Von Neumann algebra

About: Von Neumann algebra is a research topic. Over the lifetime, 3332 publications have been published within this topic receiving 64779 citations. The topic is also known as: W*-algebra.

Fundamentals of the Theory of Operator Algebras

Abstract: Comparison theory of projections--exercises and solutions Normal states and unitary equivalence of von Neumann algebras--exercises and solutions The trace--exercises and solutions Algebra and commutant--exercises and solutions Special representations of $C^*$-algebras--exercises and solutions Tensor products--exercises and solutions Approximation by matrix algebras--exercises and solutions Crossed products--exercises and solutions Direct integrals and decomposiitons--exercises and solutions Bibliography Index.

C*-Algebras and Operator Theory

Abstract: Elementary Spectral Theory C*-Algebras and Hilbert Space Operators Ideals and Positive Functionals Von Neumann Algebras Representations of C*-Algebras Direct Limits and Tensor Products K-Theory of C*-Algebras

Theory of operator algebras

Abstract: I Fundamentals of Banach Algebras and C*-Algebras.- 0. Introduction.- 1. Banach Algebras.- 2. Spectrum and Functional Calculus.- 3. Gelfand Representation of Abelian Banach Algebras.- 4. Spectrum and Functional Calculus in C*-Algebras.- 5. Continuity of Homomorphisms.- 6. Positive Cones of C*-Algebras.- 7. Approximate Identities in C*-Algebras.- 8. Quotient Algebras of C*-Algebras.- 9. Representations and Positive Linear Functional.- 10. Extreme Points of the Unit Ball of a C*-Algebra.- 11. Finite Dimensional C*-Algebras.- Notes.- Exercises.- II Topologies and Density Theorems in Operator Algebras.- 0. Introduction.- 1. Banach Spaces of Operators on a Hilbert Space.- 2. Locally Convex Topologies in ?(?).- 3. The Double Commutation Theorem of J. von Neumann.- 4. Density Theorems.- Notes.- III Conjugate Spaces.- 0. Introduction.- 1. Abelian Operator Algebras.- 2. The Universal Enveloping von Neumann Algebra of a C*-Algebra.- 3. W*-Algebras.- 4. The Polar Decomposition and the Absolute Value of Functionals.- 5. Topological Properties of the Conjugate Space.- 6. Semicontinuity in the Universal Enveloping von Neumann Algebra*.- Notes.- IV Tensor Products of Operator Algebras and Direct Integrals.- 0. Introduction.- 1. Tensor Product of Hilbert Spaces and Operators.- 2. Tensor Products of Banach Spaces.- 3. Completely Positive Maps.- 4. Tensor Products of C*-Algebras.- 5. Tensor Products of W*-Algebras.- Notes.- 6. Integral Representations of States.- 7. Representation of L2(?,?) ? ?, L1(?,?) ?y? *, and L(?,?) ?? ?.- 8. Direct Integral of Hubert Spaces, Representations, and von Neumann Algebras.- Notes.- V Types of von Neumann Algebras and Traces.- 0. Introduction.- 1. Projections and Types of von Neumann Algebras.- 2. Traces on von Neumann Algebras.- Notes.- 3. Multiplicity of a von Neumann Algebra on a Hilbert Space.- 4. Ergodic Type Theorem for von Neumann Algebras*.- 5. Normality of Separable Representations*.- 6. The Borel Spaces of von Neumann Algebras.- 7. Construction of Factors of Type II and Type III.- Notes.- Appendix Polish Spaces and Standard Borel Spaces.- Monographs.- Papers.- Notation Index.

On the equilibrium states in quantum statistical mechanics

Abstract: Representations of theC*-algebra\(\mathfrak{A}\) of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant. This means that there is an equivalent anti-linear representation of\(\mathfrak{A}\) in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.