Polynomials in operator space theory: matrix ordering and algebraic aspects

TL;DR

This paper extends the $bbbb-theory of operator spaces to matrix ordered spaces and Banach $*$-algebras, introducing cones related to $bbbb for tensor products and analyzing ideal structures.

Contribution

It generalizes the $bbbb-theory to new algebraic and ordered contexts, including matrix regular operator spaces and operator systems.

Findings

Introduces cones related to $bbbb that respect matricial structures.

Discusses the ideal structure of $bbbb$-tensor products of $C^*$-algebras.

Extends the tensor norm concepts to matrix ordered spaces and Banach $*$-algebras.

Abstract

We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $\lambda$ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of $\lambda$-tensor product of $C^*$-algebras has also been discussed.

Equations

$${\lambda }_k:M_k\times \cdots \times M_k\to M_{\tau (k)},$$

$${\varphi }_{{\lambda }_k}:={\lambda }_k\otimes \varphi :M_k(V_1)\otimes \cdots \otimes M_k(V_m)\to M_{\tau (k)}(W),$$

$$({\alpha }_1\otimes v_1,\cdots ,{\alpha }_m\otimes v_m)\mapsto {\lambda }_k({\alpha }_1,\cdots ,{\alpha }_m)\otimes \varphi (v_1,v_2,\cdots ,v_m).$$

$$\Vert \varphi {\Vert }_{cb,\lambda }:=\underset{k\in \mathbb{N}}{\sup }\{\Vert {\varphi }_{{\lambda }_k}(x_1,\dots ,x_m){\Vert }_{M_{\tau (k)}(W)}:\Vert x_i{\Vert }_{M_k(V_i)}\le 1\}$$

$$C{B}_{\lambda }(V_1,\dots ,V_m;W):=\{\varphi \in L(V_1,\dots ,V_m;W):\Vert \varphi {\Vert }_{cb,\lambda }<\infty \}.$$

$${\otimes }_{{\lambda }_k}:M_k(V_1)\otimes \cdots \otimes M_k(V_m)\to M_{\tau (k)}({\otimes }_{i=1}^m{V}_i),$$

$$({\alpha }_1\otimes v_1,\cdots ,{\alpha }_m\otimes v_m)\mapsto {\lambda }_k({\alpha }_1,\cdots ,{\alpha }_m)\otimes v_1\otimes v_2\otimes \cdots \otimes v_m.$$

$$\Vert u{\Vert }_{\lambda ,k}=\inf \{\Vert \alpha \Vert \Vert v_1\Vert \Vert v_2\Vert \cdots \Vert v_m\Vert \Vert \beta \Vert \}$$

$$S\lambda_{p}(a_{j_{1}},\cdots,a_{j_{m}})T=\left\{\begin{array}[]{ll}\varepsilon_{j}^{[k]}&\qquad\text{if}\;\;j_{1}=j_{2}=\cdots=j_{m}=j,\\ 0&\qquad\text{otherwise.}\end{array}\right.$$

$$\displaystyle\begin{split}&P\lambda_{r+s}(\varepsilon_{i_{1},j_{1}}^{[r+s]},\ldots,\varepsilon_{i_{m},j_{m}}^{[r+s]})P^{*}\\ &=\mathrm{diag}\Big{(}\lambda_{r}(\varepsilon_{i_{1},j_{1}}^{[r]},\cdots,\varepsilon_{i_{m},j_{m}}^{[r]}),\lambda_{s}(\varepsilon_{i_{1}-r,j_{1}-r}^{[s]},\cdots,\varepsilon_{i_{m}-r,j_{m}-r}^{[s]})\Big{)}.\end{split}$$

$$\gamma =P{\lambda }_p(I_p,\cdots ,I_p,\underset{\text{j-th position}}{\underbrace{\gamma }},I_p,\cdots ,I_p)Q$$

$${\lambda }_p({\alpha }_1,\cdots ,{\alpha }_m)\otimes {\lambda }_q({\beta }_1,\cdots ,{\beta }_m)=S{\lambda }_{pq}({\alpha }_1\otimes {\beta }_1,\cdots ,{\alpha }_m\otimes {\beta }_m)T$$

$${\Phi }^{(j)}:(V_1\otimes \cdots \otimes M_p(V_j)\otimes \cdots \otimes V_m,\Vert \cdot {\Vert }_{\lambda })\to M_p({\otimes }_{\lambda }{V}_i)$$

$$v_1\otimes \cdots \otimes (\alpha \otimes v_j)\otimes \cdots v_m\mapsto \alpha \otimes (v_1\otimes \cdots \otimes v_m)$$

$${\lambda }_r({\epsilon }_{i_1,j_1}^{[r]},{\epsilon }_{i_2,j_2}^{[r]},\cdots ,{\epsilon }_{i_m,j_m}^{[r]})={\lambda }_r({\epsilon }_{j_1,i_1}^{[r]},{\epsilon }_{j_2,i_2}^{[r]},\cdots ,{\epsilon }_{j_m,i_m}^{[r]}{)}^{\ast }\in M_{\tau (r)},$$

$$\displaystyle\begin{split}&P\lambda_{2r}(\varepsilon_{i_{1},j_{1}}^{[2r]},\cdots,\varepsilon_{i_{m},j_{m}}^{[2r]})P^{*}\\ &=\mathrm{adiag}\big{(}\lambda_{r}(\varepsilon_{i_{1},j_{1}-r}^{[r]},\cdots,\varepsilon_{i_{1},j_{1}-r}^{[r]}),\lambda_{r}(\varepsilon_{i_{1}-r,j_{1}}^{[r]},\cdots,\varepsilon_{i_{m}-r,j_{m}}^{[r]})\big{)};\end{split}$$

$$\stackrel{p_1\dots p_m}{{\otimes }_{{\lambda }_r}}=\stackrel{p_1\dots p_m}{\otimes }\otimes {\lambda }_r:M_r(M_{p_1})\otimes M_r(M_{p_2})\otimes \dots \otimes M_r(M_{p_m})\to M_{\tau (r)}(M_{p_1{p}_2\dots p_m})$$

$$\stackrel{p_1\dots p_m}{\otimes }:M_{p_1}\times M_{p_2}\times \dots \times M_{p_m}\to M_{p_1{p}_2\dots p_m}$$

$$({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m)\mapsto {\alpha }_1\otimes {\alpha }_2\otimes \dots \otimes {\alpha }_m,$$

$$\stackrel{p_1\dots p_m}{{\otimes }_{{\lambda }_r}}({\alpha }_1\otimes {\beta }_1,\dots ,{\alpha }_m\otimes {\beta }_m)={\lambda }_r({\beta }_1\otimes \dots \otimes {\beta }_m)\otimes {\alpha }_1\otimes \dots \otimes {\alpha }_m\in M_{\tau (r)p_1{p}_2\dots p_m}^{+},$$

$$\big{(}\alpha\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\beta\big{)}^{*}=\beta^{*}\otimes_{\lambda_{r}}\big{(}(v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*}\big{)}\alpha^{*},$$

$$\displaystyle\begin{split}&\begin{pmatrix}\otimes_{\lambda_{r}}(u^{(1)},u^{(2)},\ldots,u^{(m)})&\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\\ \otimes_{\lambda_{r}}\big{(}(v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*}\big{)}&\otimes_{\lambda_{r}}(\tilde{u}^{(1)},\tilde{u}^{(2)},\ldots,\tilde{u}^{(m)})\end{pmatrix}\\ &=P\otimes_{\lambda_{2r}}\Bigg{(}\begin{pmatrix}u^{(1)}&v^{(1)}\\ (v^{(1)})^{*}&\tilde{u}^{(1)}\end{pmatrix},\ldots,\begin{pmatrix}u^{(m)}&v^{(m)}\\ (v^{(m)})^{*}&\tilde{u}^{(m)}\end{pmatrix}\Bigg{)}P^{*}.\end{split}$$

$$\displaystyle\big{(}\otimes_{\mu_{p}}(\phi^{(1)},\ldots,\phi^{(m)})\big{)}_{n}(\alpha\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\alpha^{*})\in M_{np_{1}\ldots p_{m}}^{+}.$$

$$\displaystyle\big{(}(\phi^{(1)}\otimes\ldots\otimes\phi^{(m)})\big{)}_{n}(\alpha\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\alpha^{*})\in M_{np_{1}\ldots p_{m}}^{+}.$$

$$\displaystyle\begin{split}&\begin{pmatrix}\otimes_{\lambda_{r}}(u^{(1)},u^{(2)},\ldots,u^{(m)})&\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\\ \otimes_{\lambda_{r}}((v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*})&\otimes_{\lambda_{r}}(\tilde{u}^{(1)},\tilde{u}^{(2)},\ldots,\tilde{u}^{(m)})\end{pmatrix}\\ &=\mathrm{diag}\big{(}\otimes_{\lambda_{r}}(u^{(1)},u^{(2)},\ldots,u^{(m)}),\otimes_{\lambda_{r}}(\tilde{u}^{(1)},\tilde{u}^{(2)},\ldots,\tilde{u}^{(m)}\big{)}+\\ &\qquad\mathrm{adiag}\big{(}\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)}),\otimes_{\lambda_{r}}((v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*})\big{)}.\end{split}$$

$$x^{(t)}_{k,l}=\left\{\begin{array}[]{ll}u^{(t)}_{k,l}&\qquad\text{if}\;\;(k,l)\in R_{1},\\ \tilde{u}^{(t)}_{k,l}&\qquad\text{if}\;\;(k,l)\in R_{4}\end{array}\right\}\>\;\text{and}\;\;y^{(t)}_{k,l}=\left\{\begin{array}[]{ll}v^{(t)}_{k,l}&\qquad\text{if}\;\;(k,l)\in R_{2},\\ (v^{(t)})^{*}_{k,l}&\qquad\text{if}\;\;(k,l)\in R_{3}\end{array}\right\}.$$

