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Showing papers on "Operator algebra published in 1968"




Journal ArticleDOI
TL;DR: In this paper, a brief summary of recently derived general results relating to the mapping of functions of noncommuting operators on functions of c-numbers is given which describe the time evolution of the c-number equivalents (phase-space representations) of the density operator and of a Heisenberg operator.
Abstract: After a brief summary of recently derived general results relating to the mapping of functions of noncommuting operators on functions of c-numbers, equations are given which describe the time evolution of the c-number equivalents (phase-space representations) of the density operator and of a Heisenberg operator. The evaluation of time-ordered functions of operators by c-number techniques is also briefly discussed.

26 citations


Journal ArticleDOI
TL;DR: In this article, a non-commutative version of probability theory is outlined, based on the concept of a Σ*-algebra of operators (sequentially weakly closed C *-al algebra of operators).
Abstract: A non-commutative version of probability theory is outlined, based on the concept of aΣ*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory ofΣ*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. TheΣ*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering theΣ*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.

22 citations


Journal ArticleDOI
TL;DR: An example of a local rings system where the quasilocal algebra is a simple countably generated C*-algebra with unit is provided by the local observables for the free Fermi field as mentioned in this paper.
Abstract: An example of a local rings system where the quasilocal algebra is a simple countably generatedC*-algebra with unit is provided by the “local observables” for the free Fermi field.

13 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that non-unitary unimodular contractions exist only in infinite von Neumann algebras and that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum.
Abstract: Let T be a unitary operator on a Hubert space H Then in particular, (i) T is a contraction, ie || T\\ 1; and (ii) The spectrum of T is a subset of the unit circle, ie Sp (T)cC, where C denotes the set of complex numbers of absolute value one Call an arbitrary operator T a unimodular contraction if it satisfies conditions (i) and (ii) above Then several questions immediately come to mind Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, eg what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras This result is applied to a mapping problem of operator algebras

10 citations



Journal ArticleDOI
TL;DR: In this paper, a von Neumann algebra Mv is given on each Hilbert space and each operator T in Mv can be extended over all of H, which will be denoted as T. All these operators T together form a von NEUMAN algebra on H and are referred to as the infinite tensor product of Mv.
Abstract: (Hv, ev) is formed. Suppose further that a von Neumann algebra Mv be given on each Hv. Each operator T in Mv is extended over all of H, which will be denoted as T. All these operators T together form a von Neumann algebra on H, which we write as Mv. The von Neumann algebra M on H generated by these Mv's (z/ = l, 2, • • • ) is what we call as the infinite tensor product of Mv. Suppose now that each Mv given is a factor of type I. Then, Hv can be decomposed as a direct product of two Hilbert spaces HV1, HV 2 : HV = HV1®HV2, and Mv is thereby identified with B(R^)®IV2, where B(HV1) is the total operator algebra on HV1 and JV2 is a von Neumann algebra consisting of all scalar multiples of the identity operator on HV2. With respect to this direct product decomposition, 0V can be expressed as

8 citations




Journal ArticleDOI
TL;DR: In this paper, the authors apply a result of Herstein to simple C*-algebras which arise as quotients of properly infinite von Neumann algebrains.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the space of bounded linear functionals on M(G) can be represented as a semigroup of bounded operators on M (G), and that the Taylor structure semigroup is a compact Abelian semigroup with identity.
Abstract: Introduction. In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G). Let A denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of A in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative J3*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space. In § III, it is shown that B, with a natural multiplication, is a compact Abelian semigroup and that M(G) is topologically embedded in M(B). This gives a simplified construction of the Taylor structure semigroup for M(G). I am indebted to F. Birtel who directed this research. I. Let G be a locally compact Abelian group and T its dual group. Let CQ(G) denote the Banach algebra of continuous functions on G which vanish at infinity. Let M(G) denote the Banach algebra of bounded Borel measures on G and M(G)* its topological dual space. Let M(G)A denote the algebra of Fourier-Stieltjes transforms on T. For ju (E M(G), juA is defined on T by MA(T) =JVy(*) dn(x),y e r. For F G M(G)*, let EF denote the bounded operator on M(G) defined by (£ FM ) A (T) = F(ydn), where 7 £ r and M £ M(G). That (EF»)A £ M(G) A follows by Eberlein's theorem (5, p. 465) since

Book ChapterDOI
TL;DR: In this article, the authors present the methods of functional analysis that provide an essential insight in the mechanism of mathematical manipulations employed in quantum theory, including bounded linear operators, spectral theory of bounded self-adjoint operators, unbounded linear operators and unitary operators.
Abstract: Publisher Summary The chapter presents the methods of functional analysis that provides an essential insight in the mechanism of the mathematical manipulations employed in quantum theory. It discusses bounded linear operators, spectral theory of bounded self-adjoint operators, spectral theory of unbounded linear operators, and unitary operators, equivalence of operators. The dagger† instead of the star* is used to indicate an adjoint operator. The customary symbols of set theory, ∈ and ⊂, are employed to indicate that an element is a member of a set and that a set is included in another set, respectively.