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)