$$\displaystyle\begin{split}&\mathrm{adiag}\big{(}\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)}),0)\\ &=\mathrm{adiag}\big{(}\sum_{(k_{m},l_{m})\in R_{2}}\cdots\sum_{(k_{1},l_{1})\in R_{2}}\lambda_{r}(\varepsilon^{[r]}_{k_{1},l_{1}-r},\cdots,\varepsilon^{[r]}_{k_{m},l_{m}-r})\otimes v_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes v_{k_{m},l_{m}}^{(m)},0\big{)}\\ &=\sum_{(k_{m},l_{m})\in R_{2}}\cdots\sum_{(k_{1},l_{1})\in R_{2}}\mathrm{adiag}\big{(}\lambda_{r}(\varepsilon^{[r]}_{k_{1},l_{1}-r},\cdots,\varepsilon^{[r]}_{k_{m},l_{m}-r}),0\big{)}\otimes y_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes y_{k_{m},l_{m}}^{(m)}\\ &\overset{\text{{(O2)}}}{=}\sum_{(k_{m},l_{m})\in R_{2}}\cdots\sum_{(k_{1},l_{1})\in R_{2}}P\lambda_{2r}(\varepsilon_{k_{1},l_{1}}^{[2r]},\cdots,\varepsilon_{k_{m},l_{m}}^{[2r]})P^{*}\otimes y_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes y_{k_{m},l_{m}}^{(m)}\end{split}$$

$$\displaystyle\begin{split}&\mathrm{adiag}\big{(}0,\otimes_{\lambda_{r}}((v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*}))\\ &=\mathrm{adiag}\big{(}0,\sum_{(k_{m},l_{m})\in R_{3}}\cdots\sum_{(k_{1},l_{1})\in R_{3}}\lambda_{r}(\varepsilon^{[r]}_{k_{1}-r,l_{1}},\cdots,\varepsilon^{[r]}_{k_{m}-r,l_{m}})\otimes(v_{k_{1},l_{1}}^{(1)})^{*}\otimes\cdots\otimes(v_{k_{m},l_{m}}^{(m)})^{*}\big{)}\\ &=\sum_{(k_{m},l_{m})\in R_{3}}\cdots\sum_{(k_{1},l_{1})\in R_{3}}\mathrm{adiag}\big{(}0,\lambda_{r}(\varepsilon^{[r]}_{k_{1}-r,l_{1}},\cdots,\varepsilon^{[r]}_{k_{m}-r,l_{m}}\big{)}\otimes y_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes y_{k_{m},l_{m}}^{(m)}\\ &\overset{\text{{(O2)}}}{=}\sum_{(k_{m},l_{m})\in R_{3}}\cdots\sum_{(k_{1},l_{1})\in R_{3}}P\lambda_{2r}(\varepsilon_{k_{1},l_{1}}^{[2r]},\cdots,\varepsilon_{k_{m},l_{m}}^{[2r]})P^{*}\otimes y_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes y_{k_{m},l_{m}}^{(m)}\end{split}$$

$$\displaystyle\begin{split}&\mathrm{adiag}\big{(}\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)}),\otimes_{\lambda_{r}}((v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*})\big{)}\\ &=\sum_{(k_{m},l_{m})\in R_{2}\cup R_{3}}\cdots\sum_{(k_{1},l_{1})\in R_{2}\cup R_{3}}P\lambda_{2r}(\varepsilon_{k_{1},l_{1}}^{[2r]},\cdots,\varepsilon_{k_{m},l_{m}}^{[2r]})P^{*}\otimes y_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes y_{k_{m},l_{m}}^{(m)}\end{split}$$

$$\displaystyle\begin{split}&\mathrm{diag}\big{(}\otimes_{\lambda_{r}}(u^{(1)},u^{(2)},\ldots,u^{(m)}),\otimes_{\lambda_{r}}(\tilde{u}^{(1)},\tilde{u}^{(2)},\ldots,\tilde{u}^{(m)}\big{)}\\ &=\sum_{(k_{m},l_{m})\in R_{1}\cup R_{4}}\cdots\sum_{(k_{1},l_{1})\in R_{1}\cup R_{4}}P\lambda_{2r}(\varepsilon_{k_{1},l_{1}}^{[2r]},\cdots,\varepsilon_{k_{m},l_{m}}^{[2r]})P^{*}\otimes x_{k_{1},l_{1}}^{(1)}\otimes\cdots\otimes x_{k_{m},l_{m}}^{(m)}\end{split}$$

$$\displaystyle\begin{split}&\begin{pmatrix}\otimes_{\lambda_{r}}(u^{(1)},u^{(2)},\ldots,u^{(m)})&\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\\ \otimes_{\lambda_{r}}((v^{(1)})^{*},(v^{(2)})^{*},\ldots,(v^{(m)})^{*})&\otimes_{\lambda_{r}}(\tilde{u}^{(1)},\tilde{u}^{(2)},\ldots,\tilde{u}^{(m})\end{pmatrix}\\ &=P\Big{(}\sum_{(k_{m},l_{m})\in\cup_{i=1}^{4}R_{i}}\cdots\sum_{(k_{1},l_{1})\in\cup_{i=1}^{4}R_{i}}\otimes_{\lambda_{2r}}\big{(}\varepsilon_{k_{1},l_{1}}^{[2r]}\otimes x_{k_{1},l_{1}}^{(1)}+\varepsilon_{k_{1},l_{1}}^{[2r]}\otimes y_{k_{1},l_{1}}^{(1)},\cdots\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdots,\varepsilon_{k_{m},l_{m}}^{[2r]}\otimes x_{k_{m},l_{m}}^{(m)}+\varepsilon_{k_{m},l_{m}}^{[2r]}\otimes y_{k_{m},l_{m}}^{(m)}\big{)}\Big{)}P^{*}\\ &=P\otimes_{\lambda_{2r}}\Bigg{(}\begin{pmatrix}u^{(1)}&v^{(1)}\\ (v^{(1)})^{*}&\tilde{u}^{(1)}\end{pmatrix},\ldots,\begin{pmatrix}u^{(m)}&v^{(m)}\\ (v^{(m)})^{*}&\tilde{u}^{(m)}\end{pmatrix}\Bigg{)}P^{*}.\end{split}$$

$$\displaystyle\begin{split}&(\otimes_{\mu_{p}}(\phi^{(1)},\ldots,\phi^{(m)})_{\tau_{\mu}(p)}(\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\cdots,v^{(m)}))\\ &=\sum_{k_{m},l_{m}}\ldots\sum_{k_{1},l_{1}}(\otimes_{\mu_{p}}(\phi^{(1)},\ldots,\phi^{(m)})_{\tau_{\mu}(p)}\big{(}\lambda_{r}(\varepsilon_{k_{1},l_{1}},\ldots,\varepsilon_{k_{m},l_{m}})\otimes v^{(1)}_{k_{1},l_{1}}\otimes\ldots\otimes v^{(m)}_{k_{m},l_{m}}\big{)}\\ &=\sum_{k_{m},l_{m}}\ldots\sum_{k_{1},l_{1}}\lambda_{r}(\varepsilon_{k_{1},l_{1}},\ldots,\varepsilon_{k_{m},l_{m}})\otimes\big{(}\otimes_{\mu_{p}}(\phi^{(1)},\ldots,\phi^{(m)})(v^{(1)}_{k_{1},l_{1}}\otimes\ldots\otimes v^{(m)}_{k_{m},l_{m}})\big{)}\\ &=\sum_{k_{m},l_{m}}\ldots\sum_{k_{1},l_{1}}\lambda_{r}(\varepsilon_{k_{1},l_{1}},\ldots,\varepsilon_{k_{m},l_{m}})\otimes\mu_{p}\big{(}\phi^{(1)}(v^{(1)}_{k_{1},l_{1}})\otimes\ldots\otimes\phi^{(m)}(v^{(m)}_{k_{m},l_{m}})\big{)}\\ &=\sum_{k_{m},l_{m}}\ldots\sum_{k_{1},l_{1}}(\mu_{p})_{\lambda_{r}}\big{(}\varepsilon_{k_{1},l_{1}}\otimes\phi^{(1)}(v^{(1)}_{k_{1},l_{1}}),\ldots,\varepsilon_{k_{m},l_{m}}\otimes\phi^{(m)}(v^{(m)}_{k_{m},l_{m}})\big{)}\\ &=\sum_{k_{m},l_{m}}\ldots\sum_{k_{1},l_{1}}(\mu_{p})_{\lambda_{r}}\big{(}\phi^{(1)}_{r}(\varepsilon_{k_{1},l_{1}}\otimes v^{(1)}_{k_{1},l_{1}}),\ldots,\phi^{(m)}_{r}(\varepsilon_{k_{m},l_{m}}\otimes v^{(m)}_{k_{m},l_{m}}))\big{)}\\ &=(\mu_{p})_{\lambda_{r}}\big{(}\phi^{(1)}_{r}(v^{(1)}),\ldots,\phi^{(m)}_{r}(v^{(m)})\big{)}\\ &\in M_{\tau(r)p_{1}p_{2}\ldots p_{m}}^{+}\end{split}$$

Full text

Mathematics Subject Classification (2010): Primary 46L06, 46L07; Secondary 46L05, 47L25

11footnotetext: Corresponding author

1. Introduction

$C^{*}$-algebras are rich objects as they come along with matrix norms that are not only uniquely related to algebraic structure but are also known to have matricial cone structures being closely related to those norm. Although, operator spaces and their tensor products are primarily defined in terms of appropriate matrix norms, over the years it has been observed that some operator space tensor products of $C^{*}$-algebras still possess few algebraic properties that can be characterized in terms of the individual algebras ([1, 14]). Regarding ordering, although operator spaces may possess some order structure unrelated to the matrix norms, it was Schreiner [18] who defined matrix regular operator spaces to be the spaces where there is a relationship between norm and order. In matrix regular operator spaces, there are enough positive elements so that each element can be written as a linear combination of positive elements. Recently introduced tensor product theory for (unital) operator systems category ([13]) shows that this matrix order-matrix norm relation is successfully carried over.

Defant and Wiesner in [4] (see also [19]) have given a $\lambda$-theory which generalizes the definitions of the projective, Haagerup and Schur tensor norm for operator spaces. It is thus natural to ask for appropriate matrix ordering and algebraic structure that is compatible with this generalized $\lambda$-theory. In [9] and [17], the projective and Schur operator space tensor product of matrix ordered operator spaces are shown to be matrix ordered respectively. Further, Han in [8] successfully introduced cones at each matrix level of the tensor product of operator spaces that are closely related to projective and injective operator space tensor norms thereby, constructing two extremal tensor products of matrix regular operator space.

