Banach Algebra Norms on q-Gaussian Operators

• The paper demonstrates that the Banach algebra norm framework captures q-deformed spectral scaling and positivity via operator representations.
• Detailed methods reveal the distinction between sums of squares and broader positivity cones, crucial for norm estimation.
• Analytical techniques extend classical Gaussian theory to q-Gaussian operators, informing moment problems and C*-algebra closures.

Banach algebra norms on $q$-Gaussian operators arise from noncommutative analogues of classical Gaussian operator theory, taking into account deformed commutation relations parametrized by $q$. The Banach algebra norm is central for understanding analytic, spectral, and positivity properties of $q$-Gaussian algebras and their representations, especially in the context where the basic commutation relation is $xx^* = q x^*x$ for $q > 0$. This theory serves as a foundation for characterizing positivity, spectral scaling, sums of squares, and deformed moment problems within the framework of noncommutative real algebraic geometry.

1. $q$-Normal Operators and Algebraic Relations

Let $\mathcal{A}$ denote the unital $*$-algebra generated by $x$ with the defining relation

$xx^* = q x^*x \quad (q > 0).$

A densely defined closed operator $X$ on a Hilbert space is called $q$-normal if

• $\operatorname{Dom}(X) = \operatorname{Dom}(X^*)$,
• $\|X^*f\| = q^{1/2} \|Xf\|$ for all $f$ in the domain.

This generalizes the classical notion of normal operators ($q=1$) and further satisfies

$XX^* = q X^*X$

in operator form. The polar decomposition $X = UC$ yields scaling relations:

$UCU^* = q^{1/2} C,$

inducing a nontrivial intertwining between the partial isometry $U$ and the modulus $C$.

The noncommutative nature is reflected in the graded structure of $\mathcal{A}$: with $\deg(x) = 1$ and $\deg(x^*) = -1$, monomials $x^{*m} x^n$ form a natural basis.

2. Spectral and Representation-Theoretic Consequences

The $q$-defining relation in $\mathcal{A}$ yields several key implications:

• Modified spectra: The scaling intertwining

$U f(C) U^* = f(q^{1/2}C)$

for Borel functions $f$ implies a $q$-deformed spectral theorem: the spectrum of $C$ is scaled by $q^{1/2}$ under conjugation.

• Intertwining of projections: The unitary equivalence between the spectral projections of $C$ and those of $q^{-1/2}C$ reflects a scaling behavior absent in the classical case.
• Graded algebraic structure: The structure of $\mathcal{A}$ as spanned by monomials in $x$ and $x^*$ supports analytic decompositions (such as sums of squares and positivity cones) important for norm characterization.

3. The Complex $q$-Moment Problem and Strong Positivity

For a linear functional $F$ on $\mathcal{A}$, the $q$-moment problem concerns whether $F$ admits a "moment" representation

$F(a) = (T(a)\varphi, \varphi)$

for all $a \in \mathcal{A}$, where $T$ is a well-behaved $*$-representation arising from a $q$-normal operator $X$; that is, $T(x)$ acts as $X$ on an appropriate core.

Equivalent formulations:

• In terms of the monomial basis $\{x^{*k} x^l\}$, $F$ corresponds to a two-sequence $(a_{kl})$ with

$$a_{kl} = (X^{*k} X^l \varphi, \varphi), \quad \forall k,l \in \mathbb{N}_0.$$

• Spectral representation via positive Borel measures on $\mathbb{R}_+$, reflecting the $q$-deformed spectral data.

Strong positivity (Theorem 3): $F$ is a $q$-moment functional if and only if $F(a) \geq 0$ for all $a$ in the cone

$$\mathcal{A}^+ := \{ f = f^* \in \mathcal{A} : (f(X,X^*)\psi, \psi) \geq 0 \text{ for all well-behaved } T, \,\forall \psi \in \operatorname{Dom}(T) \}.$$

This yields the $q$-analogue of Haviland's theorem: Banach algebra norm positivity is tied not just to sums of squares but to the broader cone $\mathcal{A}^+$.

4. Positive Elements, Sums of Squares, and Banach Norm Structure

For $f = f^* \in \mathcal{A}$, $f$ is a sum of squares, $f \in \Sigma \mathcal{A}^2$, if

$f = \sum_j a_j^* a_j, \quad a_j \in \mathcal{A}.$

However, being positive in every representation ($f \in \mathcal{A}^+$) does not in general imply being a sum of squares—Theorem 2 constructs explicit polynomials in $x + x^*$ that are positive but not sums of squares in $\mathcal{A}$.

This distinction influences Banach algebra norm construction:

• In many noncommutative settings, the Banach $*$-algebra (or C*-algebra) norm is defined via the supremum over all $*$-representations.
• If every positive element were a sum of squares, norm and positivity properties could be fully controlled via quadratic forms, simplifying the norm completion.
• In the $q$-Gaussian context, failure of this property leads to subtleties in C*-norm closure and the structure of positive cones—directly affecting spectral properties, state extensions, and norm bounds.

5. Interplay with $q$-Gaussian Operators

$q$-Gaussian operators are $q$-deformations of classical Gaussians (arising in quantum probability), and in the setting of $\mathcal{A}$ are typically $q$-normal. Their analysis leverages the above moment/cone structure.

• The Banach algebra norm of $q$-Gaussian operators is influenced by the positivity/sum-of-squares distinction in $\mathcal{A}$.
• Analogues of Hilbert's 17th problem (positivity versus sums of squares) play a role in understanding norm-complete Banach algebra structures in the $q$-deformed setting.
• Norm estimates and positivity properties transfer to analytic questions about $q$-moment sequences, functional calculus in operator algebras, and possible C*-completions.

6. Deformed Real Algebraic Geometry and Operator Theory

The framework established by the $q$-normal relation and associated cones is central to noncommutative real algebraic geometry in the $q$-deformed operator setting. Banach algebra norms, positivity cones, and sums of squares amalgamate to provide:

• A rigorous foundation for extending classical moment and positivity theory to $q$-Gaussian operator algebras.
• Structural understanding of the analytic and spectral behavior as $q \to 1$ (classical) or $q \to 0$ (free probabilistic) limits.
• Techniques for operator norm estimation and C*-algebraic closure, vital for quantum probability and noncommutative harmonic analysis.

Summary Table: Core Structures and Implications

Concept	Description	Norm/Positivity Implication
$q$-normal operator (XX* = q X*X)	Generalized normal operator with $q$-dependent scaling	Spectrum and norm scaling under $q$-transformation
Cone $\mathcal{A}^+$	Positivity cone via representations of $\mathcal{A}$	Determines admissible positive functionals in norm closure
Sums of squares $\Sigma \mathcal{A}^2$	Self-adjoint elements as sums of squares in $\mathcal{A}$	Not all elements of $\mathcal{A}^+$ are sums of squares; affects norm completeness
$q$-Gaussian operators	$q$-deformed analogues of classical Gaussian elements	Structure of Banach algebra norm depends on positivity property and sums of squares decomposition

The synthesis of $q$-normality, the $q$-moment problem, and the distinction between positivity and sums of squares within $\mathcal{A}$ creates the analytic and algebraic backbone for Banach algebra norms on $q$-Gaussian operators. This is fundamental for the development of noncommutative geometry, C*-algebra theory, and operator analysis in the $q$-deformed setting.