=A⊃A which can be obtained from A by completing it with respect to the weak topology. Unlike the "eigenvectors", which have coordinates from A , the "eigenvalues" are continuous, i.e. lie in the C*-algebra A . We discuss here the non-commutative analog of this well-known fact. Here the "eigenvalues" are defined not uniquely but in some cases they can also be taken from the initial C*-algebra instead of the bigger W*-algebra. We prove here that such is the case for some continuous fields of real rank zero C*-algebras over a one-dimensional manifold and give an example of a C*-algebra A for which the "eigenvalues" cannot be chosen from A , i.e. are discontinuous. The main point of the proof is connected with a problem on almost commuting operators. We prove that for some C*-algebras if h∈A is a selfadjoint, u∈A is a unitary and if the norm of their commutant [u,h] is small enough then one can connect u with the unity by a path u(t) so that the norm of [u(t),h] would be also small along this path.

Investigating the direct integral decomposition of von Neumann algebras of bounded module operators on self-dual Hilbert W*-moduli an equivalence principle is obtained which connects the theory of direct disintegration of von Neumann algebras on separable Hilbert spaces and the theory of von Neumann representations on self-dual Hilbert {\bf A}-moduli with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module over commutative separable W*-algebras {\bf A}. Examples show posibilities and bounds to find more general relations between these two theories, (cf. R. Schaflitzel's results). As an application we prove a Weyl--Berg--Murphy type theorem: For each given commutative W*-algebra {\bf A} with a special approximation property (*) every normal bounded {\bf A}-linear operator on a self-dual Hilbert {\bf A}-module with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module is decomposable into the sum of a diagonalizable normal and of a ''compact'' bounded {\bf A}-linear operator on that module.