next up previous contents index
Next: Tensor Products Up: Operator algebras Previous: Operator algebras   Contents   Index


Examples

In the sequel we will equip the spaces $ \ell_p$ ( $ 1 \leq p \leq \infty$) with the pointwise product and consider them as Banach algebras. We will further consider the Schatten classes $ S_p$ ( $ 1 \leq p \leq \infty$) endowed with either the usual multiplication or the Schur product.
  1. The space $ \ell_2$ [BLM95, Thm. 2.1]

    With the following operator space structures , $ \ell_2$ is completely isometrically isomorphic to an operator algebra: $ R_{\ell_2}$, $ C_{\ell_2}$, $ {\mathit{OH}_{\ell_2}}$, $ R_{\ell_2} \cap C_{\ell_2}$, $ \mathit{MAX}_{\ell_2}$.
    More generally: The space $ \ell_2$ is completely isometrically isomorphic to an operator algebra, if endowed with an operator space structure which dominate dominates both $ R_{\ell_2}$ and $ C_{\ell_2}$.

    With the following operator space structures, $ \ell_2$ is not completely isomorphic to an operator algebra: $ R_{\ell_2}+C_{\ell_2}$, $ \mathit{MIN}_{\ell_2}$.
    More generally: The space $ \ell_2$ is not completely isomorphic to an operator algebra, if endowed with an operator space structure which is dominate dominated by both $ R_{\ell_2}$ and $ C_{\ell_2}$.

  2. The spaces27 $ {\mathit{MIN}}(\ell_p)$, $ {\mathit{MAX}}(\ell_p)$ and $ O\ell_p=({\mathit{MIN}}(\ell_{\infty}), {\mathit{MAX}}(\ell_1))_{\frac{1}{p}}$

    In the extreme cases $ p=1$ resp. $ p=\infty$, we have two opposite results [BLM95, Thm. 3.1]:

    1. Equipped with any operator space structure, $ \ell_1$ is completely isometrically isomorphic to an operator algebra.
    2. $ {\mathit{MIN}}(\ell_{\infty})$ is, up to complete isomorphy, the only operator algebra structure on $ \ell_{\infty}$.

    For $ 1 \leq p \leq \infty$ the following holds true (cf. [BLM95, Thm. 3.4]):

    1. $ {\mathit{MIN}}(\ell_p)$ is completely isomorphic to an operator algebra if and only if $ p=1$ or $ p=\infty$.
    2. $ {\mathit{MAX}}(\ell_p)$, in case $ 1 \leq p \leq 2$, is completely isometrically isomorphic to an operator algebra. In all the other cases $ {\mathit{MAX}}(\ell_p)$ is not completely isomorphic to an operator algebra.

    On the contrary, the operator space structure on the $ \ell_p$ spaces obtained via complex interpolation always defines an operator algebra structure. More precisely [BLM95, Cor. 3.3]:
    For each $ 1 \leq p \leq \infty$, $ O\ell_p$ is completely isometrically isomorphic to an operator algebra.

  3. The Schatten classes $ S_p$

    We write $ OS_p$ for the operator space structure defined on $ S_p$ by G. Pisier. This operator space structure is obtained by complex interpolation between $ S_{\infty}=K(\ell_2)$ and $ S_1 = K(\ell_2)^*$.

    (a)
    Let us first consider the usual product on the Schatten classes $ S_p$. Here we have the following negative result [BLM95, Thm. 6.3]: For each $ 1 \leq p < \infty$, the operator space $ OS_p$ with the usual product is not completely isomorphic to an operator algebra.
    (b)
    Consider now the Schur product on the Schatten classes $ S_p$. Here we obtain positive results, even for different operator space structures:
    (b1)
    $ {\mathit{MAX}}(S_p)$ with the Schur product is, in case $ 1 \leq p \leq 2$, completely isometrically isomorphic to an operator algebra [BLM95, Thm. 6.1].
    (b2)
    $ OS_p$ with the Schur product is, in case $ 2 \leq p \leq \infty$, completely isometrically isomorphic to an operator algebra [BLM95, Cor. 6.4].

    Caution is advised: The space $ OS_1$ (and likewise $ OS_1^{\mathrm{op}}$), whether endowed with the usual or the Schur product, is not completely isomorphic to an operator algebra [BLM95, Thm. 6.3].



Footnotes

... spaces27
For the construction of the operator spaces $ O\ell_p$ compare the chapter on complex interpolation .

next up previous contents index
Next: Tensor Products Up: Operator algebras Previous: Operator algebras   Contents   Index
Prof. Gerd Wittstock 2001-01-07