Section 2 discusses the prerequisites. Next, we introduce conditions (O1)-(O3) in Section 3 that enables generalization of Han’s ([8]) operator space tensor product matrix regularity results to $\lambda$-theory of operator spaces. In Section 4, we show that the cones defined in Section 3 also preserve the operator system structure. Finally in Section 5, we show that the techniques to study ideal structure of operator space tensor product of $C^{*}$-algebras can be extended to $\lambda$-theory.

2. Preliminaries

2.1. The $\lambda$-theory [4][19]

Let $V_{1},V_{2},\ldots,V_{m};W$ be operator spaces and let $\phi$ be an $m$-linear mapping on $V_{1}\times V_{2}\times\cdots\times V_{m}$ into $W$. Given a sequence of matrix products $\lambda=(\lambda_{k}),$ for each $k$, $\lambda_{k}$ is an $m$-linear mapping:

[TABLE]

where $\tau(k)\in\mathbb{N}$ is a natural number only depending on $k$, tensorizing $\lambda_{k}$ with $\phi$ leads to the $m$-linear mapping

[TABLE]

Further

[TABLE]

and

[TABLE]

Since $m$-fold tensor product on $V_{1}\times V_{2}\times\cdots\times V_{m}$ is an $m$-linear map onto $\otimes_{i=1}^{m}V_{i}$, the natural map obtained as above by tensorizing with $\lambda_{k}$ is represented by $\otimes_{\lambda_{k}}$:

[TABLE]

In [4, 19], a tensor norm $\lambda$ was defined as:

[TABLE]

for any element $u\in M_{k}(\displaystyle\otimes_{i=1}^{m}V_{i})$, where the infimum is taken over arbitrary decompositions $u=\alpha\otimes_{\lambda_{j}}(v_{1},v_{2},\cdots,v_{m})\beta$, $\alpha\in M_{k,\tau(j)}$, $\beta\in M_{\tau(j),k}$, $v_{t}\in M_{j}(V_{t})$.

Keeping the notations from [19, 4] unchanged, e.g. $\varepsilon_{i,j}:=\varepsilon^{[k,l]}_{i,j}\in M_{k,l}$ denotes the matrix which is 1 in the $(i,j)$-th coordinate and zero elsewhere, $\varepsilon^{[k]}_{i,j}:=\varepsilon^{[k,k]}_{i,j},\;\varepsilon_{i}:=\varepsilon_{i,i},\;\varepsilon^{[k]}_{i}:=\varepsilon^{[k,k]}_{i,i}$ and $\varepsilon^{[k,l]}_{i,j}=0$ if $(i,j)\notin\{1,\ldots,k\}\times\{1,\ldots,l\}$, we state the three technical conditions (E1)-(E3) that were isolated on the family $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ to assure that the $\|\cdot\|_{\lambda,k}$, generates an operator space structure on $V_{1}\otimes V_{2}\otimes\cdots\otimes V_{m}$ [4, Proposition 4.1]:

• (E1)

For all $k\in\mathbb{N}$ there exist $p\in\mathbb{N}$ and matrices $S\in M_{k,\tau(p)}$ , $T\in M_{\tau(p),k}$, $a_{1},\cdots,a_{k}\in M_{p}$ such that for all $j_{1},\cdots,j_{m}\in\{1,\cdots,k\}$:

[TABLE]

• (E2)

For all $r,s\in\mathbb{N}$ there exist matrices $P\in M_{\tau(r)+\tau(s),\tau(r+s)}$, with $\|P\|\leq 1$ such that for all $(i_{k},j_{k})\in\{1,\cdots,r\}^{2}\cup\{r+1,\cdots,r+s\}^{2}$ with $1\leq k\leq m$:

[TABLE]

• (E3)

$\lambda_{1}(1,1,\cdots,1)=1$ and $\displaystyle\sup_{k\in\mathbb{N}}\|\lambda_{k}\|<\infty$.

If in addition $\lambda$ satisfies:

(N1)$\qquad\tau(1)=1\qquad\qquad\text{and}\qquad\qquad$(N2)$\quad\|\lambda_{j}\|=1$ for all $j\in\mathbb{N},$

then $\displaystyle\otimes_{\lambda}\mathbb{C}=\mathbb{C}$ completely isometric [19, Proposition 4.13].

For $j\in\{1,\ldots,m\}$, if $\lambda$ further satisfy conditions:

• (W1)

For all $\gamma\in M_{p}$ there exists matrices $P\in M_{p,\tau(p)}$, $Q\in M_{\tau(p),p}$ with $\|P\|,\|Q\|\leq 1$ such that

[TABLE]

• (W2)

For all $\alpha_{1},\cdots,\alpha_{m}\in M_{p}$, $\beta_{1},\cdots,\beta_{m}\in M_{q}$ there exist matrices $S\in M_{\tau(p)\tau(q),\tau(pq)},T\in M_{\tau(pq),\tau(p)\tau(q)}$ with $\|S\|,\|T\|\leq 1$ such that

[TABLE]

then the mapping

[TABLE]

is completely contractive [19, Proposition 12.2].

If $\lambda$ satisfies (N1)-(N2), (E1)-(E3) and (W1)-(W2) then $\otimes^{\lambda}V_{i}$, the completion of $\otimes_{\lambda}V_{i}$ with respect to $\|\cdot\|_{\lambda}$ norm, is an operator space tensor product denoted by $\lambda$-operator space tensor product in the sense of [3].

The Kronecker product, matrix product and mixed product fulfill all the above conditions.

We assume throughout that $\lambda$ satisfies all the prescribed conditions.

2.2. Matrix regular operator space and operator systems

An operator space $V$ is called a matrix ordered operator space if:

1. (1)

$(V,\{M_{n}(V)^{+}\}_{n=1}^{\infty})$ is a matrix ordered vector space i.e. for each $n\in\mathbb{N}$, $M_{n}(V)$ is a $*$-ordered vector space with cone $M_{n}(V)^{+}$ and $A\in\mathbb{M}_{n,m}$ implies $A^{*}M_{n}(V)^{+}A\subseteq M_{m}(V)^{+}$. 2. (2)

the $*$-operation is an isometry on $M_{n}(V)$. 3. (3)

the cones $M_{n}(V)^{+}$ are closed.

A matrix ordered operator space $V$ is called matrix regular [18, Definition 3.1.9] if for each $n\in\mathbb{N}$ and for all $v\in M_{n}(V)_{sa}$, the following conditions hold :

1. (1)

$u\in M_{n}(V)^{+}$ and $-u\leq v\leq u$ implies that $\|v\|_{n}\leq\|u\|_{n}$. 2. (2)

$\|v\|_{n}\leq 1$ implies that there exists $u\in M_{n}(V)^{+}$ such that $\|u\|_{n}\leq 1$ and $-u\leq v\leq u$.

Next result from [18] giving a necessary and sufficient for a matrix ordered operator space $V$ to be matrix regular is quite useful:

Theorem 2.1**.**

[18, Theorem 3.4]** A matrix ordered operator space V is matrix regular if and only if the following condition holds: for all $x\in M_{n}(V)$, $\|x\|_{n}<1$ if and only if there exist $a,d\in M_{n}(V)^{+}$, $\|a\|_{n}<1$ and $\|d\|_{n}<1$, such that $\begin{pmatrix}a&x\\ x^{*}&d\end{pmatrix}\in M_{2n}(V)^{+}$.

The positive cone of a matrix regular operator space is always proper.

Adopting the methodology of [8], the norms on matrix regular operator spaces are not assumed to be complete.

For a matrix ordered operator space $V$ and its dual space $V^{*},$ the positive cone on $M_{n}(V^{*})$ for each $n\in\mathbb{N}$ is defined by $M_{n}(V^{*})^{+}=CB(V,M_{n})\cap CP(V,M_{n})$. The operator space dual $V^{*}$ with this positive cone is a matrix ordered operator space [18, Corollary 3.2].

An (abstract) operator system ([13, Definition 2.2]) is a triple $(V,\{\mathcal{C}_{n}\}_{n=1}^{\infty},e$), where $V$ is a complex $*$-vector space, $\{\mathcal{C}_{n}\}_{n=1}^{\infty}$ is a matrix ordering on $V,$ and $e\in V_{sa}$ is an Archimedean matrix order unit, i.e. for all $v\in M_{n}(V)_{sa}$,

1. (1)

there exists a real number $r>0$ such that $re_{n}>v$ and 2. (2)

for each $n\in\mathbb{N}$ and $e_{n}=\begin{pmatrix}e&&\\ &\ddots&\\ &&e\end{pmatrix}$, $se_{n}+v\in\mathcal{C}_{n}$ for all $s>0$ implies $v\in\mathcal{C}_{n}.$

3. $\lambda$-theory and Matrix regularity

In this section, we provide three additional conditions on $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ to introduce an order structure to $\lambda$-theory that preserves matrix regularity. Further, using our conditions (O1)-(O3) defined below, we prove that the results of [8] hold true in a more general setting introduced by [4, 19].

For a sequence $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ of $m$-linear mappings $\lambda_{k}\in L(^{m}M_{k};M_{\tau(k)})$ consider the following three properties:

(O1) For each $r\in\mathbb{N}$,

[TABLE]

for all $(i_{k},j_{k})\in\{1,\ldots,r\}\times\{1,\ldots,r\}$, and $k=1,2,\dots m$.

(O2) For $r\in\mathbb{N}$, the permutation matrix $P\in M_{2\tau(r),\tau(2r)}$ with $\|P\|\leq 1$ obtained in (E2) and $(i_{k},j_{k})\in R\cup S$, where $R:=\{1,\cdots,r\}\times\{r+1,r+2,\cdots,2r\}$ and $S:=\{r+1,r+2,\cdots,2r\}\times\{1,\cdots,r\}$

[TABLE]

adiag being an anti-diagonal matrix, where all the entries are zero except those on the diagonal going from the upper right corner to the lower left corner.

