On C*-algebras related to constrained representations of a free group

We consider representations of the free group $F_2$ on two generators such that the norm of the sum of the generators and their inverses is bounded by $\mu\in[0,4]$. These $\mu$-constrained representations determine a C*-algebra $A_{\mu}$ for each $\mu\in[0,4]$. We prove that these C*-algebras form a continuous bundle of C*-algebras over $[0,4]$ and calculate their K-groups.Comment: 9 page

Epidemiological pattern of community-acquired respiratory tract infections of the conscripts in the North Fleet during a vaccine-challenged period

The formation of the new military units in the North fleet is accompanied by vaccination using Exhausted diphtheria tetanus vaccine, modified. The accination coincides with periods of a rising number of army conscripts being taken ill with community-acquired infection of respiratory tracts: acute tonsillitis, acute bronchitis and community-acquired pneumonia. We need to study is to ascertain whether there is the correlation between the periods of the increase in the number of ervicemen fallen ill with community-acquired infection of respiratory tracts and the diphtheria and tetanus vaccination. The study was carried out on the North fleet conscripts who were drawn blood samples from the ulnar vein before and after the vaccination using Exhausted diphtheria tetanus vaccine, modified. The blood was examined for the presence of antibodies to diphtheria and tetanus using direct hemagglutination test. The health status of the vaccinated conscripts was under observation for 4 months, during which acute illnesses (acute tonsillitis, acute bronchitis and community-acquired pneumonia) were registered. Serologic testing demonstrated a high rate of immunological protection against diphtheria and tetanus before vaccination. After the diphtheria and tetanus vaccination, the number of conscripts, who were taken ill in the first month, was significantly higher compared to the following months. The conscripts, who fell ill, had high antibody titers against diphtheria and tetanus in the vaccine-challenged period. Vaccination of the servicemen using Exhausted diphtheria tetanus vaccine, modified, is serologically unfounded; it leads to complications such as acute tonsillitis, acute bronchitis and community-acquired pneumonia during the vaccinechallenged period especially during the first month and less considerably during the following months

Diagonalizing operators over continuous fields of C*-algebras

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 $K(x)$, $x\in X$) can be diagonalized if we pass to a bigger W*-algebra $L^\infty(X)={\bf A} \supset A$ which can be obtained from $A$ by completing it with respect to the weak topology. Unlike the "eigenvectors", which have coordinates from $\bf A$, 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 $h\in A$ is a selfadjoint, $u\in A$ 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.Comment: 21 pages, LaTeX 2.09, no figure