This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space localization. The reader is supposed to know elementary functional analysis and quantum mechanics.

Theory of Operator Algebras II

Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics

We introduce the definition of totally nonfree actions of groups; such actions are naturally related to the theory of characters of groups and their factor-representations of type II. This short note is a brief exposition of a part of a more detailed paper on this subject, which is now in preparation.

background

Nonfree Actions of Countable Groups and their Characters

Let R be a von Neumann algebra acting on a Hilbert space H and let R_sa be the set of selfadjoint elements of R. It is well known that R_sa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on R_sa, the spectral order ≤_s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on abelian subalgebras, and turns R_sa into a boundedly complete lattice. The effect algebra E(R) := {A | 0 ≤ A ≤ I} is then a complete lattice and we show that the mapping A --> R(A), where R(A) denotes the range projection of A, is a homomorphism from the lattice E(R) onto the projection lattice P(R) of A if and only if R is a finite von Neumann algebra.

methods

On a canonical lattice structure on the effect algebra of a von Neumann algebra

On strongly WCG Banach spaces

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

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Theory of Operator Algebras II

Citations

Lecture notes on C-algebras, Hilbert C-modules, and quantum mechanics

Nonfree Actions of Countable Groups and their Characters

On a canonical lattice structure on the effect algebra of a von Neumann algebra

On strongly WCG Banach spaces

The smooth entropy formalism for von Neumann algebras

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Theory of Operator Algebras II

Citations

Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics

Nonfree Actions of Countable Groups and their Characters

On a canonical lattice structure on the effect algebra of a von Neumann algebra

On strongly WCG Banach spaces

The smooth entropy formalism for von Neumann algebras

Related Papers (5)

Fundamentals of the Theory of Operator Algebras

Operator algebras and quantum statistical mechanics

C-Algebras and Their Automorphism Groups

C[*]-algebras and W[*]-algebras

Completely Bounded Maps and Operator Algebras

Lecture notes on C-algebras, Hilbert C-modules, and quantum mechanics

C[]-algebras and W[]-algebras