(O3) For each $r\in\mathbb{N}$, the map

[TABLE]

obtained by tensorizing $\lambda_{r}$ with the Kronecker product on matrix algebras $M_{p_{1}},\ldots,M_{p_{m}}$

[TABLE]

[TABLE]

is positive for all $p_{i}\in\mathbb{N}$ $(i=1,2,\ldots,m)$. Thus,

[TABLE]

whenever $\alpha_{i}\otimes\beta_{i}\in(M_{p_{i}}\otimes M_{r})^{+}$, $p_{i}\in\mathbb{N}$ $(i=1,2,\ldots m).$

Recall from [18, Proposition 4.1] (see also [19, Proposition 4.2]), given a sequence $\lambda=(\lambda_{n})$ of $m$-linear maps and operator spaces $V_{1},V_{2},\ldots,V_{m}$ any element $u\in M_{k}(V_{1}\otimes\ldots V_{m})$ has a representation $u=\alpha\otimes_{\lambda_{r}}(v_{1},v_{2},\ldots,v_{m})\beta$ where $\alpha\in M_{n,\tau(r)},\beta\in M_{\tau(r),n},v_{i}\in M_{r}(V_{i}),r\in\mathbb{N}.$

Next, we analyze the above conditions in view of their applications to matrix ordered spaces:

Lemma 3.1**.**

Let $\lambda=(\lambda_{n})$ be sequence of $m$-linear maps and $V_{1},\ldots V_{m}$ be matrix ordered operator spaces. For any $\alpha\otimes_{\lambda_{r}}(v^{(1)},v^{(2)},\ldots,v^{(m)})\beta\in M_{n}(\otimes_{\lambda}V_{i})$; $\alpha\in M_{n,\tau(r)},\beta\in M_{\tau(r),n},v^{(i)}\in M_{r}(V_{i}),i=1,\ldots,m,r\in\mathbb{N},$ we have:

1. (i)

If $\lambda$ satisfies (O1), then $*$-map defined as

[TABLE]

is a well defined involution. 2. (ii)

If $\lambda$ satisfies (O2), then for $u^{(i)},\tilde{u}^{(i)}\in M_{r}(V_{i})$, $i=1,\ldots,m$

[TABLE] 3. (iii)

For $\lambda$ and $\mu$, if $(\mu_{p})_{\lambda_{r}}$ is a positive map $(p,r\in\mathbb{N})$, then for $v^{(i)}\in M_{r}(V_{i})^{+}$ and completely positive maps $\phi^{(i)}:V_{i}\rightarrow M_{p_{i}}$, $i=1,\ldots,m$, we have

[TABLE]

In particular, if $\lambda$ satisfies (O3),

[TABLE]

Proof.

To obtain (i) one can easily verify that the $*$-operation is conjugate linear and involutive.

1. (ii)

We have

[TABLE]

Setting $R_{1}:=\{1,2,\dots,r\}^{2}$, $R_{2}:=\{1,2,\dots,r\}\times\{r+1,r+2,\ldots,2r\}$,

$R_{3}:=\{r+1,r+2,\ldots,2r\}\times\{1,2,\ldots,r\}$ and $R_{4}:=\{r+1,r+2,\ldots,2r\}^{2}$, let $u^{(t)}:=\displaystyle\sum_{(k_{t},l_{t})\in R_{1}}\varepsilon^{[r]}_{k_{t},l_{t}}\otimes u_{k_{t},l_{t}}^{(t)}$, $v^{(t)}:=\displaystyle\sum_{(k_{t},l_{t})\in R_{2}}\varepsilon^{[r]}_{k_{t},l_{t}-r}\otimes v_{k_{t},l_{t}}^{(t)}$,

$(v^{(t)})^{*}=\displaystyle\sum_{(k_{t},l_{t})\in R_{3}}\varepsilon^{[r]}_{k_{t}-r,l_{t}}\otimes(v_{k_{t},l_{t}}^{(t)})^{*}$ and $\tilde{u}^{(t)}:=\displaystyle\sum_{(k_{t},l_{t})\in R_{4}}\varepsilon^{[r]}_{k_{t}-r,l_{t}}\otimes\tilde{u}_{k_{t},l_{t}}^{(t)}$.

Define $x^{(t)}:=\mathrm{diag}(u^{(t)},\tilde{u}^{(t)})$ and $y^{(t)}:=\mathrm{adiag}(v^{(t)},(v^{(t)})^{*})$, so that

[TABLE]

Then,

[TABLE]

Also,

[TABLE]

Thus,

[TABLE]

Similarly, using (E2),

[TABLE]

Therefore,

[TABLE] 2. (iii)

Note that, if $v^{(t)}:=\displaystyle\sum_{(k_{t},l_{t})\in R}\varepsilon^{[r]}_{k_{t},l_{t}}\otimes v_{k_{t},l_{t}}^{(t)};$ $R:=\{1,\ldots,r\}^{2}$, $t=1,2,\ldots,m$ then,

[TABLE]

so that,

[TABLE]

If $\lambda$ satisfies (O3), $\mu_{p}=\displaystyle\overset{p_{1}\ldots p_{m}}{\otimes}$ gives the desired result.

∎

Verification of Properties (O1)-(O3):

• •

Kronecker product: Property (O1) reduces to

[TABLE]

which is true.

To check for the condition (O3), recall that Kronecker product of two positive matrices is positive, but Kronecker product does not commute, in fact for any square matrices $A$ and $B$, there exists a permutation matrix $S$ such that $B\otimes A=S(A\otimes B)S^{*}$. Therefore, for some suitable permutation matrix $S$ we have:

[TABLE]

whenever $\alpha_{i}\otimes\beta_{i}\in M_{r}(M_{p_{i}})^{+}$, $p_{i}\in\mathbb{N}$ $(i=1,2,\ldots m).$

In order to verify (O2), we use the same notations as in proof of [4, Proposition 4.2], let $\Delta:M_{1}\rightarrow M_{1,m}$, $x\mapsto(x,x,\ldots,x)$ and set

[TABLE]

and let $P=\begin{pmatrix}P_{1}\\ P_{2}\end{pmatrix}$

[TABLE]

• •

Schur product: Here property (O1) takes the form

[TABLE]

which is true.

To check for the condition (O3), recall that Schur product of two positive matrices is positive, for any square matrices $A,B,C\;\text{and}\;D$ of order $n$, $(A\odot B)\otimes(C\odot D)=(A\otimes C)\odot(B\otimes D)$ (see [19, Proposition 10.5]) and there exist a matrix $\mathcal{E}\in M_{n,k}$, $A^{\prime},B^{\prime}\in M_{k}$ such that $(A\otimes B)=\mathcal{E}(A^{\prime}\odot B^{\prime})\mathcal{E}^{*}$. Therefore we have,

[TABLE]

whenever $\alpha_{i}\otimes\beta_{i}\in M_{r}(M_{p_{i}})^{+}$, $p_{i}\in\mathbb{N}$ $(i=1,2,\ldots m).$

Again as for (E2), for $r\in\mathbb{N}$ and $(i_{q},j_{q})\in[\{1,\cdots,r\}\times\{r+1,\cdots,2r\}]\cup[\{r+1,\cdots,2r\}\times\{1,\cdots,r\}]$, let $P:=I_{2r}$, we have

[TABLE]

implying that (O2) holds.

• •

Matrix Product: One can easily see that this product may not satisfy (O1), (O2) and (O3).

• •

Mixed Product: One can mix above listed products to construct a new one, for example [19, Chapter 9], $\lambda=(\lambda_{k})_{k}$ with

[TABLE]

Clearly, it does not satisfy any of the (O1)-(O3) as $\bullet$ does not.

One can similarly talk of $\lambda=(\lambda_{k})_{k}$ with

[TABLE]

which clearly satisfies all the conditions (O1)-(O3).                        ∎

The self-adjoint elements in $M_{n}(\otimes_{\lambda}V_{i})$ have a special representation:

Proposition 3.2**.**

Let $V_{i}$, $1\leq i\leq m$, be matrix ordered operator spaces and let $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence of $m$-multilinear mappings satisfying (O1) and (O2). If $u\in M_{n}(\otimes_{\lambda}V_{i})_{sa},$ then $u$ has a representation:

[TABLE]

where $\alpha\in M_{n,\tau(j)}$, $x_{t}\in M_{j}(V_{t})_{sa}$, $t=1,2,\cdots,m.$ Moreover,

[TABLE]

Proof.

Suppose $u\in M_{n}(\otimes_{\lambda}V_{i})_{sa}$. Given $\epsilon>0$, there exist $\alpha\in M_{n,\tau(j)}$, $\beta\in M_{\tau(j),n}$, $x_{t}\in M_{j}(V_{t})$ such that

[TABLE]

As $u$ is self adjoint by Lemma 3.1(ii), for any $\mu>0$, we have

[TABLE]

where v^{t}=\left(\begin{array}[]{cc}0&x_{t}^{*}\\ x_{t}&0\\ \end{array}\right), and \alpha=\left(\begin{array}[]{cc}\frac{\mu^{-1}\beta^{*}}{\sqrt{2}}&\frac{\mu\alpha}{\sqrt{2}}\\ \end{array}\right), $R=\{1,2,\cdots,r\}\times\{r+1,\cdots,2r\}$ and $S=\{r+1,r+2,\cdots,2r\}\times\{1,\cdots,r\}$.

Thus, $\|u\|_{\lambda,n}\leq\frac{1}{2}(\mu^{2}\|\beta\|^{2}+\mu^{-2}\|\alpha\|^{2})]\|v^{1}\|\|v^{2}\|\cdots\|v^{m}\|$, where for each $t$, $v^{t}$ is a self adjoint element.

