Noncommutative Dynamics and E-Semigroups by William Arveson

By William Arveson

The time period Noncommutative Dynamics may be interpreted in numerous methods. it really is utilized in this e-book to consult a collection of phenomena linked to the dynamics of quantum platforms of the easiest type that contain rigorous mathematical constructions linked to infinitely many levels of freedom. The dynamics of any such approach is represented by way of a one-parameter team of automorphisms of a noncommutative algebra of observables, and the writer focuses totally on the main concrete case within which that algebra comprises all bounded operators on a Hilbert space.

This topic overlaps with numerous mathematical components of present curiosity, together with quantum box conception, the dynamics of open quantum structures, noncommutative geometry, and either classical and noncommutative likelihood conception. this can be the 1st e-book to offer a scientific presentation of development up to now fifteen years at the category of E-semigroups as much as cocycle conjugacy. there are lots of new effects that can not be present in the present literature, in addition to major reformulations and simplifications of the idea because it exists today.

William Arveson is Professor of arithmetic on the collage of California, Berkeley. He has released earlier books with Springer-Verlag, a call for participation to C*-algebras (1976) and a brief path on Spectral thought (2001).

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Example text

F I U F 2 U ... ; thus it suffices to show that each Fr is closed in the product topology of (0,00) x ß(H), where of course ß(H) is endowed with its weak operator topology. Let (tn, Tn ) be a net in Fr that converges to (t, T) E (0,00) x ß(H). We claim that at n (A)Tn converges weakly to at(A)T for every fixed A E ß(H). Indeed, this is apparent from the fact that Tn is a bounded net converging weakly to T and 2. E-SEMIGROUPS 38 at n (A) is a bounded net converging strongly to at(A). Since Tn E Ca(tn) for every n, we have atJA)Tn = TnA, and the right side of the latter converges weakly to TA.

0. Then the Hilbert space H(Ua ) is separable. PROOF. Choose any to > O. Since the Hilbert space ca(to) is separable, there is a countable set ofunits 0 ~ Ua such that {T(to) : T E O} is dense in Ca (to). Let To be any particular unit and consider the function L : Ua -+ H(Ua ) defined by L(T) = OT - OTo + N, where N denotes the space of all functions in CoUa having norm zero, and OT denotes the function on Ua that is 1 at T and 0 otherwise. Obviously, L(Ua ) has H(Ua ) as its closed linear span; thus it suffices to show that L(O) is dense in L(Ua ).

In general, one obtains an orthonormal basis for summands of the form I\n H, n = 2,3, ... , from a fixed orthonormal basis el, e2, ... for H as wedge products ei, 1\ ei 2 1\ ... 1\ ein' 1 :::; i l < i 2 < ... < in. Moreover, for every fixed zEH there is a unique operator c(z) E ß(F_(H)) that acts as follows on wedge products: (i E H, n;::: O. 24 2. E-SEMIGROUPS C(Z) is the creation operator corresponding to the vector zEH. 15) c(z)c(w) + c(w)c(z) = 0, c(z)c(w)* + c(w)*c(z) = (z, w)l, for all z, wEH, and in fact, this is an irreducible representation of the canonical anticommutation relations: The only closed subspaces of F _ (H) that are invariant under c(H) U c(H)* are {O} and F_(H).

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