arXiv subjects
Publications and source records attributed to William Arveson.
These lectures concern basic aspects of the theory of semigroups of endomorphisms of type $I$ factors that relate to causal dynamics, dilation theory, and the problem of classifying $E_0$-semigroups up to cocycle conjugacy. We give only a few proofs here; full details can be found in the author's upcoming monograph {\em Noncommutative Dynamics and $E$-semigroups}, to be published in the Springer series Monographs in Mathematics.
For a fixed C*-algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an A-dynamical system is a triple (i,B,α) where $α$ is a *-endomorphism of a C*-algebra B, and i: A --> B is the inclusion of A as a C^*-subalgebra with the property that B is generated by $A\cup α(A)\cup α^2(A)\cup...$. There is a natural hierarchy in the class of A-dynamical systems, and there is a universal one that dominates all others, denoted (i,PA,α). We establish certain properties of $(i,PA,α)$ and give applications to some concrete issues of noncommutative dynamics. For example, we show that every contractive completely positive linear map $ϕ: A\to A$ gives rise to to a unique A-dynamical system (i,B,α) that is "minimal" with respect to $ϕ$, and we show that its C*-algebra B can be embedded in the multiplier algebra of $A\otimes {\mathcal K}$.
A spin system is a sequence of self-adjoint unitary operators $U_1,U_2,...$ acting on a Hilbert space $H$ which either commute or anticommute, $U_iU_j=\pm U_jU_i$ for all $i,j$; it is is called irreducible when $\{U_1,U_2,...\}$ is an irreducible set of operators. There is a unique infinite matrix $(c_{ij})$ with $0,1$ entries satisfying $$ U_iU_j=(-1)^{c_{ij}}U_jU_i, \qquad i,j=1,2,.... $$ Every matrix $(c_{ij})$ with $0,1$ entries satisfying $c_{ij}=c_{ji}$ and $c_{ii}=0$ arises from a nontrivial irreducible spin system, and there are uncountably many such matrices. Infinite dimensional irreducible representations exist when the commutation matrix $(c_{ij})$ is of "infinite rank". In such cases we show that the $C^*$-algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor product of copies of $M_2(\Bbb C)$, and we classify the irreducible spin systems associated with a given matrix $(c_{ij})$ up to approximate unitary equivalence. That follows from a structural result. The $C^*$-algebra generated by the universal spin system $u_1,u_2,...$ of $(c_{ij})$ decomposes into a tensor product $C(X)\otimes\Cal A$, where $X$ is a Cantor set (possibly finite) and $\Cal A$ is either the CAR algebra or a finite tensor product of copies of $M_2(\Bbb C)$. The Bratteli diagram technology of AF algebras is not well suited to spin systems. Instead, we work out elementary properties of the $\Bbb Z_2$-valued "symplectic" form $$ ω(x,y) =\sum_{p,q=1}^\infty c_{pq}x_qy_p, $$ $x,y$ ranging over the free infninite dimensional vector space over the Galois field $\Bbb Z_2$, and show that one can read off the structure of $C(X)\otimes\Cal A$ from properties of $ω$.
Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of $D_{\bar T}$, including the Taylor spectrum and the notion of Fredholmness. In fact, {\it all} properties of $\bar T$ derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension $d=1,2,...$) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting $d$-tuple, and that two Dirac operators are isomorphic iff their associated operator $d$-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d-contractions of finite rank. It is shown that for the subcategory of all such $\bar T$ which are a) Fredholm and and b) graded, the curvature invariant $K(\bar T)$ is stable under compact perturbations. We do not know if this stability persists when $\bar T$ is Fredholm but ungraded, though there is concrete evidence that it does.
Let P and Q be the canonical operators acting on the Hilbert space of all L^2 functions on the real line, defined appropriately on a common dense domain. The derivations D_P(A) = i(PA - AP) and D_Q(A) = i(QA - AQ) act on the *-algebra of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative "Laplacian" L(A) = D_P^2(A) + D_Q^2(A), as a linear mapping of this *-algebra into itself. L generates a semigroup of normal completely positive maps on B(H), and we establish some basic properties of this semigroup and its minimal dilation to an E_0-semigroup. In particular, we show that the minimal dilation is pure, has no normal invariant states, and we discuss the significance of those facts for the interaction theory developed in a previous paper (appearing in the current issue of Comm. Math. Phys.). There are similar results for the canonical commutation relations with n degrees of freedom, n = 2, 3, ....
A CP semigroup is a semigroup of normal unit-preserving completely positive maps acting on the algebra B(H) of all operators on a separable Hilbert space H. Such a semigroup has a natural generator L; since the individual maps of the semigroup need not be multiplicative, the domain Dom(L) of L is typically an operator system, but not an algebra. However, we show that the set of all operators A in Dom(L), with the property that both A*A and AA* belong to Dom(L), is a *-algebra, called the domain algebra of the CP semigroup. Using this algebra, it is possible to draw a very close parallel with the Laplacian of a Riemannian manifold. We discuss properties of the "symbol" of L as the noncommutative counterpart of a (semidefinite) Riemannian metric, and give examples for which the domain algebra is, and is not, strongly dense in B(H).