Since, $\displaystyle\min_{\mu>0}\frac{1}{2}(\mu^{2}\|\beta\|^{2}+\mu^{-2}\|\alpha\|^{2})=\|\beta\|\|\alpha\|$, given $\delta>0$ we can choose $\mu_{0}>0$ such that $\|\beta\|\|\alpha\|+\delta>\frac{1}{2}(\mu_{0}^{2}\|\beta\|^{2}+\mu_{0}^{-2}\|\alpha\|^{2})$. Therefore, $\|u\|_{\lambda,n}\leq\|\tilde{\alpha}\|^{2}\|v^{1}\|\|v^{2}\|\cdots\|v^{m}\|\leq\|\beta\|\|\alpha\|\|x_{1}\|\|x_{2}\|\cdots\|x_{m}\|$. Thus, we get the desired norm condition. ∎

We are now in a position to define an appropriate cone structure :

Definition 3.3**.**

Let $V_{1},\cdots,V_{m}$ be matrix ordered operator spaces and $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence satisfying (O1)-(O2). We define

1. (a)

$\mathcal{C}_{n}$=$\{\alpha\otimes_{\lambda_{j}}(v_{1},v_{2},\cdots,v_{m})\alpha^{*}:v_{t}\in M_{j}(V_{t})^{+},\alpha\in M_{n,\tau(j)},j\in\mathbb{N},t=1,2,\cdots,m\}\subset M_{n}(\displaystyle\otimes_{\lambda}V_{i})_{sa}$* and* 2. (b)

$M_{n}(\otimes_{\lambda}V_{i})^{+}:=\mathcal{C}_{n}^{-\|\cdot\|_{\lambda,n}}.$**

Proposition 3.4**.**

For matrix ordered operator spaces $V_{i},$ $1\leq i\leq m$, \big{(}\otimes_{i=1}^{m}V_{i},\{\|\cdot\|_{\lambda,n}\}_{n=1}^{\infty},M_{n}(\otimes_{\lambda}V_{i})^{+}\big{)} is a matrix ordered operator space.

Proof.

From Proposition 3.2, the involution is an isometry on $(M_{n}(\otimes_{\lambda}V_{i}))_{sa}$ hence $M_{n}(\otimes_{\lambda}V_{i})^{+}$ is a cone provided $\mathcal{C}_{n}$ is.

Let $u_{1}=\alpha_{1}\otimes_{\lambda_{k}}(v_{1},v_{2},\cdots,v_{m})\alpha^{*}_{1}\in\mathcal{C}_{n_{1}},u_{2}=\alpha_{2}\otimes_{\lambda_{l}}(w_{1},w_{2},\cdots,w_{m})\alpha^{*}_{2}\in\mathcal{C}_{n},$ then using Lemma 3.1(ii),

[TABLE]

where $x_{t}=diag(v_{t},w_{t})\in M_{k+l}(V)^{+}$ and $\alpha=\begin{pmatrix}\alpha_{1}&\alpha_{2}\\ \end{pmatrix}$, hence the family $\{C_{n}\}$ is closed under addition. Now, for $t\geq 0$

[TABLE]

Also, for $\gamma\in M_{m,n}$ and $\alpha\otimes_{\lambda_{k}}(v_{1},v_{2},\cdots,v_{m})\alpha^{*}\in\mathcal{C}_{m}$,

[TABLE]

∎

Remark 3.5**.**

Note that $\begin{pmatrix}u&z\\ z^{*}&u^{\prime}\end{pmatrix}\in\mathcal{C}_{2n}$, implies that

[TABLE]

so using Proposition 3.2

[TABLE]

Motivated by this and Han’s [8, Definition 3.2], we relate a suitable norm to the cone $\mathcal{C}_{n}$ defined above that behaves well with matrix regular operator spaces.

Definition 3.6**.**

Let $V_{i},$ $1\leq i\leq m$ be matrix ordered operator spaces and $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence satisfying (O1)-(O3). Then for $z$ in $M_{n}(\otimes_{\lambda}V_{i})$, we define:

[TABLE]

Note that if $u\in\mathcal{C}_{n}$, then

[TABLE]

Therefore $\|\cdot\|_{\Lambda,n}\leq\|\cdot\|_{\lambda,n}$ on $\mathcal{C}_{n}$. The set \bigg{\{}\max\{\|u\|_{\lambda,n},\|u^{\prime}\|_{\lambda,n}\}:\left(\begin{matrix}u&z\\ z^{*}&u^{\prime}\end{matrix}\right)\in\mathcal{C}_{2n}\bigg{\}} is non empty from Proposition 3.7(i) and (ii) proved below.

Proposition 3.7**.**

If $V_{i},$ $1\leq i\leq m$ are matrix regular operator spaces and $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ a sequence satisfying (O1)-(O3), then

1. (i)

$\mathcal{C}_{n}$* is a proper cone in $M_{n}(\otimes_{i=1}^{m}V_{i})$ for all $n\in\mathbb{N}$.* 2. (ii)

For $z\in M_{n}(\otimes_{i=1}^{m}V_{i})$, there exist elements $u_{1}$, $u_{2}$ in $\mathcal{C}_{n}$ such that $\left(\begin{matrix}u_{1}&z\\ z^{*}&u_{2}\end{matrix}\right)\in\mathcal{C}_{2n}$. 3. (iii)

$\|\cdot\|_{\Lambda,n}$* is a norm on $M_{n}(\otimes_{i=1}^{m}V_{i}).$*

Proof.

1. (i)

Let $z\in\mathcal{C}_{n}\cap-\mathcal{C}_{n}$, then $z=\alpha\otimes_{\lambda_{j}}(v^{1},v^{2},\cdots,v^{m})\alpha^{*}\;\text{with}\;v^{t}\in M_{j}(V_{t})^{+},\alpha\in M_{n,\tau(j)}.$ By Lemma 3.1(iii), for continuous c.p. maps $\phi^{t}:V_{t}\rightarrow M_{k_{t}}$, $t=1,\ldots,m$, we have

[TABLE]

Now each $V_{t}$ being matrix regular its dual $V_{t}^{*}$ is also matrix regular ([18, Corollary 6.7]), hence each completely bounded linear map from $V_{t}$ into a matrix algebra is actually a linear combination of some c.p. maps. Thus, even for c.b. maps $f^{(t)}:V_{t}\rightarrow M_{k_{t}}$, $t=1,\ldots,m$;

[TABLE]

which further implies the operator space injective tensor norm is given by,

[TABLE] 2. (ii)

Using [4, Proposition 4.1], for any $z\in M_{n}(\otimes_{i=1}^{m}V_{i})$ can be written as:

[TABLE]

As each $V_{i}$ is a matrix regular operator space, there exist $v_{t}^{1},v_{t}^{2}\in M_{j}(V_{t})^{+}$ such that

[TABLE]

Using Lemma 3.1(ii), we have

[TABLE] 3. (iii)

$\|\cdot\|_{\Lambda,n}$ clearly satisfies homogeneity.

Let $z_{1},z_{2}\in M_{n}(\otimes_{i=1}^{m}V_{i})$ be any elements, then by (ii) above there exist $u_{i},u_{i}^{\prime}\in\mathcal{C}_{n}$ such that

[TABLE]

From

[TABLE]

it follows that

[TABLE]

Let $\|z\|_{\Lambda,n}=0$. Given $\epsilon>0$, there exist $u,u^{\prime}\in\mathcal{C}_{n}$ such that

[TABLE]

Again by Lemma 3.1(iii), for c.c.p. maps $f^{(t)}:V_{t}\rightarrow M_{k_{t}}$, $t=1,\ldots,m$;

[TABLE]

It follows that

[TABLE]

Thus $(\otimes_{t=1}^{m}f^{(t)})_{n}(z)=0$ and using matrix regularity as in case (i), $\|z\|_{M_{n}(\check{\otimes}V_{i})}=0$, which implies $z=0$.

∎

The positive cones of matrix ordered operator spaces are closed. Therefore, we consider $M_{n}(\otimes_{\Lambda}V_{i})^{+}:=\mathcal{C}_{n}^{-\|\cdot\|_{\Lambda,n}}$. From the definition of $\|\cdot\|_{\Lambda,n}$, since

$\begin{pmatrix}u&z\\ z^{*}&u^{\prime}\end{pmatrix}\in\mathcal{C}_{2n}$ if and only if $\begin{pmatrix}u^{\prime}&z^{*}\\ z&u\end{pmatrix}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}u&z\\ z^{*}&u^{\prime}\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\in\mathcal{C}_{2n}$, we have $\|z^{*}\|_{\Lambda,n}=\|z\|_{\Lambda,n}$. Thus, the involution is an isometry on $(M_{n}(\otimes_{i=1}^{m}V_{i}),\|\cdot\|_{\Lambda,n})$. Hence $M_{n}(\otimes_{\Lambda}V_{i})^{+}$ is a proper cone.

Recall from [19, Definition 4.7] that an operator space matrix norm $\|\cdot\|_{\alpha}$ is said to be $\lambda$-subcross if

[TABLE]

for all $p\in\mathbb{N}$ and $v_{i}\in M_{p}(V_{i}).$ In case of equality the norm is called $\lambda$-cross.

Theorem 3.8**.**

For matrix regular operator spaces $V_{i}$ and $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ satisfying (O1)-(O3), $(\otimes_{i=1}^{m}V_{i},\{\|\cdot\|_{\Lambda,n}\}_{n=1}^{\infty},M_{n}(\otimes_{\Lambda}V_{i})^{+})$ is a matrix regular operator space with $\lambda$-subcross matrix norm.

Proof.

We first prove that $M_{n}(\otimes_{i=1}^{m}V_{i})$ is an operator space with the family $\{\|\cdot\|_{\Lambda,n}\}_{n=1}^{\infty}$ of matrix norms.

Given $z_{1}\in M_{n_{1}}(\otimes_{i=1}^{m}V_{i})$, $z_{2}\in M_{n_{2}}(\otimes_{i=1}^{m}V_{i})$ and $\epsilon>0$, choose $u_{i},u_{i}^{\prime}\in\mathcal{C}_{n}$ such that $\left(\begin{matrix}u_{i}&z_{i}\\ z_{i}^{*}&u_{i}^{\prime}\end{matrix}\right)\in\mathcal{C}_{2n_{i}}$ and $\|u_{i}\|_{\lambda,n_{i}},\|u_{i}^{\prime}\|_{\lambda,n_{i}}<\|z_{i}\|_{\Lambda,n_{i}}+\epsilon$, $i=1,2$.

