Geoffrey L. Price

Continuous spatial semigroups of *-endomorphisms of (ℌ)

To each continuous semigroup of *-endomorphisms a of Z(S)) with an intertwining semigroup of isometries there is associated a *-representation 7t of the domain 2D(6) of the generator of ac. It is shown that the Arveson index d* (a) is the number of times the representation 7t contains the identity representation of 2 (6) . This result is obtained from an analysis of the relation between two semigroups of isometries, U and S, satisfying the condition S(t)* U(t) = e -tI for t > 0 and > 0.

Index Theory and Second Quantization of Boundary Value Problems

The second quantization functor associates to each skew-symmetric operator in one-particle space a derivation 6 of the algebra which is based on the given commutation relations. In this paper, we characterize the spatial theory of 6 (in the Fock representation) by an index which generalizes the one studied earlier by Powers and Arveson in connection with the spatial cohomological obstruction for semigroups of endomorphisms of B(X). It is well known that such semigroups corresponding to one-sided boundary conditions are generated by derivations; but derivations associated to Iwo-sided boundary conditions do not generate semigroups. We show that the known index theory for semigroups generalizes to the quantization of arbitrary boundary conditions in one-particle space. Our quantized twosided abstract boundary conditions dictate representations in a certain indetinite inner product space (a Krein space), and our index is an isomorphism invariant for representation theory in Krein spaces. The representations are not unitarizable (i.e., are not equivalent to Hermitian representations in Hilbert space).

THE ENTROPY OF RATIONAL POWERS SHIFTS

The Connes-Størmer entropy of all rational Powers shifts is shown to be 1 2 log 2.

Infinite tensor products of completely positive semigroups

Abstract. We construct a new class of semigroups of completely positive maps on

On the ranks of skew-centrosymmetric matrices over finite fields☆

{\Cal B}(H)

On the ranks of Toeplitz matrices over finite fields

which can be decomposed into an infinite tensor product of such semigroups. Under suitable hypotheses, the minimal dilations of these semigroups to E0-semigroups are pure, and have no normal invariant states. Concrete examples are discussed in some detail.

Endomorphisms of B(H)

Abstract For any finite field F we determine the number of n by n matrices of skew-centrosymmetric form which are invertible over F. This result is obtained using a unimodality property of the ranks of matrices of this form. As a corollary to this result we count the n by n matrices of skew-centrosymmetric form of any specified rank.

Advances in Quantum Dynamics

Abstract Let T be a skew-symmetric Toeplitz matrix with entries in a finite field. For all positive integers n let T n be the upper n×n corner of T , with nullity ν n =ν(T n ) . The sequence {ν n :n∈ N } satisfies a unimodality property and is eventually periodic if the entries of T satisfy a periodicity condition. We compute the maximum value and the period of the nullity sequence for Toeplitz matrices of finite bandwidth. This sequence satisfies a certain symmetry condition about its maximal values. These results apply to give some information about the ranks of general skew-symmetric Toeplitz matrices with eventually periodic entries.