Tensor Products

An operator space tensor product is an operator space whose structure is deduced from the operator space structure of the factors. Operator space tensor products are defined for all operator spaces and have functorial properties. On the tensor product of two fixed operator spaces one usually considers operator space norms which are cross norms .

A lot of spaces, especially spaces of mappings, may be considered as operator space tensor products of simpler ones. The theory of operator space tensor products follows the lines of the theory of tensor products of Banach spaces. But at some points tensor products of operator spaces have new properties not found for tensor products of Banach spaces or even better properties as their counterparts. So in some cases the theory of operator space tensor products gives solutions to problems not solvable within the theory of Banach spaces (cp. [ER90a, Thm. 3.2]).

The Haagerup tensor product $\otimes_h$ has a variety of applications in the theory of operator spaces and completely bounded operators.

The injective operator space tensor product $\stackrel{\scriptscriptstyle \vee}{\otimes}$ is the least²⁸ and the projective operator space tensor product $\stackrel{\scriptscriptstyle \wedge}{\otimes}$ is the greatest ²⁹among all operator space tensor products. [BP91, Prop. 5.10].

On the algebraic tensor product $X \otimes Y$ of operator spaces $X$, $Y$ one can compare an operator space tensor norm $\Vert\cdot\Vert _\alpha$ with the injective tensor norm $\Vert\cdot\Vert _\lambda$ and the projective tensor norm $\Vert\cdot\Vert _\gamma$ of normed spaces:

$$\Vert\cdot\Vert _\lambda \leq \Vert\cdot\Vert _{\vee,1} \leq \Ve... ...rt _{\alpha,1} \leq \Vert\cdot\Vert _{\wedge,1} \leq \Vert\cdot\Vert _\gamma.$$

Footnotes

... least²⁸

i.e. the injective operator space tensor norm is minimal among all operator space tensor norms.

... greatest²⁹

i.e. the projective operator space tensor norm is maximal among all operator space tensor norms.