By definition, there exist representations

[TABLE]

Since $\|\cdot\|_{\lambda,n}$ is an operator space norm, using Ruan’s first condition (M1) [5] for operator space, we have

[TABLE]

As,

[TABLE]

we have,

[TABLE]

Let $z\in M_{n}(\otimes_{i=1}^{m}V_{i})$ and $\alpha,\beta\in M_{m,n}$, then there exist $u,u^{\prime}\in\mathcal{C}_{n}$ such that $\begin{pmatrix}u&z\\ z*&u^{\prime}\end{pmatrix}\in\mathcal{C}_{2n}$ and $\|u\|_{\lambda,n},\|u^{\prime}\|_{\lambda,n}<\|z\|_{\Lambda,n}+\epsilon.$ Assuming, $\|\alpha\|=\|\beta\|$ by homogeneity, since

[TABLE]

we have,

[TABLE]

Hence, \big{(}\otimes_{i=1}^{m}V_{i},\{\|\cdot\|_{\Lambda,n}\}_{n=1}^{\infty}\big{)} is an operator space.

Let $v_{i}\in M_{j}(V_{i})$ with $\|v_{i}\|<1$ for $i=1,\ldots,m$. Then there exist $u_{i},u_{i}^{\prime}\in M_{j}(V_{i})^{+}_{\|\cdot\|\leq 1}$ such that

[TABLE]

Now,

[TABLE]

it follows that

[TABLE]

Therefore, the family of matrix norms $\{\|\cdot\|_{\Lambda,n}\}_{n=1}^{\infty}$ is $\lambda$-subcross.

As $\|u\|_{\Lambda,n}\leq\|u\|_{\lambda,n},$ if $\|z\|_{\Lambda,n}<1$ then there exist $u,u^{\prime}\in\mathcal{C}_{n}$ such that

[TABLE]

Thus,

[TABLE]

and matrix regularity follows. ∎

4. $\lambda$-operator system tensor product

We now prove that the cones $\mathcal{C}_{n}$ associated with $\lambda$ under the conditions (O1)-(O3) also preserve the operator system structure defined in [13]. The techniques are again same as that for the $\max$ operator system tensor product defined in [13].

Theorem 4.1**.**

Let $(\mathcal{S},\{M_{n}(\mathcal{S})^{+}\}_{n=1}^{\infty},1_{\mathcal{S}})$ and $(\mathcal{T},\{M_{n}(\mathcal{T})^{+}\}_{n=1}^{\infty},1_{\mathcal{T}})$ be operator systems. The family $\{\mathcal{C}_{n}\}_{n=1}^{\infty}$ associated with a sequence $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ satisfying (O1)-(O3), is a matrix ordering on $\mathcal{S}\otimes\mathcal{T}$ with order unit $1_{\mathcal{S}}\otimes 1_{\mathcal{T}}$.

Proof.

From Proposition 3.7, we know that $\{\mathcal{C}_{n}\}_{n=1}^{\infty}$ is a family of proper compatible cones on $M_{n}(\otimes_{\lambda}\mathcal{S}_{i})$. We only need to check that $1\otimes 1$ is a matrix order unit.

Let $\alpha\otimes_{\lambda_{j}}(s,t)\alpha^{*}\in(\mathcal{S}_{1}\otimes\mathcal{S}_{2})_{sa}$ with s\in\big{(}M_{j}(\mathcal{S})\big{)}_{sa} , t\in\big{(}M_{j}(\mathcal{T})\big{)}_{sa} and $\alpha\in M_{1,\tau(j)},$ $1_{\mathcal{S}}$ and $1_{\mathcal{T}}$ being Archimedean order unit for $\mathcal{S}$ and $\mathcal{T}$ respectively, then we can find $K$ large enough such that

[TABLE]

[TABLE]

So that,

[TABLE]

Further,

[TABLE]

which proves that $1_{\mathcal{S}}\otimes 1_{\mathcal{T}}$ is an order unit. Similarly, one can prove that $1_{\mathcal{S}}\otimes 1_{\mathcal{T}}$ is in fact a matrix order unit. ∎

Definition 4.2**.**

Let $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ fulfills conditions (O1)-(O3), and

[TABLE]

be the Archimedeanization([16]) of the matrix ordering $\mathcal{C}_{n}$ for all $n\geq 1$. We call the operator system $(\mathcal{S}\otimes\mathcal{T},\{\mathcal{C}_{n}^{\lambda}\}_{n=1}^{\infty},1_{\mathcal{S}}\otimes 1_{\mathcal{T}})$ the $\lambda$- operator system tensor product of $\mathcal{S}$ and $\mathcal{T}$ and denote it by $\mathcal{S}\otimes_{\lambda}\mathcal{T}$.

Theorem 4.3**.**

The mapping $\lambda:\mathcal{O}\times\mathcal{O}\rightarrow\mathcal{O}$ sending $(\mathcal{S},\mathcal{T})$ to $\mathcal{S}\otimes_{\lambda}\mathcal{T}$ is an operator system tensor product in the sense of [13].

Proof.

Observe that,

1. (T1)

By definition $(\mathcal{S}\otimes\mathcal{T},\{\mathcal{C}_{n}^{\lambda}(\mathcal{S}\otimes\mathcal{T})\}_{n=1}^{\infty},1_{\mathcal{S}}\otimes 1_{\mathcal{T}})$ is an operator system. 2. (T2)

For $P\in M_{k}(\mathcal{S})^{+}$ and $Q\in M_{l}(\mathcal{T})^{+}$, since

[TABLE]

where $\alpha=(I_{k+l},0,\cdots,0)\in M_{kl,\tau(k+l)}$, we have property (T2). 3. (T3)

For unital completely positive maps $\phi\in\mathcal{S}\rightarrow M_{n}$ and $\psi\in\mathcal{T}\rightarrow M_{m}$, using Lemma 3.1(iii) we have $(\phi\otimes\psi)_{n}(\mathcal{C}_{n})\subseteq M_{n}^{+}$, thus (T3) follows. 4. (T4)

Let $\phi\in UCP(\mathcal{S}_{1},\mathcal{S}_{2})$ and $\psi\in UCP(\mathcal{T}_{1},\mathcal{T}_{2})$. Then for any element $A\otimes_{\lambda_{j}}(P,Q)A^{*}\in\mathcal{C}_{n}$, where $A\in M_{n,\tau(j)}$, $P\in M_{j}(\mathcal{S}_{1})^{+}$ and $Q\in M_{j}(\mathcal{T}_{1})^{+}$:

[TABLE]

Thus, $(\phi\otimes\psi)_{n}(\mathcal{C}_{n}(\mathcal{S}_{1},\mathcal{T}_{1}))\subseteq\mathcal{C}_{n}(\mathcal{S}_{2},\mathcal{T}_{2}),$ and using [13, Lemma 2.5] $\phi\otimes\psi\in UCP(\mathcal{S}_{2},\mathcal{T}_{2}).$

∎

Remark 4.4**.**

If $\lambda$ is either Kronecker or Schur Product, the cone $\mathcal{C}_{n}^{\lambda}$ coincides with $\mathcal{C}_{n}^{\max}=\mathcal{C}_{n}^{s}$ ([13, 15]).

5. $\lambda$-tensor product of $C^{*}$-algebras

We now move on to the algebraic structures for the $\lambda$-theory. For this we make use of the condition (W2) (Section 2.1).

An associative algebra $A$ over $\mathbb{C}$ is said to be a completely contractive Banach algebra if it is a complete operator space for which the multiplication map $m_{A}:A\times A\to A$ $(a,b)\to ab$ is jointly completely contractive, i.e. $\|[a_{ij}b_{kl}]\|\leq\|[a_{ij}]\|\|[b_{kl}]\|$ for all $[a_{ij}]\in M_{n}(A)$ and $[b_{kl}]\in M_{n}(A)$.

Theorem 5.1**.**

For completely contractive Banach algebras $A_{1},A_{2},\cdots,A_{m}$, $\otimes^{\lambda}A_{i}$ is a Banach algebra if $\lambda$ satisfies (W2). Further if each $A_{i}$ is a Banach $*$-algebra and $\lambda$ also satisfies (O1), then $\otimes^{\lambda}A_{i}$ is a Banach ∗-algebra. Moreover if each $A_{i}$ is approximately unital then $\otimes^{\lambda}A_{i}$ is also approximately unital.

Proof.

Let $x$, $y\in\otimes_{\lambda}A_{i}$ with

[TABLE]

where for each $t=1,2,\cdots,m$

[TABLE]

Then, using Property (W2), there exists $S\in M_{\tau(r)\tau(s),\tau(rs)},T\in M_{\tau(rs),\tau(r)\tau(s)}$ with $\|S\|,\|T\|\leq 1$ such that

[TABLE]

where $z^{(t)}=\sum_{i_{t},j_{t}}\sum_{k_{t},l_{t}}\varepsilon^{[rs]}_{(i_{t}-1)s+k_{t},(j_{t}-1)s)+l_{t}}\otimes u_{i_{1},j_{1}}^{(t)}v_{k_{t},l_{t}}^{(t)}=u^{(t)}\otimes v^{(t)};\,t=1,\ldots,m$.

Thus,

[TABLE]

making $\otimes_{\lambda}^{m}A_{i}$, and hence $\otimes^{\lambda}A_{i}$ a Banach algebra.

If $\lambda$ satisfies (O1), $*$-part follows from Lemma 3.1(i) and definition of $\|\cdot\|_{\lambda,1}$.

One can easily verify that $\|\cdot\|_{\lambda,1}\leq\|\cdot\|_{\gamma}$, implying that $\|\cdot\|_{\lambda,1}$ is an admissible cross norm on $\otimes_{m}A_{i}$ . Therefore, $\otimes^{\lambda}A_{i}$ has a bounded approximate identity whenever each $A_{i}$ is approximately unital.∎

In particular, we have the following well known result (see[14, 17]):

Corollary 5.2**.**

$\otimes^{\otimes}A_{i}$, the projective tensor product and $\otimes^{\odot}A_{i}$, the Schur tensor product are Banach ∗-algebras with a bounded approximate identity, however, $\otimes^{\bullet}A_{i}$, the Haagerup tensor product is a Banach algebra.

