## A non-self-adjoint Lebesgue decomposition.(English)Zbl 1302.47104

To formulate noncommutative generalizations of the classical function spaces $$H^2$$ and $$H^\infty$$, we begin with the the algebra of all noncommutative polynomials in $$d$$ complex variables, $$Z_1,\dots ,Z_d$$. This algebra can be completed in a natural inner product to form the noncommutative Hardy space, denoted $${F_d}^2$$. Every element $$F$$ of $${F_d}^2$$ induces an action $$L_F$$ on $${F_d}^2$$ by left multiplication, with the caveat that $$L_F$$ need not be bounded. This leads to the so-called noncommutative analytic Toeplitz algebra, denoted $${F_d}^\infty$$, consisting of those $$F$$ for which $$L_F$$ is a bounded operator on $${F_d}^2$$.
After giving suitable definitions for what it means for a bounded linear functional on $${F_d}^\infty$$ to be absolutely continuous or singular, the authors provide a generalization of the Lebesgue decomposition for the space $${F_d}^\infty$$. Specifically, Theorem 3.6 establishes that, if $$\phi$$ is a bounded linear functional on $${F_d}^\infty$$, then there are unique functionals $$\phi_a$$ and $$\phi_s$$ such that $$\phi_a$$ is absolutely continuous, $$\phi_s$$ is singular, and $$\phi=\phi_a + \phi_s$$; moreover, $$\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|$$. (For $$d=1$$, the first inequality is an equality.) This leads to a version of the F. and M. Riesz theorem in Theorem 4.1, which asserts that, if $$J$$ is a two-sided ideal in $${F_d}^\infty$$ and if the bounded linear functional $$\phi=\phi_a+\phi_s$$ vanishes on $$J$$, then each summand $$\phi_a$$ and $$\phi_s$$ also vanishes on $$J$$. These two results together lead to Theorem 5.2, a Lebesgue decomposition theorem for bounded linear functionals on quotient algebras $${F_d}^\infty/J$$, again with the relationship $$\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|$$ holding for the norms. Since the multiplier algebra of an irreducible Nevanlinna-Pick space may be viewed as the compression of $${F_d}^\infty$$ to a coinvariant subspace, and, indeed, as a quotient of $${F_d}^\infty$$ by a two-sided ideal, this generalized Lebesgue decomposition holds for bounded linear functionals on these multiplier algebras (Corollary 5.3). In the final section of the paper, the authors show that every quotient $${F_d}^\infty/J$$ (and, hence, the multiplier algebra of a Nevanlinna-Pick space) has a strongly unique predual.
This work builds on earlier work of K. R. Davidson et al. [J. Funct. Anal. 224, No. 1, 160–191 (2005; Zbl 1084.46042)] and makes important use of the universal representation theory for $${F_d}^\infty$$ put forward by D. P. Blecher and C. Le Merdy [Operator algebras and their modules – an operator space approach. Oxford University Press (2004; Zbl 1061.47002)].

### MSC:

 47L50 Dual spaces of operator algebras 47L55 Representations of (nonselfadjoint) operator algebras 47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 46B04 Isometric theory of Banach spaces

### Citations:

Zbl 1084.46042; Zbl 1061.47002
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