Tensor products are crucial for building new C*-algebras from existing ones. They come in different flavors - algebraic, minimal, maximal, and injective - each with unique properties and uses in C*-algebra theory.
Spatial tensor products, built using faithful representations on Hilbert spaces, are particularly important. They preserve key properties like nuclearity and exactness, and play a big role in quantum mechanics and information theory.
Tensor Products of C-algebras
Algebraic and Minimal Tensor Products
- Algebraic tensor product forms basis for more refined tensor products in C*-algebra theory
- Construction involves generating vector space from elementary tensors and imposing bilinearity
- Minimal tensor product completes algebraic tensor product with respect to spatial norm
- Spatial norm defined as supremum over all faithful representations of constituent algebras
- Minimal tensor product preserves injectivity and exactness properties of C*-algebras
Maximal and Injective Tensor Products
- Maximal tensor product completes algebraic tensor product with respect to largest C*-norm
- Largest C*-norm obtained by taking supremum over all C*-cross norms on algebraic tensor product
- Maximal tensor product universal for pairs of *-homomorphisms with commuting ranges
- Injective tensor product defined using completely positive maps and operator spaces
- Injective tensor product coincides with minimal tensor product for nuclear C*-algebras
Projective Tensor Product and Applications
- Projective tensor product completes algebraic tensor product with respect to projective cross-norm
- Projective cross-norm defined as infimum over all possible decompositions of tensor
- Projective tensor product useful for studying tensor products of Banach algebras
- Applications include studying crossed products and group C*-algebras
- Tensor products crucial for constructing new C*-algebras from existing ones (quantum groups)
Spatial Tensor Products
Spatial Tensor Product Construction
- Spatial tensor product defined using faithful representations on Hilbert spaces
- Construction involves tensor product of Hilbert spaces and extending to C*-algebras
- Spatial tensor product independent of choice of faithful representations
- Yields minimal tensor product in C*-algebra context
- Preserves important properties like nuclearity and exactness
Tensor Product of Hilbert Spaces
- Tensor product of Hilbert spaces forms foundation for spatial tensor product
- Construction involves completing algebraic tensor product with respect to inner product
- Inner product on tensor product defined by extending linearly from elementary tensors
- Resulting space isomorphic to space of Hilbert-Schmidt operators between dual spaces
- Tensor product of Hilbert spaces crucial in quantum mechanics (composite systems)
Tensor Product of Operators and Applications
- Tensor product of operators defined on tensor product of Hilbert spaces
- Construction extends linearly from action on elementary tensors
- Preserves important properties like self-adjointness and positivity
- Tensor product of operators used to study joint spectra and spectral theory
- Applications in quantum information theory (entanglement, quantum channels)