In general, $\lambda$-tensor product of operator spaces is not injective. Since, $(\otimes^{\lambda}A_{i})^{*}=CB_{\lambda}(A_{1}\times A_{2}\cdots A_{m},\mathbb{C})$[19, Proposition 4.11] completely isometrically, so the proof of [14, Theorem 5] can be adopted in this case to show the injectivity of $\lambda$-tensor product for the closed ideals, i.e.

Proposition 5.3**.**

Let $I_{i}$ be closed two-sided ideals in $C^{*}$-algebras $A_{i}$ for $i=1,2,\cdots,m$, then $\otimes^{\lambda}I_{i}$ is a closed two-sided ∗-ideal of $\otimes^{\lambda}A_{i}$.

Lemma 5.4**.**

Let $W_{i},$ $1\leq i\leq m$ be completely complemented subspaces of the operator spaces $V_{i},$ $1\leq i\leq m$ complemented by cb projection having cb norm equal to 1, respectively, then $\otimes^{\lambda}W_{i}$ is a closed subspace of $\otimes^{\lambda}V_{i}$.

Proof.

Using the assumption, there are cb projections $P_{1},P_{2},\cdots,P_{m}$ from $V_{1}$ onto $W_{1}$, $V_{2}$ onto $W_{2}\cdots$, $V_{m}$ onto $W_{m}$ with $\|P_{1}\|_{cb}=\|P_{2}\|_{cb}=\cdots=\|P_{m}\|_{cb}=1$. Therefore, by the functoriality of the $\lambda$- tensor product([19, Proposition 6.1], $\otimes^{m}_{i=1}P_{i}:\otimes^{\lambda}V_{i}\to\otimes^{\lambda}W_{i}$ is a completely bounded map and $\|\otimes^{m}_{i=1}P_{i}\|_{cb}\leq 1$. Since, for $u\in\displaystyle\otimes_{i=1}^{m}V_{i}$, $\otimes^{m}_{i=1}P_{i}(u)=u$, so $\|u\|_{\otimes^{\lambda}W_{i}}\leq\|u\|_{\otimes^{\lambda}V_{i}}$, hence $\otimes^{\lambda}W_{i}$ is a closed subspace of $\otimes^{\lambda}V_{i}$.∎

Since, there is a conditional expectation from a $C^{*}$-algebra $A$ onto a finite dimensional $C^{*}$-subalgebra of $A$, so by the above Lemma for finite dimensional $C^{*}$-algebras, $\lambda$-tensor product of operator spaces is injective. In general $\|\cdot\|_{\lambda}$ need not be injective.

However, for $\widehat{\otimes}$, we have something partial:

Proposition 5.5**.**

Let $A_{0}$ and $B_{0}$ be closed ∗-subalgebras of $A$ and $B$, respectively, then $A_{0}\widehat{\otimes}B_{0}$ is (isomorphic to) closed ∗-subalgebra of $A\widehat{\otimes}B$.

Proof.

Let $I$ denote the closure of $A_{0}\otimes B_{0}$ in $A\widehat{\otimes}B$, so that $I$ is a closed ∗-subalgebra of $A\widehat{\otimes}B$. We first claim that $\|u\|_{A\widehat{\otimes}B}\leq\|u\|_{A_{0}\widehat{\otimes}B_{0}}\leq 2\|u\|_{A\widehat{\otimes}B}$ for $u\in A_{0}\otimes B_{0}$. Choose $f\in(A_{0}\widehat{\otimes}B_{0})^{*}$ such that $f(u)=\|u\|_{A_{0}\widehat{\otimes}B_{0}}$ with $\|f\|=1$. Let $\phi_{0}$ be the jcb bilinear form on $A_{0}\times B_{0}$ corresponding to $f$. By ([7, Corollary 3.10]), $\phi_{0}:A_{0}\times B_{0}\to\mathbb{C}$ extends to a jcb bilinear form $\phi:A\times B\to\mathbb{C}$ such that $\|\phi\|_{jcb}\leq 2\|\phi_{0}\|_{jcb}$. Therefore $\|\tilde{f}\|\leq 2$, where $\tilde{f}$ is the linear functional on $A\widehat{\otimes}B$ corresponding to $\phi$, and thus the claim. Now consider the identity map $i:(A_{0}\otimes B_{0},\|\cdot\|_{A_{0}\widehat{\otimes}B_{0}})\to(A\otimes B,\|\cdot\|_{A\widehat{\otimes}B})$ which is linear and continuous by the last claim, so it can be extended to $\widetilde{i}:A_{0}\widehat{\otimes}B_{0}\to A\widehat{\otimes}B$. We now show that $A_{0}\widehat{\otimes}B_{0}$ is isomorphic to $I$. For the injectivity of $\widetilde{i}$, by [10, Theorem 2], it is enough to show that it is injective on $A_{0}\otimes B_{0}$ but this follows directly by the last inequality. Again, by the last inequality, $\widetilde{i}^{-1}$ is continuous. For onto-ness, let $u\in I$. There is a sequence $u_{n}\in A_{0}\otimes B_{0}$ converging to $u$ in $\|\cdot\|_{A\widehat{\otimes}B}$-norm. The sequence $\{u_{n}\}$ becomes Cauchy with respect to $\|\cdot\|_{A_{0}\widehat{\otimes}B_{0}}$-norm by the last claim, so it converges, say, to $u^{\prime}$. Clearly, $\widetilde{i}(u^{\prime})=u$. Thus $A_{0}\widehat{\otimes}B_{0}$ can be regarded as a closed ∗-subalgebra of $A\widehat{\otimes}B$. ∎

Proposition 5.6**.**

For $C^{*}$-algebras $A$ and $B$, any $\lambda$-cb bilinear form $\phi$ on $A\times B$ can be extended uniquely to $\tilde{\phi}$ on $A^{**}\times B^{**}$ such that $\|\phi\|_{\lambda}=\|\tilde{\phi}\|_{\lambda}$.

Proof.

Since $\phi:A\times B\to\mathbb{C}$ is $\lambda$-cb bilinear form. It is in particular bounded bilinear form and thus determines a unique separately normal bilinear form $\tilde{\phi}:A^{**}\times B^{**}\to\mathbb{C}$ by [6, Corollary 2.4]. For $k\in\mathbb{N}$, consider the map $\tilde{\phi}_{k}:M_{k}(A^{**})\times M_{k}(B^{**})\to M_{\tau(k)}$ taking $\tilde{\phi}_{k}(a_{1}\otimes m,a_{2}\otimes m^{\prime})=\lambda_{k}(a_{1},a_{2})\otimes\tilde{\phi}(m,m^{\prime})$. Let $a^{**}=[a_{ij}^{**}]\in M_{k}(A^{**})$ and $b^{**}=[b_{ij}^{**}]\in M_{k}(B^{**})$ with $\|a^{**}\|\leq 1$ and $\|b^{**}\|\leq 1$. Since the unit ball of $M_{k}(A)$ is $w^{*}$-dense in the unit ball of $M_{k}(A^{**})$, so we obtain a net $(a_{\lambda})=(a_{ij}^{\lambda})$ (resp., $(b_{\nu})$) in $M_{k}(A)$ (resp., $M_{k}(B)$) which is $w^{*}$-convergent to $a^{**}$ (resp., $b^{**}$) with $\|a_{\lambda}\|\leq 1$ (resp., $\|b_{\nu}\|\leq 1$). Therefore, $\widehat{a_{ij}^{\lambda}}$ is $w^{*}$-convergent to $a_{ij}^{**}$ for each $i,j$. Now by the separate normality of $\tilde{\phi}$, we have $\|\lambda_{k}\otimes\tilde{\phi}(\displaystyle\sum_{i,j}\epsilon_{ij}\otimes a_{ij}^{**},\displaystyle\sum_{p,l}\epsilon_{pl}\otimes b_{pl}^{**})\|=\|\displaystyle\sum_{i,j,p,l}\lambda_{k}(\epsilon_{ij},\epsilon_{pl})\otimes\tilde{\phi}(a_{ij}^{**},b_{pl}^{**})\|=\|\displaystyle\lim_{\lambda}\displaystyle\lim_{\nu}\displaystyle\sum_{i,j,p,l}\lambda_{k}(\epsilon_{ij},\epsilon_{pl})\otimes\tilde{\phi}(a_{ij}^{\lambda},b_{pl}^{\nu})\|=\displaystyle\lim_{\lambda}\displaystyle\lim_{\nu}\|\displaystyle\sum_{i,j,p,l}\lambda_{k}(\epsilon_{ij},\epsilon_{pl})\otimes\phi(a_{ij}^{\lambda},b_{kl}^{\nu})\|\leq\|\lambda_{k}\otimes\phi\|$ for each $k\in\mathbb{N}$. Thus $\|\tilde{\phi}_{k}\|\leq\|\phi_{k}\|\leq\|\phi\|_{\lambda}$ for every $k\in\mathbb{N}$. Clearly, $\|\phi\|_{\lambda}\leq\|\tilde{\phi}\|_{\lambda}$ as $\phi$ being the restriction of $\tilde{\phi}$. Hence $\|\phi\|_{\lambda}=\|\tilde{\phi}\|_{\lambda}$. ∎

For a tensor norm $\alpha$ and a closed ideal $J$ of $A\otimes^{\alpha}B$, we try to find out whether $a\otimes b\in J_{\min}$ implies that $a\otimes b\in J$. This question stems from the study of the elusive nature of the Haagerup tensor product of $C^{*}$-algebras, it was resolved for the Haagerup tensor product in ([1, Theorem 4.4]) and for the operator space projective tensor product in [14, Theorem 6]. We present here a unified approach.

For $C^{*}$-algebras $A$ and $B$, assume that $\|\cdot\|_{\lambda}\geq\|\cdot\|_{\min}$ on $A\otimes B$, so there will be a identity map $i$ from $A\otimes^{\lambda}B$ into $A\otimes^{\min}B$.

