On products of noncommutative symmetric quasi Banach spaces and applications (Q830233)

Let \(E\) be a quasi-Banach symmetric space on \((0,\alpha)\). Let \(\mathcal{M}\) be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace \(\tau\) such that \(\tau(1)=\alpha\). Let \(L_0(\mathcal{M})\) be the space of all \(\tau\)-measurable operators affiliated with \(\mathcal{M}\). Define \[E(\mathcal{M})=\{x\in L_0(\mathcal{M}):\mu(x) \in E \}\] and \(\|x\|_{E(\mathcal{M})}:=\|\mu(x)\|_E\), \(x\in E(\mathcal{M})\), where \(\mu(x)\) is the generalized singular value function of \(x\). Let \(E_i\), \(i=1,2,\), be quasi-Banach symmetric spaces on \((0,\alpha)\). The pointwise product space \(E_1(M)\odot E_2(M) \) is defined by \[E_1(\mathcal{M})\odot E_2(\mathcal{M}) : = \{f :f =f_1f_2,\, f_i \in E_i (\mathcal{M}), \,i=1,2\}\] with a functional \(\|f\|_{E_1(\mathcal{M}) \odot E_2(\mathcal{M})} := \inf \{ \|f_1\|_{E_1(\mathcal{M})} \|f_2\|_{ E_2(\mathcal{M})} :f =f_1f_2, \,f_i \in E_i(\mathcal{M}),\,i=1,2\}\). The Calderón space \(E_1(\mathcal{M})^\theta E_2(\mathcal{M})^{1-\theta}\), \(\theta\in (0,1)\), is defined by \[E_1(\mathcal{M})^\theta E_2(\mathcal{M})^{1-\theta} :=\{ f: |f|\le \lambda |f_1|^{1-\theta } |f_2|^\theta, \ \lambda >0, \ \|f_i\|_{E_i(\mathcal{M}) } \le 1,\ i=1,2\}. \] The authors consider relations among the pointwise product space \(E_1(\mathcal{M})\odot E_2(\mathcal{M})\), Calderón spaces, and complex (and real) interpolation spaces. It is proved that \[ (E_1(\mathcal{M}), E_2(\mathcal{M}))_\theta = E_1(\mathcal{M})^\theta E_2(\mathcal{M})^{1-\theta} = E_1^{1/\theta}(\mathcal{M})\odot E_2^{1/(1-\theta)}(\mathcal{M}) .\]