A mathematical notion of interaction is introduced for noncommutative dynamical systems, i.e., for one parameter groups of *-automorphisms of $\Cal B(H)$ endowed with a certain causal structure. With any interaction there is a well-defined "state of the past" and a well-defined "state of the future". We describe the construction of many interactions involving cocycle perturbations of the CAR/CCR flows and show that they are nontrivial. The proof of nontriviality is based on a new inequality, relating the eigenvalue lists of the "past" and "future" states to the norm of a linear functional on a certain C^*-algebra.
A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established for graded (contractive) Hilbert modules over the complex polynomial algebra in d variables, d=1,2,3,.... The curvature invariant, Euler characteristic, and degree are computed for some explicit examples based on varieties in (multidimensional) complex projective space, and applications are given to the structure of graded ideals in C[z_1,...,z_d] and to the existence of "inner sequences" for closed submodules of the free Hilbert module H^2(C^d).
A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators acting on a common Hilbert space H such that $ \|T_1ξ_1+T_2ξ_2+... +T_dξ_d\|^2\leq \|ξ_1\|^2+\|ξ_2\|^2+...+\|ξ_d\|^2 $ for all $ξ_1,ξ_2,...,ξ_d\in H$. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball B_d in complex d-space, including von Neumann's inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new $H^2$ space associated with B_d, and which is the higher dimensional counterpart of the unilateral shift. $H^2$ and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C^*-algebra we find that there is more uniqueness in dimension $d\geq 2$ than there is in dimension one.
A numerical index is introduced for semigroups of completely positive maps of $\Cal B(H)$ which generalizes the index of E_0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E_0-semigroup, provided that the dilation is minimal.
It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator $ L: B(H)\to B(H) $ in terms of natrual structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup $P={\exp{tL}: t\geq 0}$ to an \esg\ is is cocycle conjugate to a CAR/CCR flow.
An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $α_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $ω$ and which are minimal in an appropriate sense. In such cases, for every normal state $ρ$ there is convergence to equilibrium in the trace norm $\lim_{t\to\infty}\| ρ\circα_t - ω\| = 0.$ A normal state $ω$ with this property is called an absorbing state. Such E_0-semigroups mest be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists.
It is known that every semigroup of normal completely positive maps of a von Neumann can be ``dilated" in a particular way to an E_0-semigroup acting on a larger von Neumann algebra. The E_0-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of {\it minimality} are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is $\Cal B(H)$. In this paper we clarify these issues by giving a new characterization of minimality in terms projective cocycles and their limits. Our results are valid for semigroups of endomorphisms acting on arbitrary von Neumann algebras with separable predual.
We show that a noncommutative dynamical system of the type that occurs in quantum theory can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. The nature of these dynamical principles is similar to that of the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a ``Riemannian metric", and a potential. These structures are classified in terms of geometric objects which, in the simplest cases, occur in finite dimensional matrix algebras. As a consequence, we obtain a new classification of E_0-semigroups acting on type I factors.
We classify all continuous tensor product systems of Hilbert spaces which are ``infinitely divisible" in the sense that they have an associated logarithmic structure. These results are applied to the theory of E_0 semigroups to deduce that every E_0 semigroup which possesses sufficiently many ``decomposable" operators must be cocycle conjugate to a CCR flow. A *path space* is an abstraction of the set of paths in a topological space, on which there is given an associative rule of concatenation. A metric path space is a pair (P,g) consisting of a path space P and a function g:P^2 --> complex numbers which behaves as if it were the logarithm of a multiplicative inner product. The logarithmic structures associated with infinitely divisible product systems are such objects. The preceding results are based on a classification of metric path spaces.
This is a survey of four recent papers which deal with the relationship of simple C*-algebras to the problem of computing the spectra of self-adjoint operators in the general case, especially when the spectrum is not discrete. It is an expanded version of a talk presented at the 50 year C*-algebra celebration, held at the annual meeting of the AMS in San Antonio during January, 1993.
We consider problems associated with the computation of spectra of self-adjoint operators in terms of the eigenvalue distributions of their n x n sections. Under rather general circumstances, we show how these eigenvalues accumulate near points of the essential spectrum of the given operator, and we prove that their averages converge to a measure concentrated precisely on the essential spectrum. In the primary cases of interest, namely the discretized Hamiltonians of one-dimensional quantum systems, this limiting measure is associated with a tracial state on a certain simple C*-algebra. These results have led us to conclude that one must view this kind of numerical analysis in the context of C*-algebras.
We discuss some basic issues that arise when one attempts to model quantum mechanical systems on a computer, and we describe the mathematical structure of the resulting discretized cannonical commutation relations.