Lemma 5.7**.**

Let $M$ and $N$ be von Neumann algebras and let $L$ be a closed ideal in $M\otimes^{\lambda}N$, where the sequence $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ satisfies condition (W2). If $1\otimes 1\in L_{\min},$ then $1\otimes 1\in L$ and $L$ equals $M\otimes^{\lambda}N$.

Proof.

Since, $1\otimes 1\in L_{\min}$, for a given $\epsilon=\dfrac{1}{2}$, there exists $w\in L$ such that $\|i(w)-1\otimes 1\|_{\min}<\dfrac{1}{2}$. Let $w=\displaystyle\sum_{t=1}^{\infty}\alpha_{t}\otimes_{\lambda_{r_{t}}}(u^{(t)},v^{(t)})\beta_{t};$ be a norm convergent representation in $M\otimes^{\lambda}N$ where for $t=1,2,\ldots,\;\;r_{t}\in\mathbb{N}$,

[TABLE]

By [11, Theorem 8.3.5], there exist sequences $\{x_{i_{t},j_{t}}^{(t)}\}\in Z(M)$, $\{y_{i_{t},j_{t}}^{(t)}\}\in Z(N)$, $\{\phi_{n}\}\in P(M)$ and $\{\psi_{n}\}\in P(N)$ such that

[TABLE]

where $P(M)$ denotes the set of all mappings $\phi:M\to M$ such that, for $m\in M$, $\phi(m)$ is in the convex hull of the set $\{\mathrm{umu}^{*}:\mathrm{u}\in U(M)\}$.

For each $n\in\mathbb{N}$, using the contractive maps $\phi_{n}\otimes\psi_{n}$ on $M\otimes^{\lambda}N$ ([19, Proposition 6.1]) and invariance of ideal $L$ under $\phi_{n}\otimes\psi_{n}$, we have for all positive integers $k\leq l$

[TABLE]

Therefore, one can define an element

[TABLE]

For sufficiently large choice of $n$ and $\epsilon>0$, we deduce easily that

[TABLE]

Since, $L$ is left invariant by $\phi_{n}\otimes\psi_{n}$ for each $n$, so

[TABLE]

It is easy to show that $i\circ(\phi_{n}\otimes^{\lambda}\psi_{n})=(\phi_{n}\otimes^{\min}\psi_{n})\circ i$.

[TABLE]

By (4), we have

[TABLE]

Then the inequality

[TABLE]

is a consequence of (5), (6) and triangle inequality, and so $i(z)$ is invertible in $L_{\min}\cap(Z(M)\otimes^{\min}Z(N))$. Using similar arguments as in ([12, Theorem 2.11.6, Lemma 2.11.1] and the fact that $Z(M)$ is a nuclear $C^{*}$-algebra [2, Proposition 1] we get $Z(M)\otimes^{\lambda}Z(N)$ is semisimple. The regularity of $Z(M)\otimes^{\lambda}Z(N)$ follows from [12, Lemma 4.2.19]. Since $i(z)$ is invertible in $L_{\min}\cap(Z(M)\otimes^{\min}Z(N))$, i.e. invertible in both $L_{\min}$ and $Z(M)\otimes^{\min}Z(N)$, so $0\notin\sigma_{Z(M)\otimes^{\min}Z(N)}(i(z))$. So by [12, Exercise 4.8.12] $0\notin\sigma_{Z(M)\otimes^{\lambda}Z(N)}(z)$. Hence, $z$ is invertible in $Z(M)\otimes^{\lambda}Z(N)$, so there exists $w\in Z(M)\otimes^{\lambda}Z(N)$ such that $zw=wz=1\otimes 1$. Since, $z\in L$ and $L$ being an ideal, so $1\otimes 1\in L$. ∎

Theorem 5.8**.**

*Let $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ fulfills (W2), $A$ and $B$ be $C^{*}$-algebras and let $J$ be a closed ideal in $A\otimes^{\lambda}B$. If $a\otimes b\in J_{\min}$ then $a\otimes b\in J$. In particular $A\otimes^{\lambda}B$ is a -semi-simple Banach algebra provided $\lambda=(\lambda_{n})_{n\in\mathbb{N}}$ further fulfills (O1).

Proof.

Suppose that $a,b\geq 0$ and $a\otimes b\in J_{\min}$ but not in $J$. So by Hahn Banach theorem there exists $\phi\in(A\otimes^{\lambda}B)^{*}$ such that $\phi(J)=0$ and $\phi(a\otimes b)\neq 0$. Since, $(A\otimes^{\lambda}B)^{*}=CB_{\lambda}(A\times B,\mathbb{C})$ so $\phi(x\otimes y)=\tilde{\phi}(x,y)$ for some $\tilde{\phi}\in CB_{\lambda}(A\times B,\mathbb{C})$ and for all $x\in A,y\in B$. By Proposition 5.6, we have $\tilde{\phi}^{**}:A^{**}\times B^{**}\to\mathbb{C}$ a $\lambda$-completely bounded operator satisfying $\|\tilde{\phi}^{**}\|_{\lambda}=\|\tilde{\phi}\|_{\lambda}$ . Let $L$ be the closed ideal in $A^{**}\otimes^{\lambda}B^{**}$ generated by $J$. Let $u=\displaystyle\sum_{j=1}^{\infty}\alpha_{j}\otimes_{\lambda_{k_{j}}}(x_{1}^{j},x_{2}^{j})\beta_{j}$ be a norm convergent sum in $A\otimes^{\lambda}B$ representing a fixed but an arbitrary element of $J$. Since $\phi$ annihilates $J$, so $\displaystyle\sum_{j=1}^{\infty}\alpha_{j}\phi_{\lambda_{k_{j}}}(\otimes_{\lambda_{k_{j}}}(x_{1}^{j},x_{2}^{j}))\beta_{j}=\displaystyle\sum_{j=1}^{\infty}\displaystyle\sum_{i_{1},i_{2}}\alpha_{j}\lambda_{k_{j}}(a_{i_{1}},a_{i_{2}})\otimes\phi(x_{i_{1}}^{j}\otimes x_{i_{2}}^{j})\beta_{j}=0$. Let $u,v\in A$ and $s,t\in B$ then we have $\displaystyle\sum_{j=1}^{\infty}\displaystyle\sum_{i_{1},i_{2}}\alpha_{j}\lambda_{k_{j}}(a_{i_{1}},a_{i_{2}})\otimes\phi(ux_{i_{1}}^{j}v\otimes sx_{i_{2}}^{j}t)\beta_{j}=0$. Let $M=A*^{**}$ and $N=B^{**}$ be the von Neumann algebras generated by $A$ and $B$. For each $n\in\mathbb{N}$ and $u\in M$, define $w_{n}(u)=\displaystyle\sum_{j=1}^{n}\displaystyle\sum_{i_{1},i_{2}}\alpha_{j}\lambda_{k_{j}}(a_{i_{1}},a_{i_{2}})\otimes\phi(ux_{i_{1}}^{j}v\otimes sx_{i_{2}}^{j}t)\beta_{j}$. We will claim that $\{w_{n}\}$ is a Cauchy sequence. To see this, let $m<n$, $|w_{n}(u)-w_{m}(u)|=|\displaystyle\sum_{j=m+1}^{n}\displaystyle\sum_{i_{1},i_{2}}\alpha_{j}\lambda_{k_{j}}(a_{i_{1}},a_{i_{2}})\otimes\phi(ux_{i_{1}}^{j}v\otimes sx_{i_{2}}^{j}t)\beta_{j}\ |\leq\|u\|\|s\|\|t\|\|v\|\|\displaystyle\sum_{j=m+1}^{n}\alpha_{j}\otimes_{\lambda_{k_{j}}}(x_{1}^{j},x_{2}^{j})\beta_{j}\|_{\lambda}$, and so $\{w_{n}\}$ is a Cauchy sequence with limit $w\in M_{*}$ given by $w(u)=\displaystyle\sum_{j=1}^{\infty}\displaystyle\sum_{i_{1},i_{2}}\alpha_{j}\lambda_{k_{j}}(a_{i_{1}},a_{i_{2}})\otimes\phi(ux_{i_{1}}^{j}v\otimes sx_{i_{2}}^{j}t)$. Again as in [1], we obtain $\phi$ annihilates $L$.

Now for $\epsilon>0$, let $p_{\epsilon}\in M$ and $q_{\epsilon}\in N$ be the spectral projections of $a$ and $b$ respectively for the closed interval $[\epsilon,\infty)$. Since there is a conditional expectation from $M$ onto $p_{\epsilon}Mp_{\epsilon}$, so $p_{\epsilon}Mp_{\epsilon}\otimes^{\lambda}q_{\epsilon}Nq_{\epsilon}$ is a closed subalgebra of $M\otimes^{\lambda}N$ by Lemma 5.4. Let $L_{0}=L\cap(p_{\epsilon}Mp_{\epsilon}\otimes^{\lambda}q_{\epsilon}Nq_{\epsilon})$, a closed ideal in $p_{\epsilon}Mp_{\epsilon}\otimes^{\lambda}q_{\epsilon}Nq_{\epsilon}$, and so $(L_{0})_{\min}$ is a closed ideal in $p_{\epsilon}Mp_{\epsilon}\otimes^{\min}q_{\epsilon}Nq_{\epsilon}$. Now as in [1, Theorem 4.4] and [14, Theorem 6], we get $(L_{0})_{\min}$ contains $p_{\epsilon}\otimes q_{\epsilon}$ and so by the above Lemma 5.7, $p_{\epsilon}\otimes q_{\epsilon}\in L_{0}$. Hence $L_{0}=p_{\epsilon}Mp_{\epsilon}\otimes^{\lambda}q_{\epsilon}Nq_{\epsilon}$, which further implies that $p_{\epsilon}a\otimes q_{\epsilon}b\in L$, and so $\phi(p_{\epsilon}a\otimes q_{\epsilon}b)=0$. Letting $\epsilon\to 0$, we have $\phi(a\otimes b)=0$, contrary to the choice of $\phi$.

In the case when both $a$ and $b$ are arbitrary elements, then one may apply the similar technique as given in [1] to obtain the result.∎

Acknowledgements

The authors would like to thank Andreas Defant for providing a copy of [19]. The authors would also like to thank the referee for useful comments and suggestions.