Representation of ℬ, ℬ’ by Banach Spaces of Operators in a Hilbert Space

Michael P. Strand, R. Stephen Berry

Günther Ludwig

Frank Oertel

Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fréchet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces $H\otimes_2 (K \otimes_2 L)$ and $(H \otimes_2 K) \otimes_2 L$ (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces $H \otimes_2 K$ (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).

Slawomir Bugajski

Abstract It is demonstrated that if V* is not a vector lattice, where V is a base norm Banach space, then there is no commutative observable providing a classical representation for V. This observation generalizes a similar result of Busch and Lahti, obtained for V - the trace class of operators on a separable complex Hilbert space.

Frank Oertel

Starting from a thorough analysis of the conjugate $\overline{H}$ of a complex Hilbert space $H$, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fréchet-Riesz over to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).

Charles R. Giardina