We prove that the fractional BMO space on a one-dimensional manifold is an interpolation space between C and $C^1$. We also prove that $BM{O}^1$ is an interpolation space between C and $C^2$. The proof is based on some nonclassical interpolation constructions. The results obtained cannot be transferred to spaces of functions defined on manifolds of higher dimension. The interpolation description of fractional BMO spaces is used at the end of the paper for the proof of the boundedness of commutators of the Hilbert...

The normal cohomology functor $Ex{t}_{\aleph }$ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of $Ex{t}_{\aleph }$-groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov...

In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II₁ factor, equality of the induced traces implies unitary...

Let $H_p$ denote the usual Hardy space of analytic functions on the unit disc $(0\<p\le \infty )$. We prove that for every function $f\in H_1$ there exists a linear operator $T$ defined on $L_1\left(\mathbf{T}\right)$ which is simultaneously bounded from $L_1\left(\mathbf{T}\right)$ to $H_1$ and from $L_{\infty }\left(\mathbf{T}\right)$ to $H_{\infty }$ such that $T\left(f\right)=f$. Consequently, we get the following results $(1\le p_0,p_1\le \infty )$:1) $(H_{p_0},H_{p_1})$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F(H_{p_0},H_{p_1})=H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$, where$H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$ denotes the closed subspace of $F\left(L_{p_0}\right(\mathbf{T}),L_{p_1}(\mathbf{T}\left)\right)$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy...